text:
\section*{Introduction}

Ample line bundles are fundamental objects in Algebraic Geometry.
From the geometric perspective, an ample line bundle is one such that some positive multiple of the underlying divisor moves in a linear system that is large enough to give a projective embedding. In numerical terms a divisor is ample if and only if it is in the interior of the real cone generated by nef divisors (Kleiman). Equivalently, a divisor is ample if it intersects positively every closed integral subscheme (Nakai-Moishezon).
In cohomological terms, an ample line bundle is one such that a twist of a power by any coherent sheaf is generated by the global sections (Serre). Over the complex numbers, ampleness of line bundles is also equivalent to the existence of a metric with positive curvature (Kodaira).

The very ampleness of divisors on blow-ups of projective spaces and other varieties was studied
by several authors, e.g. Beltrametti and  Sommese \cite{bs}, Ballico and Coppens \cite{baco}, Coppens
\cite{coppens,coppens2},  Harbourne \cite{Ha}.
The notion of $l$-very ampleness of line bundles was introduced by Beltrametti, Francia and Sommese \cite{bfs} and $l$-very ample line bundles on del Pezzo surfaces were classified by Di Rocco \cite{dirocco}.

This paper studies ampleness, $l$-very ampleness and further positivity questions
for divisors on blow-ups of projective spaces of arbitrary dimension  in points
 in  general position.

The blown-up projective space ${\mathbb P}^n$ in $s\le n+3$ points is \emph{log Fano}, hence in particular a Mori Dream Space (see for example \cite{am,CT}). It  is related to the
 {\it moduli space of parabolic vector bundles} over ${\mathbb P}^1$ of rank $2$ for $s=n+3$
(see \cite{bau}, \cite{Mu}) or to the {\it moduli space of stable rational marked curves} for $s=n+2$

The \emph{effective cone} of divisors on  the moduli space of semistable parabolic vector
 bundles on a rational curve was computed via birational techniques by Moon
and Yoo in \cite{Moon}.
Independently, the effective cone as well as  the \emph{movable cones} of divisors on the blown-up
 ${\mathbb P}^n$ in up to $n+3$ points was investigated in \cite{bdp3};
the authors gave equations for the facets of the cones together with a geometric interpretation
in terms of joins between linear subspaces
spanned by the points and the {\it secant varieties} of
 the unique \emph{rational normal curve} of degree $n$ determined by the collection of
$n+3$ points. In this paper we use this description to give the {\it big cone} of divisors for the the blown-up ${\mathbb P}^n$ in at most $n+3$ points, Theorem \ref{big n+3 X0}.

In a very recent work, Araujo and Massarenti \cite{am} use this
description to give an explicit log Fano structure to the blow-up of ${\mathbb P}^n$
in up to $n+3$ points in general position;  they do so by studying the blow-up
of ${\mathbb P}^n$ along the joins mentioned above.

 In \cite{bdp1} the authors analysed the dimension of the space of global sections of divisors
on blown-up projective spaces in points in general position.
The dimensionality problem is important also in the algebraic framework due to its
 connections to the {\it Fr\"oberg-Iarrobino conjectures} that describe the Hilbert
series of  ideals generated by general powers of linear forms in the polynomial ring with $n+1$ variables (see \cite[Section 6]{bdp1} for an account on this).
Moreover, in \cite{bdp3} a conjectural formula for the dimension
of linear systems on the blow-up of ${\mathbb P}^n$
in up to $n+3$ points, that takes into account the contributions given
by the rational normal curve and the joins of its secants with linear subspaces, was given.
In this article we pose a series of questions
about vanishing cohomology of strict transforms of
divisors in the blow-up along the joins of which a positive answer would
imply  the dimension count, see Subsection \ref{section vanishing cohomology n+3}.

In \cite{bdp1,dp} it was shown that each divisor $D$ that is \emph{only linearly obstructed} can be birationally modified
by blowing-up its linear base locus and contracting the linear divisorial components, into a divisor $\tilde{D}$, the strict transform of $D$, such that all  higher cohomology groups of $\tilde{D}$ vanish.
In this work, we would like to emphasize that  the number of
global sections of $\tilde{D}$ gives information on
the effective cone of divisors while \emph{vanishing theorems}
for the higher cohomology groups give information on positivity properties such as
the ampleness, the nefness, and the global generation of such divisors.

The first application of the vanishing theorems that will be developed in this paper is the
description of {\it $l$-very ample divisors}, in particular {\it globally generated}
divisors and {\it very ample} divisors. This description is contained in   Theorem \ref{tva theorem X0}.

Moreover, in Theorem \ref{gg theorem Xr}, we show that if $D$ is an effective divisor on the blow-up of
${\mathbb P}^n$ in $s$ points in general position, with arbitrary $s$,
that is only linearly obstructed, the strict transforms of $D$ after
blowing-up in increasing dimension the linear cycles in the base locus, is base point free (and semi-ample).

Furthermore, in Theorem \ref{base point free tilde-sigma-D},
 we extend this result to a class of divisors with $s=n+3$ that can have non-linear base locus, by taking the blow-up along the joins between linear cycles and secant varieties to the unique rational normal curve of degree $n$.

Using these results, we prove a number of conjectures in birational geometry.

First, we establish  {\it Fujita's conjectures} for ${\mathbb P}^n$ blown-up in
$s$ points when $s\leq 2n$, Proposition \ref{fujita answer},
and for an infinite family of divisors on ${\mathbb P}^n$ blown-up in arbitrary number of points,
with a bound on coefficients, Proposition \ref{fujita answer2}.

Moreover, we prove that the \emph{(log) abundance conjecture} holds for an infinite family of log pairs given by
effective divisors on the blow-up of ${\mathbb P}^n$ in an arbitrary number of points in general position,
Theorem \ref{abundance Q}, in particular
providing  an explicit construction of \emph{good minimal models}.
For a small number of points, $s\le n+3$, an existence proof of this conjecture follows from work contained in \cite{CT, Mukai} (see also \cite{am}). In this paper we give a constructive proof that extends to an arbitrary number of points, $s$.

Finally, as explained in \cite[Subsection 6.3]{bdp1}, the {\it F-conjecture} predicting the nef
cone of $\overline{\mathcal{M}}_{0,n}$ was the original motivation for the study
of the vanishing theorems.
 In this setting, the iterated blow-up of the projective space along all linear cycles in
increasing dimension is identified  with the moduli space $\overline{\mathcal{M}}_{0,n+3}$
(see \cite{Ka}). Using vanishing theorems we prove that the conjecture holds for strict transforms of
linear systems in ${\mathbb P}^n$ interpolating multiple points in general position, under the blow-up
 of their linear base locus, Theorem \ref{fulton nef}.

This paper is organized as follows. In Section \ref{notations} we introduce the
 general construction, notation and some preliminary facts.
In Subsection \ref{section vanishing cohomology n+3} we pose a number of questions
about the cohomology of the strict transform of divisors in the iterated
blow-up of ${\mathbb P}^n$ along joins between linear cycles spanned by the points and
the secant varieties of the unique rational normal curve of degree $n$
through $n+3$ points.

Section \ref{lva section} contains one of the main result sof this article,
Theorem \ref{tva theorem X0}, that concerns  $l$-very ampleness
of line bundles on blown-up projective spaces in an arbitrary number of points in  general position.

In Section \ref{other positivity} we characterize other positivity properties of divisors on  blown-up projective spaces at points such as nefness, ampleness, bigness, and we establish Fujita's conjecture for a bounded number of points.

Section \ref{gg section} contains a description of semi-ample divisors on
the iterated  blow-up of the projective spaces along linear cycles  spanned by
subsets of the base points in increasing dimension, Theorem \ref{gg theorem Xr}. Subsection \ref{vanishing question} contains a complete description of the base locus of linear systems in ${\mathbb P}^n$ interpolating multiple points,
Theorem \ref{base locus theorem}.

In Section \ref{abundance section} we prove that the  abundance conjecture
 holds for log pairs given
by effective divisors on the blow-up of ${\mathbb P}^n$ at $s$
 points in general position.

In Section \ref{fulton}, as an application of the results contained in Section \ref{gg section},
we establish the F-conjecture  for a particular class of divisors, Theorem \ref{fulton nef}.

\subsection*{Acknowledgements}
The authors would like to thank the Research Center FBK-CIRM  Trento for the hospitality and financial support during
 their one month ``Research in Pairs'', Winter 2015. This project was carried out during this program. We would also like to express our gratitude to Edoardo Ballico, Ana-Maria Castravet, Antonio Laface, Han-Bom Moon, Brian Osserman, Giorgio Ottaviani and Stefano Urbinati for their remarks and suggestions regarding this project.

\section{Preliminary results and conjectures}\label{notations}
Let $K$ be an algebraically closed field of characteristic zero.
 Let $\mathcal{S}=\{p_1,\dots,p_s\}$ be a collection of $s$ distinct points in ${\mathbb P}^n_K$ and
let $S$ be the set of indices parametrizing $\mathcal{S}$, with $|S|=s$.

Let
\begin{equation}\label{linear system}
{\mathcal{L}}:={\mathcal{L}}_{n,d}(m_1,\dots,m_s)
\end{equation}
denote the linear system of degree-$d$ hypersurfaces of ${\mathbb P}^n$ with multiplicity at least $m_i$ at $p_i$, for $i=1,\dots,s$.

\subsection{The blow-up of ${\mathbb P}^n$ in points}\label{blow-up in points}
 We denote by $X_{s,(0)}$ the blow-up of ${\mathbb P}^n$ in the
 points of $\mathcal{S}$ and by $E_i$ the exceptional divisor of $p_i$, for all $i$.
 The index $(0)$ indicates that the space ${\mathbb P}^n$ is blown-up in $0$-dimensional cycles. The Picard group of $X_{s,(0)}$ is spanned by the class of a general hyperplane, $H$, and the classes of the
 exceptional divisors $E_{i}$, $i=1,\dots,s$.

\begin{notation}\label{generality definition}
Ffix positive integers $d, m_1, \ldots, m_s$ and define the following divisor on $X_{s,(0)}$:
  \begin{equation}\label{def D}
  dH-\sum_{i=1}^{s} m_i E_i\in\textrm{Pic}\left(X_{s,(0)}\right).
  \end{equation}
In this paper we denote by $D$  a \textit{general divisor} in $|dH-\sum_{i=1}^{s} m_i E_i|$. Notice that
the global sections of $D$ are in bijection with the elements
of the linear system ${\mathcal{L}}$ defined in \eqref{linear system}.
  \end{notation}
\begin{remark}\label{generality remark}
It is proved in \cite[Proposition 2.3]{CC} and \cite[Lemma 4.2]{CDDGP} that for any divisor $D$ of the form \eqref{def D} the general member of $|D|$
 has multiplicity equal to $m_i$ at the point $p_i$. It is important to mention here that
even if it is often omitted in the framework of classical interpolation problems in ${\mathbb P}^n$,
 the generality hypothesis of the divisor $D$ is always assumed.
\end{remark}

\begin{theorem}[{\cite[Theorem 5.3]{bdp1}, \cite[Theorem 1.6]{dp}}]\label{vanishing theorems}
Assume that $\mathcal{S}\subset{\mathbb P}^n$ is a set of points in  general position. Let $D$ be as in \eqref{def D}. Assume that
\begin{equation}\label{cond vanishings}
\begin{split}
& 0\le m_i, \ \forall i\in\{1,\dots,s\},\\
&m_i+m_j\le d+1, \ \forall i,j\in\{1,\dots,s\},\ i\ne j, \ (\textrm{if } s\ge2),\\
&\sum_{i=1}^{s} m_i\le nd+\max\{1,s-n-2\}.
\end{split}
\end{equation}
Then $h^1(X_{s,(0)},D)=0$.
\end{theorem}

\subsection{Base locus and effectivity of divisors on the blow-up of ${\mathbb P}^n$ in points}
\label{base locus and effectivity}

We adopt the same notation as in Subsection \ref{blow-up in points}.
For any integer
$0\le r\le  n-1$ and for any multi-index of cardinality $r+1$,
$$I:=\{i_1,\ldots,i_{r+1}\}\subseteq\{1,\ldots,s\},$$ we define the integer
$k_{I}$ to be the \emph{multiplicity of containment} of the strict transform in
$X_{s,(0)}$ of the linear cycle spanned by the points parametrized by $I$, $L_{I}\subset{\mathbb P}^n$, in
 the base locus of  $D$. Notice that under the generality assumption we have $L_I\cong{\mathbb P}^r$.

\begin{lemma}[{\cite[Proposition 4.2]{dp}}]\label{base locus lemma}
For any effective divisor $D$  as in \eqref{def D},
 the multiplicity of containment in ${\rm{Bs}}(|D|)$ of the strict transform in $X_{s,(0)}$ of the linear subspace
 $L_{I}$, with $0\leq r\leq n-1$, is the integer
\begin{equation}\label{mult k}
k_{I}=k_I(D):=\max \{0,m_{i_1}+\cdots+m_{i_{r+1}}-rd\}.
\end{equation}
\end{lemma}

\vskip.5em

We recall here notations and results introduced in \cite{bdp3}.
There exists a unique rational normal curve $C$ of degree $n$ through $n+3$ general points of ${\mathbb P}^n$;
this theorem is classically known and its first proof is due to Veronese \cite{veronese}, although it is often attributed to Castelnuovo.
Now, let $\sigma_t=\sigma_t(C)$ denote the $t$-th secant variety of $C$, namely the Zariski closure of the union of $t$-secant $(t-1)$-planes. In this notation we have $\sigma_1=C$.
For every $I\subset\{1,\dots,n+3\}$ with $|I|=r+1$, $-1\le r \le n$,
let $$\J(L_I,\sigma_t)$$ be  the \emph{join} of the linear cycle $L_I$ and $\sigma_t$.
We use the conventions $|\emptyset|=0$ and $\sigma_0=\emptyset$.

The dimension of the variety
$\J(L_I,\sigma_t)$ is
\begin{equation}\label{dimension join}
r=r_{I,\sigma_t}:=\dim \J(L_I,\sigma_t)=|I|+2t-1.
\end{equation}

\begin{lemma}[{\cite[Lemma 4.1]{bdp3}}]\label{base locus lemma II}
For any effective divisor $D$  as in \eqref{def D} and for $r_{I,\sigma_t}\leq n-1$,
 the multiplicity of containment in  ${\rm{Bs}}(|D|)$
of the strict transform in $X_{s,(0)}$ of the subvariety $\J(L_I,\sigma_t)$
is the integer
\begin{equation}\label{expanded}
\ \ \ k_{I,\sigma_t}=k_{I,\sigma_t}(D):=\max\left\{0, t\sum_{i=1}^{n+3} m_i+\sum_{i\in I}m_i-((n+1)t+|I|-1)d\right\}.
\end{equation}
\end{lemma}

We will now prove a stronger result that will play a crucial role in Section
\ref{abundance section}, particularly in Proposition \ref{linear series of strict transform even}.

\begin{proposition}\label{infinitesimal base locus} Let $D$ be an effective divisor on $X_{s,(0)}$. Let $p$ be a point on the variety $\J(L_I,\sigma_t)$ that does not lie on  any smaller join, namely any
$\J(L_{I'},\sigma_{t'})\subsetneq \J(L_{I},\sigma_{t})$, $I'\subset I$, $t'\le t$.
Then $p$ is contained in the base locus of the general member of the linear system $|D|$ with multiplicity precisely equal to $k_{I,\sigma_t}$.
\end{proposition}

\begin{proof} We first prove the claim for $t=0$ when the join $\J(L_I,\sigma_0)$ is the linear cycle $L_I$
that is in the base locus of the general member of $|D|$
with multiplicity of containment equal to $k_I=k_I(D)$, by Lemma \ref{base locus lemma}.

Let us assume, by contradiction, that there is a point $p$ of $L_I$ with multiplicity of containment at least $k_I+1$.
Let $I(r)$ be the largest index set such that $p\in L_{I(r)}$.
Lemma 4.2 of \cite{CDDGP} implies $r\geq 1$.
We introduce the following notation:
$$K_{I(r)}=K_{I(r)}(D):=\sum_{i\in I(r)} m_i - rd.$$

Consider first the case when $K_{I(r)}\geq 1$ and note that $k_{I(r)}=K_{I(r)}$. For $r\geq 2$, let $I(r-2)$ be a subset of $I(r)$, of cardinality equal to $r-1$, and take $H_{I(r-2)}$ a general hyperplane passing through $p$  and all points of $I(r-2)$.
For $r=1$,  $H_{I(-1)}$ denotes the hyperplane passing through the point $p$. For every $r\ge1$, consider  the divisor $D'$, defined as follows, and denote by $d'$ and $m_i'$ its corresponding degree and multiplicities:
$$D':=D+K_{I(r)} H_{I(r-2)}.$$
By assumption, the general member of $|D'|$ contains the point $p$ with multiplicity at least $1$. We can compute the multiplicity of containment of $L_{I(r)}$ in the base locus of $D'$:
\begin{equation}
\begin{split}
K_{I(r)}(D')&=\sum_{i\in I(r)} m'_i - rd'\\
&=\sum_{i\in I(r)} m_i - rd+(r-1)K_{I(r)}-rK_{I(r)}\\
&=K_{I(r)}-K_{I(r)}\\ &=0.
\end{split}
\end{equation}

Consider now the case  $K_{I(r)}<0$. Let $H_{I(r)}$ denote a hyperplane containing
all points of $I(r)$. Define the divisor
$$D'':=D-K_{I(r)} H_{I(r)}.$$
A similar computation shows the following
\begin{equation}
\begin{split}
K_{I(r)}(D'')&=\sum_{i\in I(r)} m''_i - rd''\\
&=\sum_{i\in I(r)} m_i - rd-(r+1)K_{I(r)}+rK_{I(r)}\\
&=K_{I(r)}-K_{I(r)}\\&=0.
\end{split}
\end{equation}

In the above cases we reduced to the case when $D$ is a divisor with $k_{I(r)}=0$
whose general member has a base point, $p$.
We  now prove by induction on $r$ that this leads to a contradiction.

We  discuss separately the case $r=1$ as the first induction step. The line $L_{I(1)}$ is not contained in the base locus of $D$ by Proposition \ref{base locus lemma}. However, the intersection multiplicity between the line $L_{I(1)}$ and $D$ is negative, a contradiction.

In general, we assume  that the statement holds for linear cycles of dimension $r-1$
and we prove that it holds for linear cycles of dimension $r$.
The point $p$ can not lie
on any smaller linear cycle contained in some $L_{I(r-1)}\subset L_{I(r)}$, by the induction assumption. Therefore, $p$ is a point inside the \emph{interior}
of the cycle $L_{I(r)}$, namely $p\in L_{I(r)}\setminus\bigcup_{I(r-1)\subset I(r)}L_{I(r-1)}$.
 It is easy to see that whenever $k_{I(r)}\geq 0$, then $k_J\geq 0$ for any subset $J$.
This implies that for any subset $I(r-1)\subset I(r)$ of cardinality $r$, the divisor $D$
contains the linear cycle $L_{I(r-1)}$ in its base locus with multiplicity $k_{I(r-1)}\geq 0$.
We consider $l$ a general line in $L_{I(r)}$ passing trough $p$.
We observe that the multiplicity of intersection between $l$ and the divisor $D$ is at most
\begin{equation*}
\begin{split}
d-\sum_{\substack{I(r-1)\subset I(r)}}k_{I(r-1)}-1&=
d-\sum_{\substack{I(r-1)\subset I(r)}}\left(\sum_{i\in I(r-1)}m_i - (r-1)d\right)-1\\
&=r\left(\sum_{i\in I(r)} m_i - rd\right) - 1\\
&= -1.
\end{split}
\end{equation*}
Since the family of lines passing through the point $p$ covers the linear
 cycle $L_{I(r)}$ one obtains that $L_{I(r)}$ is in the base locus of the divisor $D$
that is a contradiction with Lemma \ref{base locus lemma}, since $k_{I(r)}=0$.

If $t\geq 1$,  the proof follows by a similar argument to the one used \cite[Proposition 4.2]{dp}.

\end{proof}

\vskip.3em

We also recall here the classification of effective and movable divisors on the blow-up  of ${\mathbb P}^n$ at
$s\le n+3$ points in general position, based on the study of the base locus, Lemma \ref{base locus lemma},
  that was given  in \cite{bdp3}. For $s\leq n+2$ these cones of divisors were originally discussed in \cite[Theorem 4.9]{CDDGP}.

\begin{theorem}[{\cite[Theorems 5.1 and 5.3]{bdp3}}]\label{effectivity theorem}
We use the same notation as above.
\begin{enumerate}\item
 Assume that $s=n+2$.
The divisor $D$ is effective if and only if
\begin{equation}\label{effectivity condition n+2}
\begin{split}
&  d \ge 0, \\
& m_i \le d,  \ \forall i \in\{1,\dots,s\} , \\
& \sum_{i\in I}m_i \le nd, \ \forall I \subset \{1,\dots,s\}, \ |I|=n+1,n+2.
\end{split}
\end{equation}
Moreover, $D$ is movable if and only if it is effective and
\begin{equation}\label{movable condition n+2}
\begin{split}
& m_i\ge 0, \ \forall i \in\{1,\dots,s\},\\
 & \sum_{i\in I}m_i \le (n-1)d, \ \forall I \subset \{1,\dots,s\}, \ |I|=n.\quad \quad
\end{split}
\end{equation}
\item
Assume that $s=n+3$.
The divisor $D$ is effective if and only if \eqref{effectivity condition n+2} and
\begin{equation}\label{effectivity condition n+3}
\begin{split}
&  k_{I(n-2t),\sigma_t} \le0, \ \forall I(n-2t),\  t\ge 1. \quad\quad\quad\quad\quad\quad
\end{split}
\end{equation}
Moreover, $D$ is movable if and only if it is effective and the following conditions
are satisfied: \eqref{movable condition n+2} and

\begin{equation}\label{movable condition n+3}
\begin{split}
& k_{I(n-2t-1),\sigma_t} \le0,\ \forall I(n-2t-1),\ t\ge 1.\ \quad\quad\quad
\end{split}
\end{equation}
are satisfied.
\end{enumerate}

\end{theorem}

\subsection{The blow-up of ${\mathbb P}^n$ along linear cycles.}\label{higher blow ups}
For an effective divisor $D$ on $X_{s,(0)}$, we can  resolve its linear base locus by
constructing a birational morphism $X_{s,(r)}\rightarrow X_{s,(0)}$
(\cite[Section 4]{bdp1} and \cite[Section 2]{dp}).
For $r\leq \min\{s,n\}-1$, we denote by $X_{s,(r)}$ the iterated
 blow-up of ${\mathbb P}^n$ along the strict transform of the linear subspaces $L_I$
of dimension at most $r$ spanned by sets of points $I\subset\mathcal{S}$,
$|I|\le r+1$, with  $k_{I}>0$, ordered by increasing dimension.
We further denote by $E_I$ the (strict transform of) the exceptional divisor in $X_{s,(r)}$ of the linear space $L_I$, for every such an $I$.

We point out that the space $X_{s,(r)}$ depends on the divisor $D$, precisely on the integers
$k_I$, but for
the sake of simplicity we omitted this dependency from the notation.

\begin{remark}\label{linear cycles}
Abusing notation, we will denote by $L_I$ the linear subspace of ${\mathbb P}^n=:X_{s,(-1)}$
spanned by the set of points parametrized by $I$ as well as its strict transform in the
 blown-up spaces $X_{s,(r)}$, for every $r\ge0$.
\begin{enumerate}
\item  For $|I|\geq r+2$, $L_I$ on $X_{s,(r)}$ represents a blown-up projective space of dimension $|I|-1$ along linear cycles of dimension at most $r$.
\item If $|I|\leq r+1$, the strict transform of $L_I$ will be the exceptional divisor $E_I$ that is a product of blown-up projective spaces. The full description of this space and its intersection theory is explicitly given in \cite[Section 2]{dp}.
\end{enumerate}
\end{remark}

Denote by $D_{(r)}$ the strict transform of the divisor $D$ on $X_{s, (r)}$. One has
\begin{equation}\label{strict transform}
D_{(r)}:=dH - \sum_{\substack{I\subset\{1,\dots,s\}:\\0\le |I|\le r+1}} k_{I} E_{I}.
\end{equation}

\begin{remark}\label{remark divisorial contraction}
If $r=n-1$, then $D_{(n-1)}$ is the strict transform of  $D_{(n-2)}$
via the divisorial contraction $X_{s,(n-2)}\dashrightarrow X_{s,(n-2)}$ obtained by removing
$k_{I(n-1)}$ times the  strict transform of the hyperplane spanned by the points
parametrized by $I(n-1)$, for any $I(n-1)\subset\{1,\dots,s\}$.
\end{remark}

Set $\bar{r}$ to be the maximal dimension of the linear cycles of the
 base locus of $D$. We will also use $\tilde{D}:=D_{(\bar{r})}$ to denote the strict transform of $D$ under the blow-up of all its linear base locus (including hyperplanes for $\bar{r}=n-1$). To simplify notation we will abbreviate $h^i(X_{s,(r)},\mathcal{O}_{X_{s,(r)}}(D_{(r)}))$ by $h^i(D_{(r)})$.

\begin{theorem}[{\cite[Theorem 4.6]{bdp1}}]\label{extended vanishing} Let $\mathcal{S}$ be a collection of points of ${\mathbb P}^n$ in  general position. Let $D_{(r)}$ be as in \eqref{strict transform}. Assume that
\begin{equation}\label{cond vanishing}
\begin{split}
&0\leq m_i\leq d + 1, \\
&\sum m_i \leq nd+ \max\{1,s-n-2\}.
\end{split}
\end{equation}
Then  $h^i(D_{(r)})=0$, for every $i\ne 0,r+1$.
Moreover, $h^i(\tilde{D})=0$ for every $i\geq 1$.
\end{theorem}

Theorem \ref{extended vanishing} states that if $D$ is a \emph{special} divisor on $X_{s,(0)}$, i.e. one for which the first cohomology groups does not vanish, and if it satisfies the  bounds on the coefficients \eqref{cond vanishing}, then its  strict transform $\tilde{D}$ is no more special, i.e.  it has vanishing higher cohomology groups.
In \cite{bdp1} the following question was posed, namely whether a similar statement   is true for all cycles $-$linear and not linear$-$ of ${\rm{Bs}}(|D|)$.

\begin{question}[{\cite[Question 1.1]{bdp1}}]\label{question}
Consider any effective divisor $D$ in the blown-up ${\mathbb P}^{n}$ at general points.
Let $\widetilde{X}$ be the smooth composition of blow-ups of ${\mathbb P}^n$ along the
(strict transforms of the)
cycles of the base locus of $|D|$, ordered in increasing dimension.
Denote by $\widetilde{{\mathcal{D}}}$ the strict transform of the general divisor of ${\mathcal L}$  in $\widetilde{X}$.
  Does $h^i(\widetilde{X},\mathcal{O}_{\widetilde{X}}(\widetilde{\mathcal{D}}))$ vanish for all $i\ge 1$?
\end{question}

Notice that we adopt the notation  $\tilde{D}$ for the strict transform after
the blow-up of the linear base locus and $\widetilde{\mathcal{D}}$
 for the strict transform after the blow-up of the whole base locus.
An affirmative answer to Question \ref{question} would imply that $h^0(D)=\chi(Y, \mathcal{O}_{Y}(\widetilde{\mathcal{D}})).$

One may ask the following slightly different question.
\begin{question}\label{question2}
Let $X$ be  the blown-up ${\mathbb P}^{n}$ at general points and $D$ an effective divisor on $X$.
Let $(\widetilde{X},\widetilde{D})$ be obtained with a log resolution of the pair $(X,D)$.
   Does $h^i(\widetilde{X},\mathcal{O}_{\widetilde{X}}(\widetilde{\mathcal{D}}))$ vanish for all $i\ge 1$?
\end{question}
Notice that the assumption that  there exists  a log resolution of the pair $(X,D)$ of Question \ref{question2} is stronger than  that of Question \ref{question}; in fact proving that the strict transform $\widetilde{D}$ is smooth is an extremely difficult task in general.

In Section \ref{gg section} for $s\leq n+2$ and $D$ effective, or for $s\geq n+3$ and $D$ effective having only linear base locus, Theorem \ref{question answer} establishes that $\widetilde{\mathcal{D}}$ and $\tilde{D}$ coincide, answering positively Question \ref{question} and Question \ref{question2}.

\subsection{The blow-up of ${\mathbb P}^n$ along joins of
secant varieties of the rational normal curve and linear cycles}\label{section vanishing cohomology n+3}
In this section we extend Question \ref{question} to the case of effective divisors on
the blow-up of ${\mathbb P}^n$ at $n+3$ general points. Take
\begin{equation}\label{divisor n+3 points}
D=dH-\sum_{i=1}^{n+3}m_iE_i\ge0,
\end{equation}
 a divisor on $X_{n+3,(0)}$, the blown-up ${\mathbb P}^{n}$ at $n+3$ base points, with $d,m_i\ge0$.

We will iteratively blow-up $X_{n+3,(0)}$ along the subvarieties that are contained in the base locus of $D$, that are the strict
transforms of $\J(L_{I},\sigma_{t})$, see Lemma \ref{base locus lemma II}.
The pairwise intersections of such subvarieties  with some constraints on the index sets parametrizing the vertices,
is computed in \cite[Proposition 5.6]{am}. For the sake of completeness we state below this result in our notation.

\begin{proposition}[{\cite[Proposition 5.6]{am}}]\label{intersection of joins}
Let $I_1,I_2\subset\{1,\dots,n+3\}$ be index sets such that
$I_1\cap I_2=\emptyset$. Let $t_1,t_2\ge0$
 be integers such that
\begin{equation}\label{conditions intersection of joins}
\begin{split}
&r_{I_1,\sigma_{t_1}}=r_{I_2,\sigma_{t_2}}, \\
&r_{I_i,\sigma_{t_i}}\le n-1, \ \forall i\in\{1,2\},\\
& 2r_{I_i,\sigma_{t_i}}\le 2n-(|I_1|+|I_2|),\ \forall i\in\{1,2\}.
\end{split}
\end{equation}
Then $$\J(L_{I_1},\sigma_{t_1})\cap\J(L_{I_2},\sigma_{t_2})=\bigcup_{J}\J(L_J,\sigma_{t_J}),
$$
where the union is taken over all subsets $J\subseteq I_1\cup I_2$ satisfying
$$2|(I_i\cup J)\setminus(I_i\cap J)|=|I_1|+|I_2|,\ \forall i\in\{1,2\},$$
and for every such $J$, $t_J$ is the integer defined by the following equation
$$
2r_{J,\sigma_{t_J}}=2r_{I_i,\sigma_{t_i}}-(|I_1|+|I_2|).
$$
\end{proposition}

For any effective divisor $D$,
let $\pi^\sigma:Y^\sigma\to X$ be the iterated blow-up of $X:=X_{n+3,(0)}$
 along (the strict transforms of) all varieties $\J(L_I, {\sigma}_{t})$,
$t\ge 0$, $|I|\ge0$, such that
 $r_{I, {\sigma}_{t}}\le n-2$ and $k_{I,\sigma_t}>0$ in increasing dimension, composed with
a contraction of the strict transforms of the divisors $\J(L_I, {\sigma}_{t})$
with $r_{I, {\sigma}_{t}}= n-1$ and $k_{I, {\sigma}_{t}}>0$.
The latter divisors were described in \cite[Section 3.2]{bdp3}.
The space $Y^\sigma$ is constructed by Araujo and Massarenti in their recent preprint \cite[Section 5]{am} in order to give explicit log
Fano structures on  $X_{n+3,(0)}$ (see \cite[Propositions 5.8,5.11]{am}).

We denote by $E_{I,\sigma_t}$ the exceptional divisors, for
all $t\ge 0$, $|I|\ge0$, $r_{I, {\sigma}_{t}}\le n-1$.
It is immediate to see, using \cite[lemma 4.1]{bdp3},
that the strict transform on $Y^\sigma$ of $D$ is given by
\begin{equation}\label{definition D sigma}
\tilde{D}^{\sigma}:=dH-\sum_{i}m_i E_i-\sum_{r=1}^{n-1}\sum_{\substack{I,t:\\ r_{I,\sigma_t}=r}}k_{I,\sigma_t}E_{I,\sigma_t}.
\end{equation}
We stress the fact that the space $Y^\sigma$ depends on the divisor $D$.

\subsection{Conjectures on vanishing cohomology
on the blow-up along joins of
secant varieties of the rational normal curve and linear cycles}\label{conjectures}
Let $D$ be an effective divisor on ${\mathbb P}^n$ blown-up in $n+3$ general points.

\begin{question}\label{question sigma}
Consider the divisor $\tilde{D}_{\sigma}$ defined in \eqref{definition D sigma}
as the strict transform of $D$ in $Y^\sigma$.
Does $h^i(Y^\sigma,\mathcal{O}_{Y^\sigma}(\tilde{D}^{\sigma}))$ vanish for all $i\ge 1$?
\end{question}

\subsubsection{Related questions}

Another challenge would be  to compute the Euler characteristic of  $\tilde{D}^{\sigma}$.
In what follows we would like to propose a candidate for such a number, namely
the so called \emph{secant linear virtual dimension} for
linear systems of hypersurfaces of ${\mathbb P}^n$ interpolating $n+3$ general points with
assigned multiplicity, or equivalently of linear systems $|D|$. This number  was introduced in \cite{bdp3}.

\begin{definition}[{\cite[Definition 6.1]{bdp3}}]\label{new definition rnc}
Let $D$ be a divisor on $X_{n+3,(0)}$ as in \eqref{divisor n+3 points}.
The {\em secant linear virtual dimension} of $|D|$ is the number
\begin{equation}\label{RNC expected dimension}
\sldim(D):=\sum_{I,\sigma_t}(-1)^{|I|}{{n+k_{I,\sigma_t}-r_{I,\sigma_t}-1}\choose n},
\end{equation}
where the sum ranges over all indexes $I\subset\{1,\dots,n+3\}$ and $t$ such that
$0\le t \le l+\epsilon$, $n=2l+\epsilon$  and $0\le|I|\le n-2t$. The integers $k_{I,\sigma_t}$
and $r_{I,\sigma_t}$ are defined in \eqref{expanded} and in \eqref{dimension join} respectively.
\end{definition}

\begin{conjecture}\label{conjecture chi sigma}
The Euler characteristic of  the divisor $\tilde{D}^{\sigma}$,
defined in \eqref{definition D sigma} as the strict transform of $D$ in $Y^\sigma$, is
$$
\chi(Y^\sigma,\mathcal{O}_{Y^\sigma}(\tilde{D}^{\sigma}))=\sldim(D).
$$
\end{conjecture}

\vskip.3cm

The above questions are related to the \emph{dimensionality problem} for linear systems of
the form \eqref{linear system}
and in particular to the  Fr\"oberg-Iarrobino   conjectures
\cite{Froberg, Iarrobino}, which give a predicted value for the Hilbert series of
an  ideal generated by $s$ general powers of linear forms  in the polynomial ring
 with $n+1$ variables. We refer to \cite[Section 2.1]{bdp3} for a more detailed  account on this.
In \cite{bdp3} the following conjectural answer to this problem was given in terms of Definition \ref{new definition rnc}.

\begin{conjecture}[{\cite[Conjecture 6.4]{bdp3}}]\label{conjecture}
Let $D$ be as in \eqref{divisor n+3 points}. Then
$$h^0(X,\mathcal{O}_X(D))=\max\{0,\sldim(D)\}.$$
\end{conjecture}

Previous work \cite{bdp1,dp} contains a proof that
Question \ref{question sigma} admits an affirmative answer, as well as proofs of Conjectures
\ref{conjecture chi sigma} and \ref{conjecture},
 for divisors satisfying the bound \eqref{cond vanishing}, namely those that do not
contain positive multiples of the rational normal curve of degree $n$ in the base locus, nor joins $\J(I,\sigma_t)$.
The approach adopted was based on
 the study of the normal bundles of the exceptional divisors of linear cycles, $L_I$, and
vanishing cohomologies of strict transforms
on $X_{s,(n-2)}$, the blow-up along the linear cycles, see  Theorem
\ref{extended vanishing}.

We believe that a proof of the above conjectures for the general case for $s=n+3$
would rely on the study of the normal
bundles to the joins $\J(I,\sigma_t)$. We plan to develop
this approach in future work.

On a more general note,  we would like to point out that the construction of $Y^{\sigma}$ and
Question \ref{question sigma} and Conjectures
\ref{conjecture} and \ref{conjecture chi sigma} could be generalised to ${\mathbb P}^n$
blown-up in arbitrary number of points in linearly general position.

\section{$l$-very ample divisors on $X_{s,(0)}$}\label{lva section}

\begin{definition}[\cite{bfs}]\label{definition k-va}
Let $X$ be a complex projective smooth variety.
For an integer $l\ge0$, a line bundle  $\mathcal{O}_X(D)$ on $X$ is said to be \textit{$l$-very ample},
if for any $0$-dimensional subscheme $Z\subset X$ of weight $h^0(Z,\mathcal{O}_Z)=l+1$,
 the restriction map $H^0(X,\mathcal{O}_X(D))\to H^0(Z, \mathcal{O}_X(D)|_Z)$ is surjective.
\end{definition}

We will now recall some of the results obtained in the study of positivity of blown-up spaces or surfaces. Di Rocco
 \cite{dirocco} classified  $l$-very ample line bundles on del Pezzo surfaces, namely for ${\mathbb P}^2$ blown-up at $s\le 8$ points in general position.
For general surfaces, very ample divisors on rational surfaces were considered by Harbourne \cite{Ha}.
De Volder and Laface \cite{DeLa1} classified
 $l$-very ample divisors, for $l=0,1$, on the blow-up of ${\mathbb P}^3$ at $s$ arbitrary general points.
Ampleness and very ampleness properties of divisors on blow-ups at points of higher dimensional
projective spaces in the  case of points of multiplicity one
were studied by Angelini \cite{angelini}, Ballico \cite{ba} and Coppens \cite{coppens2}.

 Positivity properties for blown-up ${\mathbb P}^{n}$ in general points were considered
by Castravet and Laface. In particular, for small number of points in general position,
$s\leq 2n$, the semi-ample and nef cones, that we describe in this paper in Corollary \ref{nef cone}, were obtained via a different technique (private communication).

\vskip.3cm

For any effective divisor on $X_{s,(0)}$ denote by $s(d)$ the number of points in $\mathcal{S}$
of which the multiplicity equals $d$.
We  introduce the following integer (see \cite[Theorem 5.3]{bdp1}):
\begin{equation}\label{b bound}
b=b(D):= \min\{n -s(d); s-n-2\}.
\end{equation}

We can describe $l$-very ample line bundles over $X_{s,(0)}$, the blown-up
projective space at $s$ points in  general position,  whose underlying divisor is of the form
\eqref{def D} $$D=dH-\sum_{i=1}^sm_iE_i,$$
as follows.

\begin{theorem}[$l$-very ample line bundles]\label{tva theorem X0}
Assume that $\mathcal{S}\subset{\mathbb P}^n$ is a collection of points in general position.
Let $l$ be a non-negative integer. Assume that either $s\le 2n$ or  $s\ge 2n+1$ and $d$ large enough, namely
\begin{equation}\label{tva equations X0 large s}
\sum_{i=1}^s m_i -nd \le b-1-l,
\end{equation}
where $b$ is defined as in \eqref{b bound}.
Then a divisor $D$  of the form \eqref{def D}  is $l$-very ample if and only if
\begin{equation}\label{tva equations X0}
\begin{split}
&l\leq m_i, \ \forall i\in\{1,\dots,s\},\\
&l \le d-m_i-m_j, \ \forall i,j\in\{1,\dots,s\},\ i\ne j. \\
\end{split}
\end{equation}

\end{theorem}

\begin{remark}
When $l=0$ ($l=1$), $l$-very ampleness corresponds to \emph{global generation}, or \emph{spannedness}
(resp. very ampleness).
 \end{remark}

\begin{remark}
One can view conditions \eqref{cond vanishings} as particular case of \eqref{tva equations X0}, by setting $l=-1$.
\end{remark}

\begin{corollary}[Globally generated line bundles]\label{gg corollary}
In the same notation of Theorem \ref{tva theorem X0},
assume that either $s\le 2n$ or  $s\ge 2n+1$ and
$$\sum_{i=1}^sm_i -nd \le b-1.$$
Then $D$
 is globally generated if and only if
\begin{equation}\label{gg equations X0}
\begin{split}
& 0 \leq m_i, \ \forall i\in\{1,\dots,s\},\\
& 0 \le d-m_i-m_j, \ \forall i,j\in\{1,\dots,s\},\ i\ne j.\\
\end{split}
\end{equation}
\end{corollary}

\begin{corollary}[Very ample line bundles]\label{va corollary}
In the same notation of Theorem \ref{tva theorem X0},
assume that either $s\le 2n$ or  $s\ge 2n+1$ and
$$\sum_{i=1}^sm_i -nd \le b-2.$$
Then $D$ is very ample if and only if
\begin{equation}\label{va equations X0}
\begin{split}
&1\leq m_i, \ \forall i\in\{1,\dots,s\},\\
& 1\le d-m_i-m_j, \ \forall i,j\in\{1,\dots,s\},\ i\ne j.\\
\end{split}
\end{equation}
\end{corollary}

\begin{example}\label{example}
We present  an example where the bound \eqref{tva equations X0} of  Theorem \ref{tva theorem X0} is sharp. Let us consider the anticanonical divisor of the blown-up ${\mathbb P}^2$ in eight points in general position
$$D:=3H-E_1-\ldots - E_8.$$
Sections of $D$ correspond to planar cubics passing through eight simple points. It is well-known that all such cubics meet at a ninth point, therefore $D$ is not a globally generated divisor. However, $D$ is nef.
\end{example}

\subsection{Proof of Theorem \ref{tva theorem X0}}

In order to give a proof of Theorem \ref{tva theorem X0}, we first give the following vanishing theorem, that has its own intrinsic interest.

\begin{theorem}\label{toric tensored}
In the same notation as Theorem \ref{tva theorem X0},
fix integers $d,m_1,\dots,m_s,l\ge0$, $s\ge1$.
Assume that either $s\le 2n$ or that $s\ge 2n+1$ and  that
 \eqref{tva equations X0 large s} is satisfied.
Moreover, assume that
\begin{equation}
\begin{split}
&l\leq m_i, \ \forall i\in\{1,\dots,s\},\\
&l \le d-m_i-m_j, \ \forall i,j\in\{1,\dots,s\},\ i\ne j.
\end{split}
\end{equation}
Then $h^1(D\otimes {\mathcal I}_{\{q^{l+1}\}})=0$ for any $q\in X_{s,(0)}$.
\end{theorem}

\begin{proof}
Case (1).
Assume first of all that  $q\in E_i$, for some $i\in\{1,\dots,s\}$.
We claim that
\begin{equation}\label{transferring h^1}
h^1(D\otimes {\mathcal I}_{\{q^{l+1}\}})\le h^1(D-(l+1)E_i).
\end{equation} Hence
we conclude because the latter vanishes, by  Theorem \ref{vanishing theorems}.
We now prove that \eqref{transferring h^1} holds. Let $\pi$ be the blow-up of $X_{s,(0)}$ at $q\in E_i$ and let $E_q$ be the exceptional divisor created.
By the \emph{projection formula} we have $H^i(D\otimes{\mathcal I}_{\{q^{l+1}\}})\cong H^i(\pi^*(D)-(l+1)E_q)$.
For $l=0$, consider the  exact sequence
\begin{equation}\label{seq l=0}
0\to \pi^*(D)-\pi^*(E_i)\to\pi^*(D)-E_q\to (\pi^*(D)-E_q)|_{\pi^*(E_i)-E_q}\to0.
\end{equation}
 Notice that $\pi^*(E_i)-E_q$ is the blow-up of $E_i\cong{\mathbb P}^{n-1}$ at the point $q$:
 denote by $h,e_q$ the generators of its Picard group.
We have $(\pi^*(D)-E_q)|_{\pi^*(E_i)-E_q}\cong m_ih-e_q$,
 in particular it has vanishing first cohomology group.
Hence, looking at the long exact sequence in cohomologies associated with \eqref{seq l=0},
one gets that the map $$H^1(\pi^*(D)-\pi^*(E_i))\to H^1(\pi^*(D)-E_q)$$ is surjective, therefore $h^1(\pi^*(D)-\pi^*(E_i))\ge h^1(\pi^*(D)-E_q)$. Finally,
by the projection formula one has $H^i(\pi^*(D)-\pi^*(E_i))=H^i(D-E_i)$, so we conclude. For $l\ge 1$, one can iterate $l$ times the above argument and conclude.

\vskip.3cm

Case (2). Assume $q\in X_{s,(0)}\setminus\{E_1,\dots,E_s\}$. Hence  $q$  is the pull-back of a point $q'\in {\mathbb P}^n\setminus\{p_1,\dots,p_s\}$.

We will prove the statement by induction on $n$. The case $n=1$ is obvious. Indeed, any such $D\otimes{\mathcal I}_{\{q^{l+1}\}}$ corresponds to a linear series on the projective line given by three points whose sum of the multiplicities is bounded above as follows $m_1+m_2+(l+1)\le d+1$. Hence the first cohomology group vanishes. From now on we will assume $n\ge2$.

\vskip.3cm

Recall that a set of  points $\mathcal{S}$ of ${\mathbb P}^n$ is said to be in
\textit{linearly general position} if for each integer $r$ we have $\sharp (S\cap L) \le r+1$, for all $r$-dimensional linear subspaces $L$ in ${\mathbb P}^n$.

\vskip.3cm

Case (2.a).
Assume first that the points in $\mathcal{S}\cup\{ q'\}$ are not in linearly general position in ${\mathbb P}^n$.
If $s\ge n$, $q'$ lies on a hyperplane $H$ of ${\mathbb P}^n$ spanned by $n$ points of $\mathcal{S}$. Reordering the points if necessary, assume that $q'\in H:=\langle p_1,\dots,p_n\rangle$.
If $s< n$, let $H$ be any hyperplane containing $\mathcal{S}\cup\{q'\}$.
Let $\bar{H}$ denote the pull-back of $H$ on $X_{s,(0)}$. By Remark \ref{linear cycles} part (1),
$\bar{H}$ is isomorphic to the space ${\mathbb P}^{n-1}$ blown-up at $\bar{s}:=\min\{s,n\}$ distinct points in general position, that we  may denote by $\bar{H}\cong X^{n-1}_{\bar{s},(0)}$. Its Picard group is generated by $h:=H|_H$, $e_i:=E_i|_{H}$.
As a divisor, we have $\bar{H}=H-\sum_{i=1}^{\bar{s}} E_i.$
Consider the restriction exact sequence of line bundles
\begin{equation}\label{restriction to hyperplane}
0\to (D-\bar{H})\otimes{\mathcal I}_{\{q^{l}\}}\to D\otimes{\mathcal I}_{\{q^{l+1}\}} \to (D\otimes{\mathcal I}_{\{q^{l+1}\}})|_{\bar{H}}\to0.
\end{equation}
We iterate this restriction procedure $l+1$ times.
The restriction of the $(\lambda+1)$st exact sequence, $0\le \lambda\le l$,
is the complete linear series on $X^{n-1}_{\bar{s},(0)}$ given by
\begin{equation}\label{restriction lambda}
\left((d-\lambda)h-\sum_{i=1}^{\bar{s}}(m_i-\lambda) e_i\right)\otimes{\mathcal I}_{\{q^{l+1-\lambda}\}|_H}.
\end{equation}
We leave it to the reader to verify that it satisfies the hypotheses of the theorem, for every $0\le\lambda\le l$. Hence we conclude, by induction on $n$, that the first cohomology group vanishes.

Moreover, the (last) kernel is the line bundle associated with the following divisor:
\begin{equation}\label{last kernel}
d'H-\sum_{i=1}^dm'_iE_i:=(d-l-1)H-\sum_{i=1}^{\bar{s}}(m_i-l-1)E_i-\sum_{i=\bar{s}+1}^s m_i E_i.
\end{equation}
It is an easy computation to verify that the
 conditions of Theorem \ref{vanishing theorems} are verified.
Indeed, if $\bar{s}=s<n$ we have
$$\sum_{i=1}^{\bar{s}}m'_i-nd'=\sum_{i=1}^{\bar{s}}(m_i-l-1)-n(d-l-1)\le0,$$
because $m_i\le d$. Otherwise, if $\bar{s}=n\le s$, we compute
$$
\sum_{i=1}^s m'_i-nd'=\sum_{i=1}^{\bar{s}}(m_i-l-1)+\sum_{i=\bar{s}+1}^s m_i-n(d-l-1)=
\sum_{i=1}^sm_i-nd.$$
The above number is bounded above by $0$ whenever $s\le 2n$, and by $b-1-l$ whenever $s\ge 2n+1$, by the hypotheses.
Moreover, in all cases one has $m'_i+m'_j-d'\le1$, for all $i\neq j$. Hence we conclude in this case, using Theorem \ref{vanishing theorems}.

\vskip.3cm

Case (2.b).
Lastly, assume that $\mathcal{S}\cup\{ q'\}$ is in linearly general position in ${\mathbb P}^n$.
If $s\ge n-1$, let $H$ denote the hyperplane $\langle p_1,\dots,p_{n-1},q'\rangle$.  If $s< n-1$, let $H$ be any hyperplane containing $\mathcal{S}\cup\{q'\}$. In both cases such an $H$ exists by the assumption that points of
 $\mathcal{S}$ are in general position.
As in the previous case, let $\bar{H}$ denote the pull-back of $H$ on $X_{s,(0)}$.
 It is isomorphic to the space ${\mathbb P}^{n-1}$ blown-up at $\bar{s}:=\min\{s,n-1\}$ distinct points in general position,
that we  may denote by $\bar{H}\cong X^{n-1}_{\bar{s},(0)}$.

We iterate the same restriction procedure shown in \eqref{restriction to hyperplane} $l+1$ times as before.
As before the restriction of the $(\lambda+1)$st exact sequence, that is of the form \eqref{restriction lambda} with $\bar{s}$ differently defined here, verifies the hypotheses of the theorem,  so  it has vanishing first cohomology group by induction on $n$.

Furthermore, the (last) kernel, that is in the shape \eqref{last kernel}, with $\bar{s}, d', m'_i$ as defined here, verifies the
 conditions of Theorem \ref{vanishing theorems}. Indeed, if
 $\bar{s}=s<n-1$ then it is the same computation as before. While if  $\bar{s}=n-1\le s$ we have
 $$
\sum_{i=1}^sm'_i-nd'= \sum_{i=1}^{\bar{s}}(m_i-l-1)+\sum_{i=\bar{s}+1}^sm_i-n(d-l-1)=\sum_{i=1}^sm_i-nd+l+1.
 $$
The number on the right hand side of the above expression is bounded above by $0$ is $s\le 2n$ and by $b$ if $s\ge 2n+1$. Hence we conclude.
\end{proof}

Let ${\mathcal{L}}={\mathcal{L}}_{n,d}(m_1,\dots,m_s)$ be the linear system of the form \eqref{linear system}.

\begin{corollary}
Assume that ${\mathcal{L}}={\mathcal{L}}_{n,d}(m_1,\dots,m_s)$ satisfies the conditions of  Theorem \ref{toric tensored}. Then the linear system of elements of ${\mathcal{L}}$ that vanish with multiplicity $l+1$ at an arbitrary extra point, ${\mathcal{L}}_{n,d}(m_1,\dots,m_s,l+1)$, is non-special.

\end{corollary}
\begin{proof}
The projection formula implies that, for all $i\ge 0$,  $H^i(X_{s,(0)}, D\otimes{\mathcal I}_{\{q^{l+1}\}})\cong H^i({\mathbb P}^n,{\mathcal{L}}_{n,d}(m_1,\dots,m_s,l+1))$. Therefore ${\mathcal{L}}_{n,d}(m_1,\dots,m_s,l+1)$ has the expected dimension.
\end{proof}

Before we proceed with the proof of the main result, Theorem \ref{tva theorem X0}, we need the following lemmas.

\begin{lemma}\label{flat degeneration}
Let $X$ be a complex projective smooth variety and $\mathcal{O}_X(D)$ a line bundle. Let $Z$ be a $0$-dimensional subscheme of $X$ and let $Z_0$ be a flat degeneration of $Z$.
Then $h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_Z)\le h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_0})$.
\end{lemma}
\begin{proof}
It follows from the property of \emph{upper semicontinuity} of cohomologies, see e.g, \cite[Sect. III.12]{hartshorne}.
\end{proof}

\begin{lemma}\label{inclusion}
In the same notation of Lemma \ref{flat degeneration}, let $Z_1\subseteq Z_2$ be an inclusion of $0$-dimensional schemes.
Then $h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_1})\le h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_2})$.
\end{lemma}
\begin{proof}
If $Z_1=Z_2$ then equality obviously holds. We will assume $Z_1\subsetneq Z_2$.
Consider the following short exact sequence
$$0\stackrel{}{\rightarrow}\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_2}\stackrel{}{\rightarrow}\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_1}\stackrel{}{\rightarrow}\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_1}|_{Z_2\setminus Z_1}\stackrel{}{\rightarrow}0.$$
Consider the associated long exact sequence in cohomology. Since
$h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_1}|_{Z_2\setminus Z_1})=0$, the map $H^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_2})\to H^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{Z_1})$ is surjective and this concludes the proof.
\end{proof}

Lemma \ref{flat degeneration} and Lemma \ref{inclusion} will allow to reduce the proof of $l$-very ampleness for divisors $D$ to the computation of vanishing theorems of the first cohomology group of the sheaf associated with $D$ tensored by the ideal sheaf of a collection of fat points, whose multiplicities sum up to $l+1$.

\begin{proof}[Proof of Theorem \ref{tva theorem X0}]
We first prove that  \eqref{tva equations X0} is sufficient condition for $D$ to be $l$-very ample. For every $0$-dimensional scheme $Z\subset X_{s,(0)}$ of weight
 $l+1$, consider the exact sequence of sheaves
\begin{equation}\label{sequence for lva}
0\to \mathcal{O}_{X_{s,(0)}}(D)\otimes{\mathcal I}_Z\to \mathcal{O}_{X_{s,(0)}}(D)\to \mathcal{O}_{X_{s,(0)}}(D)|_Z\to0.
\end{equation}
We will prove that $h^1(X_{s,(0)},\mathcal{O}_{X_{s,(0)}}(D)\otimes{\mathcal I}_Z)=0$.
This will imply the surjectivity of the map
$H^0(X_{s,(0)},\mathcal{O}_{X_{s,(0)}}(D))\to H^0(Z, \mathcal{O}_{X_{s,(0)}}(D)|_Z)$, by taking the
long exact sequence in cohomology associated with \eqref{sequence for lva}.

Let $Z_0$ denote a flat degeneration of $Z$ with support at the union of points $q_1,\dots,q_s,q_{s+1}\in  X_{s,(0)}$ with  $q_i\in E_i$, for all $i=1,\dots,s$, and $q_{s+1}\in X_{s,(0)}\setminus\{E_1,\dots,E_s\}$.
Since every exceptional divisor $E_i$ as well as $X_{s,(0)}\setminus\{E_1,\dots,E_s\}$ are homogeneous spaces, in order to prove that $h^1(D\otimes{\mathcal I}_Z)=0$ for all $Z\subset X_{s,(0)}$ $0$-dimensional schemes of weight $l+1$, by Lemma \ref{flat degeneration}
 it is enough to prove that the same statement holds for every such $Z_0$.

Set  $\mu_i$ to be the weight of the irreducible component of $Z_0$ supported at $q_i$,
for all $i=1,\dots,s,{s+1}$. One has that $\mu_i\ge0$, $\sum_{i=1}^{s+1}\mu_i=l+1$.
In order to prove that $h^1(D\otimes{\mathcal I}_{Z_0})=0$, by Lemma \ref{inclusion}, it suffices to prove the a priori stronger statement that
 $h^1(\mathcal{O}_X(D)\otimes{\mathcal I}_{\bar{Z}_0})=0$, for every $\bar{Z}_0$ collection of fat points $\{q_1^{\mu_1},\dots, q_s^{\mu_s},q_{s+1}^{\mu_{s+1}}\}$. Indeed, the following containment of schemes supported at $q_i$ holds for all $i=1,\dots,s,{s+1}$: $Z_i|_{q_i}\subseteq \{q_i^{\mu_i}\}$.

Assume first of all that $\bar{Z}_0$ has support in one point $q=q_i$, namely $\mu_i=l+1$ for some $i\in\{1,\dots,s,{s+1}\}$ and $m_j=0$ for all $j\neq i$. We have $h^1(D\otimes{\mathcal I}_{\bar{Z}_0})=0$ by Theorem \ref{toric tensored}.

Assume now that  $\bar{Z}_0$ is supported in several points $q_1,\dots,q_s,q_{s+1}$.
We want to prove that  $h^i(X_{s,(0)},D\otimes{\mathcal I}_{\bar{Z}_0})\le h^i(X_{s,(0)},\left(D-\sum_{i=1}^s \mu_i E_i\right)\otimes{\mathcal I}_{\{q_{s+1}^{\mu_{s+1}}\}})=0$. The first inequality follows from \eqref{transferring h^1}.  The  equality follows from Theorem \ref{toric tensored};
we leave it to the reader to verify that the conditions are indeed satisfied.

\vskip.3cm

We now prove that \eqref{tva equations X0} is necessary condition for $D$ to be $l$-very
ample, by induction on $l$.

Let us first assume $l=0$, namely that $D$ is base point free.
If $m_i<0$ then $m_iE_i$ would be contained in the base locus of $D$.
If $m_i+m_j> d$ for some $i\ne j$, then the strict transform of the line $\langle p_i,p_j\rangle\subset{\mathbb P}^n$
would be contained in the base locus of $D$.
In both cases we would obtain a contradiction.

Assume that $l=1$, namely that $D$ is very ample. If $m_i\le0$ (or $0\le d-m_i-m_j$ for some $i\ne j$),
then $E_i$ (resp. the strict transform of the line through $p_i$ and $p_j$) would be contracted by $D$, a contradiction.

More generally, assume that $D$ is $l$-very ample and $l\ge2$.
Then conditions \eqref{tva equations X0} are satisfied. Indeed, if $m_i\le l-1$ for some $i$,
we can find a $0$-dimensional scheme, $Z$, of weight $l+1$ such that
 $h^1(D\otimes{\mathcal I}_Z)>0$. Let $Z\subset E_i$ be a $l$-\emph{jet scheme} centred at $q\in E_i$ (see \cite{mustata}).
Consider the restriction $D\otimes{\mathcal I}_Z|_{E_i}\cong m_1h\otimes{\mathcal I}_Z$, where $h$
 is the hyperplane class of $E_i\cong{\mathbb P}^{n-1}$. We have $h^1(E_i,D\otimes{\mathcal I}_Z|_{E_i})\ge1$, hence
$h^1(X_{s,(0)},D\otimes{\mathcal I}_Z)\ge1$. To see this, let $x_1\dots,x_{n-1}$ be
affine coordinates for an affine chart $U\subset E_i$ and let  $Z$ be the jet-scheme
with support
$q=(0,\dots,0)\in U$
given by the tangent directions up to order $l$ along $x_1$.
The space of global sections of $D\otimes{\mathcal I}_Z|_{E_i}$
is isomorphic to the set of degree-$m_i$ polynomials $f(x_1,\dots, x_{n-1})$,
 whose partial derivatives $\partial^\lambda f/\partial x_1^{\lambda}$
 vanish at $q$, for $0\le \lambda\le l$. On the other hand,
$H^1(E_i,D\otimes{\mathcal I}_Z|_{E_i})$
is the space of linear dependencies among the $l+1$ conditions imposed by the vanishing of
 the partial derivatives
to the coefficients of $f$. Since $m_i\leq l-1$ then $f$ is a polynomial of degree bounded by $l-1$, therefore $\partial^{l} f/\partial x_1^{l}\equiv0$ for every such a polynomial, and we conclude.

Similarly, if $d-m_i-m_j\le l-1$ for some $i,j,i\ne j$, then one finds a jet-scheme $Z$ contained in the pull-back of the line through $p_i$ and $p_j$, $L$, for which $h^1(X_{s,(0)},D\otimes{\mathcal I}_Z)\ge1$. Indeed, if $Z$ is such a scheme, then the restriction is
 $D\otimes{\mathcal I}_Z|_L\cong (d-m_i-m_j)h\otimes{\mathcal I}_Z|_{L}$, where in this case $h$ is  the class of a point on $L$, and $Z|_L$ is a fat point of multiplicity $l$ on $L$. One concludes by the Riemann-Roch Theorem that $h^1(L,D\otimes{\mathcal I}_Z|_L)\ge1$ because $\chi(L,D\otimes{\mathcal I}_Z|_L)=(d-m_i-m_j)-l\le-1$ and $h^0(L,D\otimes{\mathcal I}_Z|_L)=0$.

\end{proof}

\subsection{$l$-jet ampleness}
In \cite{bfs}, Beltrametti, Francia and Sommese introduced  notions of
higher order embeddings, one of these
being $l$-very ampleness (Definition \ref{definition k-va}) with the aim of studying the \emph{adjoint bundle} on surfaces.

\begin{definition}
In the same notation as Definition \ref{definition k-va},
 if for any fat point $Z=\{q^{l+1}\}$, $q\in X$,  the natural restriction map to $Z$,
 $H^0(X,\mathcal{O}_X(D))\to H^0(Z, \mathcal{O}_X(D)|_{Z})$, is surjective, then $D$
 is said to be $l$-\emph{jet spanned}.

Moreover, if for any collection of \emph{fat points}
$Z=\{q_1^{\mu_1},\dots,q_\sigma^{\mu_{\sigma}}\}$ such that $\sum_{i=1}^{\sigma}\mu_i=l+1$,
 the restriction map to $Z$ is surjective, then $D$
 is said to be $l$-\emph{jet ample}.
\end{definition}

\begin{remark}\label{jet spannedness}
Theorem \ref{toric tensored} can be restated in terms of $l$-jet spannedness. Namely
 any divisor $D$ satisfying the hypotheses is $l$-jet spanned.

\end{remark}

\begin{proposition}[{\cite[Proposition 2.2]{belsom}}]\label{jet implies very}
In the above notation, if $D$ is $l$-jet ample, then $D$ is $l$-very ample.
\end{proposition}

The converse of Proposition \ref{jet implies very} is  true for the projective space ${\mathbb P}^n$ and for curves, but not in general.
In this section we proved that the converse is true for lines bundle $D$ on $X_{s,(0)}$,
 that satisfy the hypotheses of Theorem \ref{tva theorem X0}.

\begin{theorem}\label{jet ampleness}
Assume that $s\le 2n$, or $s\ge 2n+1$ and \eqref{tva equations X0 large s}. Assume that
 $D$ is a line bundle on $X_{s,(0)}$ of the form \eqref{def D}. The following are equivalent:
\begin{enumerate}
\item $D$ satisfies \eqref{tva equations X0};
\item $D$ is  $l$-jet ample;
\item $D$ is $l$-very ample.
\end{enumerate}
\end{theorem}
\begin{proof}
We proved that the natural restriction map of the global sections of $D$ to the any fat point of multiplicity $l+1$ is surjective
 in Theorem \ref{toric tensored}, see also Remark \ref{jet spannedness}. We showed that the same is true in the case of
arbitrary collections of fat points whose multiplicity sum up to
$l+1$ in the first part of the proof of Theorem
\ref{tva theorem X0}.  This proves that $(1)$ implies $(2)$. Moreover, $(2)$ implies $(3)$
 by Proposition \ref{jet implies very}.
Finally, that $(3)$ implies $(1)$ was proved in the second part of the proof of Theorem \ref{tva theorem X0}.
\end{proof}

\section{Other positivity properties of divisors on $X_{s,(0)}$}\label{other positivity}

In this section we will apply Theorem \ref{tva theorem X0} to establish further positivity
properties of divisors on $X_{s,(0)}$. All results
we prove in this section apply to $\mathbb{Q}-$divisors on the blown-up projective space.

We recall here the notation introduced in Section \ref{lva section}: for a given divisor $D$
of the form $D=dH-\sum_{i=1}^sm_iE_i,$ (cfr. \eqref{def D}),
we will use the integer $b=b(D):= \min\{n -s(d); s-n-2\}$, defined in (\ref{b bound})
and the bound \eqref{tva equations X0 large s}:
$$
\sum_{i=1}^s m_i -nd \le b-1-l.
$$

 \subsection{Semi-ampleness and ampleness}\label{section semiampleness}

A line bundle is ample if some positive  power is very ample.
It is known that for smooth toric varieties a divisor is ample if and only if is very ample and nef if and only if is globally generated.
  From Corollary \ref{gg corollary} and Corollary \ref{va corollary}, we obtain
that this holds for a small number of points $s\le 2n$ too, as well as for
arbitrary $s$ under a bound on the coefficients.

A line bundle is called \emph{semi-ample}, or \emph{eventually free}, if some positive  power is globally generated. By \eqref{gg equations X0}, one can see that a divisor is semi-ample if and only if it is globally generated.

\begin{theorem}
\label{ample cone}
 Let $X_{s,(0)}$ be defined as in Section \ref{notations}. Assume $s\le 2n$.
\begin{enumerate}
\item
The cone of semi-ample divisors in ${\rm{N}}^1(X_{s,(0)})_{\mathbb{R}}$  is given by
\eqref{gg equations X0}.
\item
The cone of ample divisors in ${\rm{N}}^1(X_{s,(0)})_{\mathbb{R}}$ is given by \eqref{va equations X0}.
\end{enumerate}

Assume $s\ge 2n+1$.
\begin{enumerate}
\item   Divisors satisfying
\eqref{tva equations X0 large s} with $l=0$
are semi-ample if and only if \eqref{gg equations X0}.
\item
Divisors satisfying \eqref{tva equations X0 large s} with $l=1$
are ample if and only if \eqref{va equations X0}.
\end{enumerate}

\end{theorem}

\subsection{Nefness}\label{section nefness}

For any projective variety, Kleiman \cite{kleiman} showed that a divisor is ample if and only if its numerical equivalence class lies in the interior of the nef cone
 (see also \cite[Theorem 1.4.23]{lazarsfeld}).

For a line bundle, being  generated by the global sections implies being nef, but the opposite is not true in general,
see e.g. Example \ref{example}. However for line bundles on $X_{s,(0)}$, with $s\le 2n$,
 or with arbitrary $s$ under a bound on the coefficients,
 these two properties are equivalent.

\begin{theorem}\label{nef cone}
In the same notation as Theorem \ref{tva theorem X0}, assume that for $D$ of the form
\eqref{def D} we have that either $s\le 2n$ or  $s\ge 2n+1$ and
\eqref{tva equations X0 large s} with $l=0$
is satisfied.
Then $D$  is nef if and only if is globally generated.

In particular if $s\le 2n$, the nef cone of $X_{s,(0)}$ is given by
\eqref{gg equations X0}.
\end{theorem}

\begin{proof}
If $D$ is nef, then for effective $1$-cylce $C$, $D\cdot C\geq 0$.
In particular the divisor $D$ intersects positively the classes of lines through
 two points and classes of lines in the exceptional divisors. This means inequalities
\eqref{gg equations X0} hold and therefore the divisor $D$ is globally generated by
Corollary \ref{gg corollary}.
\end{proof}

\begin{corollary}\label{nef = sa}
The nef cone and the cone of semi-ample divisors on $X_{s,(0)}$, for $s\le 2n$, coincide.
\end{corollary}

\begin{corollary}\label{mori cone}
The Mori Cone of curves of the blown-up ${\mathbb P}^n$ in $s\leq 2n$ points,  ${\rm{NE}}(X_{s,(0)})$,
  is generated by the classes of lines through two points and the classes of lines in the exceptional divisors.
\end{corollary}

\subsection{Bigness}\label{bigness section}
The \emph{pseudo-effective cone} is the closure of the effective cone and the  \emph{big cone} is the interior of the pseudo-effective cone (see e.g. \cite[Theorem 2.2.26]{lazarsfeld}).
For Mori Dream Spaces, the effective cone is polyhedral and closed, hence it coincides with the pseudo-effective cone.
In particular, when equations for the supporting hyperplanes of such a cone are known, one can recover the big cone.

We consider here ${\mathbb P}^n$ blown-up in $n+3$ general points, that is known to be a Mori Dream Space (see e.g. \cite{CT}).
For $|I|\le n-2t$ the number $k_{I,\sigma_t}$ \eqref{expanded}, introduced in
Section \ref{base locus and effectivity}, is the multiplicity of containment of the join $\J(I,\sigma_t)$
 in the base locus of $D$, see Lemma \ref{base locus lemma II}.
Theorem \ref{effectivity theorem} states that the equation $k_{I,\sigma_t}=0$, for $|I|= n-2t+1$, gives a supporting
hyperplane  for the pseudo-effective cone of $X_{n+3,(0)}$, while $k_{I,\sigma_t}=0$, for $|I|= n-2t$,
gives a supporting hyperplane  for the  movable cone of $X_{n+3,(0)}$.
The next results shows that $k_{I,\sigma_t}=-1$, for $|I|= n-2t+1$, is a supporting hyperplane for the big cone of $X_{n+3,(0)}$.

\begin{theorem}\label{big n+3 X0}
Assume $s\le n+2$. The big cone of $X_{s,(0)}$ is described by
\begin{equation}\label{bigness n+2 X0}
\begin{split}
&d> 0,\\
&m_i< d,\ \forall i\in \{1,\ldots,s\},\\
&\sum_{i\in I}m_i< nd,\ \forall I\subseteq\{1,\ldots,s\},\ |I|=n+1,n+2.
\end{split}
\end{equation}

Assume $s= n+3$.
The big cone of $X_{s,(0)}$ is described by
\begin{equation}\label{bigness n+3 X0}
\begin{split}
&d> 0,\\
&m_i< d,\ \forall i\in \{1,\ldots,s\}, \\
&\sum_{i\in I}m_i< nd,\ \forall I\subseteq\{1,\ldots,s\},\ |I|=n+1,n+2,\\
& k_{I, \sigma_t}<0,\  \forall |I|=n-2t+1,\  1\leq t\leq \nu+\epsilon,\\
\end{split}
\end{equation}
where $n=2\nu+\epsilon$ with $\epsilon\in \{0, 1\}$.
\end{theorem}

\subsection{Fujita's conjectures hold the blown-up ${\mathbb P}^n$ in points}

\begin{conjecture}[Fujita's conjectures, \cite{fujita}]\label{fujita}
Let $X$ be an $n$-dimensional projective algebraic variety, smooth or with mild singularities.
Let $K_X$ be the canonical divisor of $X$ and  $D$ an ample divisor on $X$. Then the following holds.
\begin{enumerate}
\item For $m\ge n+1$, $mD+K_X$ is globally generated.
\item For $m\ge n+2$, $mD+K_X$ is very ample.
\end{enumerate}
\end{conjecture}

In \cite{payne} Payne established the Fujita's conjecture  for toric varieties, in particular for $X_{s,(0)}$, for $s\le n+1$.
Using the results from this article, we can extend the above to $s\le2n$.

\begin{proposition}\label{fujita answer}
Let $X_{s,(0)}$ be the blown-up ${\mathbb P}^{n}$ at $s$ points in  general position with $s\le 2n$. Conjecture \ref{fujita} holds for $X_{s,(0)}$.
\end{proposition}

\begin{proof}
For smooth varieties, a consequence of the Mori's Cone Theorem is that the same statement of Conjecture  \ref{fujita} with globally generated (very ample) replaced by nef (resp. ample) holds (see also \cite{fujita}).

For $X_{s,(0)}$, $s\le 2n$, global generation (very ampleness) is equivalent to nefness (resp. ampleness), by Theorem \ref{ample cone} and Theorem \ref{nef cone}. This concludes the proof.
\end{proof}

\begin{proposition}\label{fujita answer2}
Let $X_{s,(0)}$ be the blown-up ${\mathbb P}^{n}$ in an arbitrary number of points in
general position, $s$, and let $D$ be a divisor on $X_{s,(0)}$ such that
\begin{equation}\label{zero bound}
\sum_{i=1}^s m_i\leq nd
\end{equation}
Then Conjecture \ref{fujita} holds for  $D$.
\end{proposition}

\begin{proof}
It is enough to consider the case $s\geq 2n+1$.
Write $X=X_{s,(0)}$. Notice that the divisor $mD+K_X$ has the following properties
\begin{align*}
\sum_{i=1}^s (mm_i - n+1) - n(md-n-1) &=
m(\sum_{i=1}^s m_i - n d)+n(n+1) - s(n-1)\\
&\leq n(n+1)-(2n+1)(n-1)\\
&=(-n^2+n)+(n+1)\\
&\leq -2+n+1\\
&=n-1.\\
\end{align*}

Notice that $b(mD+K_X)=n-1$, for $s=2n+1$ and $b(D)=n$ for $s\geq 2n+2$, using
the definition \eqref{b bound}.
Therefore $mD+K_X$ satisfies conditions of
Theorem \ref{ample cone}. We now leave it to the reader to check that if $D$ is ample then the divisor $mD+K_X$ satisfies conditions \eqref{gg equations X0} and \eqref{va equations X0}.

\end{proof}

\section{Globally generated divisors on $X_{s,(r)}$}\label{gg section}

  Let $D$ be a divisor on $X_{s,(0)}$ of the form \eqref{def D} and let $D_{(r)}$ be its strict transform of in $X_{s, (r)}$, as in \eqref{strict transform}:
$$
D_{(r)}=dH - \sum_{\substack{I\subset\{1,\dots,s\}:\\0\le |I|\le r+1}} k_{I} E_{I}.
$$
If $\bar{r}$ is the dimension of the linear base locus of $D$, denote $\tilde{D}:=D_{(\bar{r})}$ (see also Section \ref{notations}).

\begin{theorem}\label{gg theorem Xr}
Assume that $\mathcal{S}\subset{\mathbb P}^n$ is a collection of points in  general position.
 Assume that
 $s\le n+1$ or that $s\ge n+2$ and
 $d$ is large enough, namely
\begin{equation}\label{gg equations Xr large s}
\sum_{i=1}^s m_i -nd \le  \max\{0,s-n-3\}.
\end{equation}
Then for any $0\leq r\leq n-1$ the divisor $D_{(r)}$ on $X_{s,(r)}$ is globally generated  if and only if
\begin{equation}\label{gg equations Xr}
\begin{split}
&0\le m_i \le d, \ \forall i\in\{1,\dots,s\},\\
&0\le (r+1)d-\sum_{i\in I} m_i,\ \forall I\subseteq\{1,\dots,s\},\ |I|=r+2.\\
\end{split}
\end{equation}
\end{theorem}

\begin{remark}
Notice that if $s\le \left\lfloor\frac{r+2}{r+1}n\right\rfloor$, then
 the condition on the degree \eqref{gg equations Xr large s} is always satisfied.
Hence in this range, Theorem \ref{gg theorem Xr} provides
a complete classification of divisors $D$ on $X_{s,(0)}$ whose strict transform $D_{(r)}$  in $X_{s,(0)}$ is  globally generated.
\end{remark}

\begin{remark}\label{gg for Dn-1}
If $\bar{r}=n-1$, then $\tilde{D}$ is the strict transform of  $D_{(n-2)}$
via the contraction
of strict transforms of hyperplanes $L_{I(n)}\subset{\mathbb P}^n$,
$X_{s,(n-2)}\dashrightarrow X_{s,(n-2)}$, cfr. Remark \ref{remark divisorial contraction}.
Therefore in the same assumptions of Theorem \ref{gg theorem Xr}, we have that $\tilde{D}$
is globally generated if and only if
\begin{equation}\label{gg equations Xn-1}
\begin{split}
&0\le m_i \le d, \ \forall i\in\{1,\dots,s\},\\
&0\le nd-\sum_{i\in I} m_i,\ \forall I\subseteq\{1,\dots,s\},\ |I|=n+1.\\
\end{split}
\end{equation}
\end{remark}

\vskip.3cm

In order to prove the main result of this section, Theorem \ref{gg theorem Xr}, we need the following result.

\begin{lemma}\label{vanishing prelim}
In the above notation, assume
\begin{equation}\label{gg equations Xr large}
\sum_{i=1}^s m_i -nd \leq \max\{0,  s-n-3\}.
\end{equation}
Then for any point $q\in E_I$, for some $E_I$, we have
$$h^1(D_{(r)}\otimes{\mathcal I}_q)\leq h^1(D_{(r)}-E_I).$$
\end{lemma}

\begin{proof}
We prove the statement by induction on $n$. If $n=2$, it follows from Theorem \ref{toric tensored} applied in the case with $l=0$. We will assume $n\ge 3$.
Consider the short exact sequence
\begin{equation}\label{sequence}
0\stackrel{}{\rightarrow}D_{(r)}-E_I\stackrel{}{\rightarrow}D_{(r)}\otimes{\mathcal I}_q\stackrel{}{\rightarrow}D_{(r)}\otimes{\mathcal I}_q|_{E_I}\stackrel{}{\rightarrow}0.
\end{equation}
We claim that
\begin{equation}\label{claim 0}
h^1(D_{(r)}\otimes{\mathcal I}_q|_{E_I})=0.
\end{equation}
The statement will follow from this, by looking at the long exact sequence in cohomology associated with \eqref{sequence}.

We now prove the claim, namely that \eqref{claim 0} holds.
Set $\rho+1:=\#I$.
We recall that $E_I$ is a product,
whose  second factor is isomorphic to $X^{n-\rho-1}_{\alpha(I),(r-\rho-1)}$, the blown-up projective space of dimension $n-\rho-1$ along linear cycles up to dimension $r-\rho-1$ spanned by $\alpha(I)$ points. We refer to \cite[Lemma 2.5]{dp} for details.
 We introduce  the positive integer
$$\alpha(I):=\#\{I(\rho+1): k_{I(\rho+1)}\geq 1,\  I(\rho+1)\supset I\},$$
where $I(\rho+1)$ denotes an index set contained in $\{1,\dots, s\}$ of cardinality $\rho+2$ and $k_{I(\rho+1)}$ is defined as in \eqref{mult k}.
Moreover, let $F$
be a divisor on the blow-up of ${\mathbb P}^{n-\rho-1}$ at $\alpha(I)$  points in linearly general position, $X^{n-\rho-1}_{ \alpha(I),(0)}$, of the following form:
\begin{equation}\label{definition F}
F:=k_{I}h -\sum_{\substack { I(\rho+1)\supset I:\\k_{I(\rho+1)}\geq 1}} k_{I(\rho+1)}\cdot e_{I(\rho+1)|{I}}.
\end{equation}
We claim the following hold.
\begin{enumerate}
\item The restriction of $D_{(r)}$ to $E_I$ is
$D_{(r)}|_{E_I}=(0, F_{(r-\rho-1)})$, where $F_{(r-\rho-1)}$ denotes the strict transform of $F$ in $X^{n-\rho-1}_{\alpha(I),(r-\rho-1)}$.
\item If $D$ satisfies the bound \eqref{gg equations Xr large}, so does $F$.
\end{enumerate}

The proof of \eqref{claim 0} follows from these two claims. Indeed
(1) implies that
$$\mathcal{O}_{X_{(r)}}(D_{(r)})\otimes{\mathcal I}_q|_{E_I}\cong \mathcal{O}_{X^{n-\rho-1}_{\alpha(I),(r-\rho-1)}} (F_{(r-\rho-1)})\otimes{\mathcal I}_{q_b},$$
where $q=(q_b,q_f)\in E_I$. This has vanishing first cohomology group, by induction on $n$, since $F$ satisfies \eqref{gg equations Xr large}, by (2).

\vskip.3cm

We are left to proof the two statements (1) and (2).

We first prove (1). The fact that the first factor of the restriction $D_{(r)}|_{E_I}$ is zero follows from the computation of normal bundles of the $E_I$'s and from intersection theory on $X_{s,(r)}$, see \cite[Section 4]{bdp1}  or \cite[Section 2]{dp}.
We now compute the second factor.  We have
\begin{align*}
E_I|_{E_I}& =(*, h)\\
 E_I|_{E_{I(j)}}& =(*, e_{{J}|I}), \textrm{ for all }J\supset I,
\end{align*}
where $*$ denotes the appropriate divisor on the first factor, as we are only interested here in the second factor.
The classes of the divisors $h$ and $e_{J|I}$ generate the Picard group of $X^{n-\rho-1}_{\alpha(I),(r-\rho-1)}$.
Therefore one can compute
$$\ \ \ \ \ D_{(r)}|_{E_I}=\left(0, k_{I}h -\sum_{ J\supset I} k_{J}e_{J|{I}}\right).$$
To conclude, we need to prove that the second factor on the right hand side of the above expression equals the strict transform $F_{(r-i-1)}$.
Notice first that for any integer $\tau\ge 1$, one has $|I(\tau+\rho+1)\setminus I|=\tau+1$. Hence it is enough to prove that
$$\sum_{\substack {I(\rho+1)\supset I:\\
I(\rho+1)\subset I(\tau+\rho+1)}} k_{I(\rho+1)}-\tau k_I=k_{I(\tau+\rho+1)}.$$
By definition, the left hand side equals
\begin{equation*}\label{}
\begin{split}
&\sum_{\substack {I\subset I(\rho+1)\subset I(\tau+\rho+1)}} k_I+\sum_{j\in I(\tau+\rho+1)\setminus I}m_j-(\tau+1)d -\tau k_I=\\
&=(1+\tau)k_I+\sum_{j\in I(\tau+\rho+1)\setminus I}m_j-(\tau+1)d-\tau k_I\\
&=k_I+\sum_{j\in I(\tau+\rho+1)\setminus I}m_j-(\tau+1)d\\
&=k_{I(\tau+\rho+1)}.\\
\end{split}
\end{equation*}

\vskip.3cm

We now  prove (2). We need to show that the following inequality holds
\begin{equation}\label{ineq for 2}
\sum_{ I(\rho+1)\supset I}  k_{I(\rho+1)}\le (n-\rho-1)k_I+ \max\{0,\alpha(I)-(n-\rho-1)-3\}.
\end{equation}
Set  $A(I)=\{j\in S\setminus I: k_{I\cup \{j\}}\ge 1\}$. Notice that $\# A(I)=\alpha(I)\le s-\rho-1$.
Since the left hand side of \eqref{ineq for 2}  equals
$$
\sum_{j\in A(I)} \left(k_I+m_j-d\right)
=\alpha(I)k_I+\sum_{j\in A(I)} m_j - \alpha(I)d,
$$
 we need to prove the equivalent inequality
\begin{equation}\label{final ineq}
 \sum_{j\in A(I)} m_j - \alpha(I)d\leq (n-\rho-1-\alpha(I))k_I + \max\{0,\alpha(I)-(n-\rho-1)-3\}.
\end{equation}
For $\alpha(I)\le n-\rho-1$, this  holds since $|A(I)|=\alpha(I)$ and $m_j\le d$ so that the left hand side of \eqref{final ineq} is a non-positive integer, while the right hand side is  non-negative because $k_I\ge0$.
For $\alpha(I)\ge n-\rho$, the left hand side of \eqref{final ineq} equals
$$
\sum_{j\in S} m_j - nd  - k_I-\sum_{j\in S\setminus (I\cup A(I))}m_j - (\alpha(I)-n+\rho)d,
$$
therefore \eqref{final ineq} is equivalent to
\begin{align*}
&(\alpha(I)-n+\rho)(k_I-d)+\left(\sum_{j\in S} m_j-nd\right)
-\sum_{j\in S\setminus (I\cup A(I))}m_j \\
&\le \max\{0,\alpha(I)-(n-\rho-1)-3\}.
\end{align*}
One concludes by noticing that
$k_I\le d$ and by using \eqref{gg equations Xr large s}; we leave the details to the reader.

\end{proof}

\begin{proof}[Proof of Theorem \ref{gg theorem Xr}]
The case $r=0$ is contained in Corollary \ref{gg corollary}. We will assume $r\ge 1$.
We assume first that the  inequalities \eqref{gg theorem Xr} are satisfied and we claim $h^1(D_{(r)}\otimes{\mathcal I}_q)=0$, for all points $q$.  Hence $D_{(r)}$ is globally generated.

In order to show  this, we distinguish the following two cases:
\begin{enumerate}
\item $q$ is on some exceptional divisor $E_I$,
\item $q$ is the pull-back of a point outside of the union $\sum_{I\subseteq\{1,\dots,s\}, |I|\le r+1} L_I$.
\end{enumerate}

Case (1). Assume $q\in E_I$ and  write $\rho+1:=|I|$, $\rho\le r$.
We want to prove that $h^1(D_{(r)}-E_I)=0$.  Then the conclusion will follow from  Theorem \ref{vanishing prelim}.

Reordering the points if necessary, we may assume that $1\in I$.
Define the divisor $D'=D-E_1$ in $X_{s,(0)}$.
 One can easily check that $D'$ satisfies the hypotheses of  Theorem
 \ref{extended vanishing}, therefore $h^1(D'_{(r)})=0$.

Consider the set $\mathcal{J}$ of all indices $J$ of cardinality $1\le |J|\le r+1$ such that $1\in J$ and such that $\sum_{i\in J}m_i-|J|d\ge0$.
Let us endow $\mathcal{J}$ with the \emph{graded lexicographical order} on the index sets, namely   if $|J_1|<|J_2|$  then $J_1\prec J_2$, while if  $|J_1|=|J_2|$ we use the lexicographical order.

Notice that $I\in \mathcal{J}$ and that $D'_{(r)}=D_{(r)}-\sum_{J\in\mathcal{J}}E_J$.
In other words, $$D_{(r)}-E_I=D'_{(r)}+\sum_{\substack{J\in\mathcal{J}:\\ J\ne I}}E_J.$$

We can obtain $D'_{(r)}$ as the residual of iterative applications of exact sequences starting from $D'_{(r)}+\sum_{J\in\mathcal{J}, J\ne I}E_J$ by restrictions to the exceptional divisors $E_J$, with $J\in\mathcal{J}$, $J\ne I$, following the order on $\mathcal{J}$.
 More precisely, one starts from $D'_{(r)}+\sum_{J\in\mathcal{J}, J\ne I}E_J$ and restricts to $E_1$,  then to all $E_J$'s, with $J\in \mathcal{J}\setminus\{I\}$, $|J|=2$, then to all $E_J$'s, with $J\in \mathcal{J}\setminus\{I\}$, $|J|=3$, etc.
We claim that each restriction has vanishing first cohomology group. This gives a proof of the statement, since the last kernel, that is $D'_{(r)}$, has vanishing first cohomology too, by Theorem \ref{extended vanishing}.

To prove the claim, recall that each exceptional divisor $E_J$ is the product of two blown-up projective spaces of dimension $|J|-1$ and $n-|J|$ respectively, see \cite[Lemma 2.5]{dp}. In particular the first component is isomorphic to ${\mathbb P}^{|J|-1}$ blown-up along linear cycles (see Remark \ref{linear cycles}). Let us denote by $h$ the class of a general hyperplane on the first component and by $e_{J'}$ the class of the restriction of the exceptional divisors $E_{J'}$, namely $E_{J'}|_{E_J}=(e_{J'},0)$, for all $J'\subset J$. Let
$\Cr(h)$ be the proper transform of the \emph{standard Cremona transformation}
 of the hyperplane class $h$ on ${\mathbb P}^{|J|-1}$, i.e.
\begin{equation}\label{cremona}
\Cr(h)=(|J|-1)h-\sum_{\substack{J'\subset J:\\ |J'|<|J|-1}}(|J|-|J'|-1)e_{J'}.
\end{equation}
We have
\begin{equation*}
\begin{split}
D'_{(r)}|_{E_{J}}&=(0,*)  \textrm{ for every } E_{J},\\
{E_{J}}|_{E_{J}}&=(- \Cr(h),*), \\
{E_{J'}}|_{E_{J}}&=(0,*)  \textrm{ for all } J'\supset J,
\end{split}
\end{equation*}
where we use $*$ to denote the appropriate
divisor on the second factor. See \cite[Sect. 2-3]{dp} for details.

Therefore each of the above restrictions is
$$\left(D'_{(r)}+\sum_{\substack{J'\in\mathcal{J}:\\ J'\ne I, J\prec J'}}E_{J'}\right)|_{E_{J}}=(-\Cr(h), *).$$ It  has vanishing first cohomology group by \cite[Theorem 3.1]{dp}.
This concludes the proof of Case (1).

\vskip.3cm

Case (2). In this case $q$ is the pull-back of a point $q'\in{\mathbb P}^n\setminus\bigcup_{I\subset\{1,\dots,s\}, |I|\le r}L_I$.
As in the proof of Theorem \ref{toric tensored} we distinguish two subcases and we prove the claim by induction on $n$. The case $n=1$ is obvious. Assume $n\ge2$.

\vskip.3cm

Case (2.a).
Let us assume first that the points $p_1, \ldots, p_s, q'$ are not in linearly general position. If $s\ge n$, $q'$ lies on a hyperplane $H$ of ${\mathbb P}^n$ spanned by $n$ points of
$\mathcal{S}$. Reordering the points if necessary, assume that $q'\in H:=\langle p_1,\dots,p_n\rangle$.
If $s< n$, let $H$ be any hyperplane containing $\mathcal{S}\cup\{q'\}$.
Let $\bar{H}$ denote the pull-back of $H$ on $X_{s,(r)}$. It is isomorphic to the space ${\mathbb P}^{n-1}$ blow-up along linear cycles of dimension up to $\min\{r,n-2\}$,
spanned by $\bar{s}:=\min\{s,n\}$ points in general position, that we may denote by $\bar{H}\cong X^{n-1}_{\bar{s},(r)}$ as in Remark \ref{linear cycles}, part (1). As a divisor, we have
\begin{equation}\label{bar H r}
\bar{H}=H-\sum_{i=1}^{\bar{s}-1} E_i-\sum_{\substack {I\subset \{1,\ldots, \bar{s}\}:, \\ 1\leq |I|\leq \min\{r,n-2\}}}E_{I}.
\end{equation}

Consider the restriction exact sequence of line bundles
\begin{equation}\label{restriction hyperplane r}
0\to D_{(r)}-\bar{H}\to D_{(r)}\otimes {\mathcal I}_{q} \to (D_{(r)}\otimes {\mathcal I}_{q})|_{\bar{H}}\to0.
\end{equation}
The restriction, $D_{(r)}|_{\bar{H}}\otimes {\mathcal I}_{q}$, is a \textit{toric divisor} on the blown-up space $\bar{H}\cong X^{n-1}_{\bar{s},(r)}$, with a point, $q$, in possible special linear configuration with the other points.
As in the proof of Theorem \ref{toric tensored}, Case (2.a), we conclude that it has vanishing first cohomology group by induction on $n$.
The kernel also has vanishing first cohomology because it has possibly only simple linear obstructions, see \cite[Theorem 1.5]{dp}.
We  conclude that $h^1(D_{(r)}\otimes {\mathcal I}_{q})=0.$

\vskip.3cm

Case (2.b).
Let us assume now that the points $p_1, \ldots, p_s, q'$ are in linearly general position.
If $s\ge n-1$, let $H$ denote the hyperplane $\langle p_1,\dots,p_{n-1},q'\rangle$.  If $s< n-1$, let $H$ be any hyperplane containing
 $\mathcal{S}\cup\{q'\}$. In both cases such an $H$ exists by the assumption that
 $\mathcal{S}$ is a set of points in  general position. Let $\bar{H}$ denote the pull-back of $H$ on $X_{s,(r)}$, as in \eqref{bar H r} and consider the corresponding restriction sequence as in \eqref{restriction hyperplane r}.

As in Case (2.a), we conclude by induction on $n$ and by noticing that the kernel has only possibly simple linear obstructions.

\vskip.3cm

Assume now that one of the inequalities in \eqref{gg equations Xr} does not hold.  We claim that $D_{(r)}$ is not globally generated. Indeed, if $m_i\geq d+1$ then the divisor $D$ is not effective therefore $D_{(r)}$ is not globally generated.
If $m_i\leq -1$ then the divisor $E_i$ is in the base locus of $D_{(r)}$. If $k_I\geq1$
for some $I$ such that $|I|=r+2$ and $r\leq n-1$, then the divisor $D_{(r)}$ contains in its base locus the
strict transform of the linear cycle $L_{I}$ by Lemma \ref{base locus lemma}, therefore is not globally generated.

\end{proof}

\begin{remark}
The strict transform $\tilde{D}=D_{(\bar{r})}$ of ${\mathcal{L}}$ is base point free if \eqref{gg equations Xr} is
 satisfied with $r=\bar{r}$.
\end{remark}

\subsection{Vanishing Cohomology of strict transforms}\label{vanishing question}
In this section we will determine the base locus and their intersection multiplicity for divisors satisfying condition \eqref{gg equations Xr large}. Furthermore, Theorem \ref{question answer} answers Question 1.1 posed in \cite{bdp1} in this range.

Recall that a line bundle is globally generated if and only if the associated linear system is base point free.
We are now ready to prove that Theorem \ref{gg theorem Xr} implies a complete description of the
base locus of all non-empty linear systems in
${\mathbb P}^n$ of the form ${\mathcal{L}}={\mathcal{L}}_{n,d}(m_1,\dots, m_s)$ \eqref{linear system}
that are only linearly obstructed.

\begin{theorem}[Base locus of linear systems]\label{base locus theorem}
The base locus of the divisor $D_{(r)}$ on $X_{s,(r)}$, strict transform of
$D=dH-\sum_{i=1}^{s}m_iE_i$ with \eqref{gg equations Xr large s}, namely
$$
\sum_{i=1}^s m_i\le nd+\max\{0,s-n-3\},
$$
 is the formal sum \begin{equation}\label{formal sum linear base locus}
\sum_{\substack{I\subseteq\{1,\dots,s\}:\\ |I|\ge r+1}}k_I L_I\in A^\ast(X_{s,(r)}).
\end{equation}
In particular, if $s\le n+2$, the sum \eqref{formal sum linear base locus} with $r=-1$, describes the base locus of all non-empty linear systems ${\mathcal{L}}_{n,d}(m_1,\dots,m_s)$, while if $s\geq n+3$ the sum \eqref{formal sum linear base locus} with $r=-1$, describes the base locus of non-empty linear systems satisfying the bound
\eqref{gg equations Xr large s}.
\end{theorem}
\begin{proof}
By Lemma \ref{base locus lemma},  each linear subspace $L_{I}$ with $k_{I}>0$ is a base locus cycle for ${\mathcal{L}}$ and  $k_{I}$ is its exact multiplicity of containment. Therefore
$$ \bigcup_{\substack{I\subseteq\{1,\dots,s\}: \\|I|\ge r+1}}k_I L_I\subset {\rm{Bs}}(|D_{(r)}|).$$

By
Theorem \ref{gg theorem Xr}, the strict transform $D_{(r)}$ of an element of ${\mathcal{L}}$ is base point free as soon as no higher dimensional cycle, i.e. no $(r+1)$-plane, is contained in the base locus, namely when $k_{I}\le0$ for all $I\subseteq\{1,\dots,s\}$ of cardinality $r+1$. In particular, if $\bar{r}$ is the dimension of the linear base locus of $D$, then  $\tilde{D}=D_{(\bar{r})}$ is base point free.

Since the total transform of $D_{(r)}$ in $X_{s,(\bar{r})}$ equals
$$\tilde{D}+ \sum_{\substack{I\subseteq\{1,\dots,s\}: \\|I|\ge r+1}} k_ I E_I,$$
one concludes that the base locus of $D$ is supported only along linear cycles
$${\rm{Bs}}(|D_{(r)}|)\subset \bigcup_{\substack{I\subseteq\{1,\dots,s\}: \\|I|\ge r+1}}k_I L_I.$$
\end{proof}

\begin{remark}
The unique rational normal curve $C$ of degree $n$  through $n+3$ points of ${\mathbb P}^n$ and its secant varieties  were studied in \cite{bdp3} as cycles of the base locus of non-empty linear systems ${\mathcal{L}}$, see also Section \ref{bigness section}.
Condition \eqref{gg equations Xr large s} for $s=n+3$ says that neither $C$ nor $\sigma_t(C)$ are contained in the base locus of ${\mathcal{L}}$. Hence Theorem \ref{base locus theorem} states that  if $C$ is not in the base locus of ${\mathcal{L}}$, then nothing else is, besides the linear cycles.
\end{remark}

\begin{remark}\label{stable base locus}
Recall that for a line bundle $D_{(r)}$ as above, the \emph{stable base locus} is defined as $$\mathbb{B}(D_{(r)})=\cap_{m\in\mathbb{N}}{\rm{Bs}}(|mD_{(r)}|).$$
The obvious equality $k_I(mD)=mk_{I}(D)$, for any integer $m\ge1$, and Theorem \ref{base locus theorem} show that ${\rm{Bs}}(|mD_{(r)}|)=m\cdot{\rm{Bs}}(|D_{(r)}|)$. Therefore the base locus of $D_{(r)}$ is stable, namely $$\mathbb{B}(D_{(r)})={\rm{Bs}}(|D_{(r)}|).$$

Because the stable base locus is invariant under taking multiples, we can extend this definition to the case of  $\mathbb{R}$-divisors, see \cite[Lemma 3.5.3]{BCHM}. In particular we can consider the stable base locus of $\epsilon D_{(r)}$, for any $\epsilon \in\mathbb{R},$ and obtain
$$\mathbb{B}(\epsilon D_{(r)})= \mathbb{B}(  D_{(r)})= {\rm{Bs}}(|D_{(r)}|).$$

\end{remark}

We now can prove that the strict transform of $D$ in the iterated blow-up along
its base locus has vanishing cohomology groups. This answers affirmatively Question \ref{question} .
\begin{theorem}\label{question answer}
Let $\mathcal{S}$ be a collection of points in  general position and $D$
 be any effective divisor on $X_{s, {(0)}}$. Assume that one of the following holds:
\begin{enumerate}
\item  $s\leq n+2$,
\item  $s\geq n+3$ and $\sum_{i=1}^n m_i-nd\le s-n-3$.
\end{enumerate}
 Then Question \ref{question} has affirmative answer.
\end{theorem}

\begin{proof}
Since all cycles of the base locus of the divisors $D$ are linear by Theorem \ref{base locus theorem}, we conclude that $\widetilde{\mathcal{D}}$ equals $\tilde{D}$. The claims follow from Theorem \ref{extended vanishing}.
\end{proof}

\subsection{Log resolutions for divisors on blown-up projective spaces in points.}

Let $D$ be any divisor on $X_{s, {(0)}}$ in the hypothesis of Theorem \ref{question answer} and let $\bar{r}$ be the maximum dimension of its linear base locus.
 If the divisor $D$ has $\bar{r}<n-1$, Theorem \ref{question answer} implies that the map $X_{s, (\bar{r})}\stackrel{}{\rightarrow} X_{s, (0)}$ is a resolution of singularities of $D$.

Moreover, we recall that a \emph{log resolution} of the pair $(X_{s,(0)},D)$ is a birational morphism
$\pi:Y\to X_{s,(0)}$ such that the pair $(Y, \tilde{D})$ is \emph{log smooth}, where
$\tilde{D}=\pi_\ast^{-1}D$ is the strict transform of $D$, namely such that
$Y$ is smooth and the sum $\tilde{D}+\textrm{Exc}(\pi)$, where
$\textrm{Exc}(\pi)$ is the sum of exceptional divisors of $\pi$, is simple normal crossing.

For $D$ a general divisor of the linear system $|dH-\sum_{i=1}^sm_i E_i|$ as in
 Notation \ref{generality definition}, we obtain the following result.

\begin{corollary}\label{log resolutions}

If $D$ satisfies relation \eqref{gg equations Xr large s}, then
the pair $(X_{s,(\bar{r})},D_{(\bar{r})})$ is log smooth and the birational morphism
$X_{s, (\bar{r})}\stackrel{}{\rightarrow} X_{s, (0)}$
is a \emph{log resolution} of the singularities of the pair $(X_{s,(0)},D)$.
\end{corollary}
\begin{proof}
The variety  $X_{s,(\bar{r})}$ is smooth.
Indeed at each step of the blow-up of $X$,
$X_{s,(r)}\to X_{s,(r-1)}$, $r\le\bar{r}$, the center of the blow-up is a disjoint union of smooth
subvarieties, namely the strict transforms of the liner cycles $L_{I(r)}$ with $k_{I(r)}>0$
 (see \cite[Subsection 4.2]{dp} for details). This, together with the fact that each exceptional divisor is smooth, also proves that the sum $\Exc(\pi)=\sum_{r=1}^{n-2}\sum_{I(r)}E_{I(r)}$ is simple normal crossing.
 Moreover $D_{(\bar{r})}$ is base locus free and its  support
intersects transversally all exceptional divisors $E_{I(r)}$, by  Theorem \ref{base locus theorem}. Finally since $D$ is general, then $D_{(\bar{r})}$ is general too hence it is smooth by Bertini's theorem.

\end{proof}

\section{Log abundance for pairs on blow-ups of ${\mathbb P}^n$}\label{abundance section}

In this section we  construct an infinte family of log canonical pairs  given
by effective divisors on the blow-up of ${\mathbb P}^n$ at $s$ points in general position,
$X_{s,(0)}$, with arbitrary $s$. Moreover
we prove that the (log) abundance conjecture holds for these pairs.

In the case $s\le n+3$, the blown-up space $X_{s,(0)}$ is log Fano (see \cite{CT, Mukai}
and the recent
preprint \cite{am} for an explicit proof), hence a Mori Dream Space.
Therefore it is a well-known fact that all nef divisors are semi-ample.
Moreover this statement hold for $s\le 2n$, see Theorem \ref{nef cone} and \cite[Proposition 1.4]{am}.
Our contribution in this section is to provide an explicit constructive proof
of the abundance conjecture for all effective divisors (Section \ref{section abundance n+3}).

For $s\ge n+4$, the result is new (Section \ref{section abundance n+2}). This implies that
it is possible to run the Minimal Model Program and find good minimal models.

\subsection{Preliminary definitions}
For an introduction to singularities in the Minimal Model Program
and in particular to the abundance conjecture, we refer for instance to \cite{KM}.
Let $(X,\Delta)$ be a log pair, with $X$ a normal variety and
$\Delta=\sum_j a_j\Delta_j$ a formal
$\mathbb{Q}$-linear combination of prime divisors.
Let $\pi:Y\to X$ be a log resolution of $(X,\Delta)$, denote by
 $\tilde{\Delta}:=\pi_*^{-1}\Delta$ the strict transform of $\Delta$ and by

$E_i$ the exceptional divisors.
Write
$$K_Y+\tilde{\Delta}=\pi^*(K_X+\Delta)+\sum_i a(E_{i},X,\Delta) E_i,$$
 with $a(E_{i},X,\Delta)\in\mathbb{Q}.$
By \cite[Corollary 2.32]{KM}, if $a_j\le 1$, then the \emph{discrepancy} of the pair
 $(X,\Delta)$ can be computed as
\begin{equation}\label{discrepancy}
\textrm{discrep}(X,\Delta)=\min_i\left\{a(E_{i},X,\Delta), \min_j\{1-a_j\},1\right\}.
\end{equation}
The pair $(X,\Delta)$ is  said to be \emph{log canonical} (\emph{lc})
if  $\textrm{discrep}(X,\Delta)\ge-1$.

\begin{conjecture}[{\cite[Conjecture 3.12]{KM}}]\label{abundance}
Let $(X,\Delta)$ be lc, $\Delta$ effective. Then $K_X+\Delta$
is nef if and only if it is semi-ample.
\end{conjecture}

The abundance conjecture is known to hold for surfaces and threefolds (see \cite{KMM}) and for fourfolds with
positive Kodaira dimension (see \cite{Fk}).  Very little is known in higher dimension.

\subsection{Abundance theorem for blown-up projective spaces}

As an application of the results of this paper, we prove that the log abundance conjecture
holds for $(X_{s,(0)},\Delta)$, where $\Delta:=\epsilon D$ with
$0\le \epsilon\le1$ and $D\ge 0$ effective divisor on $X_{s,(0)}$,
by explicitly constructing  a log resolution of the pair.

We recall here that effective divisors on $X_{s,(0)}$,
 with $s\le n+3$ were classified in \cite{bdp3}, see also Theorem
\ref{effectivity theorem}.
Moreover for arbitrary $s$, if \eqref{gg equations Xr large s} holds, the divisor $D$
is effective, by \cite[Theorem 5.3]{bdp1}.

For a small number of points, $s\leq n+3$, or an arbitrary number
 of points with a bound of the coefficients of the divisors, we construct an infinite family of log canonical pairs  given by  effective divisors $D$
on $X_{s,(0)}$.

\begin{theorem}\label{abundance Q}
Fix integers $n> 3$, arbitrary $s$, and $\epsilon\in\mathbb{Q}$,
with $0\le \epsilon\ll1$. Let
$D=dH-\sum_{i=1}^sm_iE_i$ be a general effective
divisor on $X=X_{s,(0)}$ with $s\le n+3$ or with $s\ge n+4$ and  satisfying
\eqref{gg equations Xr large s}, i.e.
$$\sum_{i=1}^{s}m_i-nd\le \max\{0,s-n-3\}.$$
Assume that
\begin{equation}\label{abundance condition}
\begin{split}
\ & \epsilon m_i \ge n-1,\ \forall i\in\{1,\dots,s\}\\
\ & \epsilon (m_i+m_j-d)\le n-3, \ \forall i,j\in\{1,\dots,s\}, \ i\neq j.
\end{split}
\end{equation}
Then the  pair $(X,\Delta)$ is lc.
\end{theorem}

Recall the classes of the canonical divisors on $X$,
$$
K_X=-(n+1)H+(n-1)\sum E_i,
$$ and consider the $\mathbb{Q}$-divisor
$$K_X+\Delta=(\epsilon d-n-1)H-\sum_i (\epsilon m_i-n+1) E_i.$$

\begin{proposition}\label{abundance A}
In the same hypothesis of Theorem \ref{abundance Q},
Conjecture \ref{abundance} holds for the pair $(X,\Delta)$, namely
 if $K_X+\Delta$ is nef then it is semi-ample.
\end{proposition}
\begin{proof}
By Theorem \ref{abundance Q}, the pair $(X,\Delta)$ is lc. To conclude,
it is easy to see that under the condition \eqref{gg equations Xr large s}, then the divisor $K_X+\Delta$ is nef (equiv. semi-ample)
if and only if the
conditions \eqref{abundance condition} of Theorem \ref{abundance Q} are verified.
This follows from Theorem  \ref{ample cone}.
In fact, if $s\le 2n$, the nef and the semi-ample cone coincide,
and the thesis follows trivially.
 Otherwise, if $s\ge 2n+1$,
we just have to check that \eqref{gg equations Xr large s} implies condition
 \eqref{tva equations X0 large s} of
of Theorem \ref{ample cone} (and of Theorem \ref{tva theorem X0}) with $l=0$. We leave the details of this to the reader.
\end{proof}

\begin{corollary}
In the same notation as Theorem \ref{abundance Q}, the canonical ring
$$\bigoplus_{l\ge0}H^0(X,\mathcal{O}_X(l K_X+\lfloor l\Delta \rfloor))$$
is finitely generated.
\end{corollary}
\begin{proof}
See \cite[Section 3.13]{KM}.
\end{proof}

Recall that in the space ${\rm{N}}^1(X_{s,(0)})_\mathbb{R}$, the N\'eron-Severi group tensored with the real numbers,
the semi-ample divisors on $X_{s,(0)}$ are described in Theorem \ref{ample cone} respectively.
Also the notion of stable base locus of line bundles, that in our case corresponds to that of base locus,  can be extended to the case of $\mathbb{R}$-divisors by taking real multiples of line bundles, see Remark \ref{stable base locus}.
 Hence we can extend  Proposition \ref{abundance A} to $\mathbb{R}$-divisors.

\begin{proposition}\label{abundance R}
Let $0\le \epsilon\ll1$, $\epsilon\in\mathbb{R}$. Let $\Delta=\epsilon D\in N^1(X_{s,(0)})_\mathbb{R}$  be an $\mathbb{R}$-divisor on $X_{s,(0)}$ satisfying the assumption of Theorem \ref{abundance Q}.
Then Conjecture \ref{abundance} holds for the pair $(X,\Delta)$.
\end{proposition}

\begin{remark}
In the notation of Theorem \ref{abundance Q}, if $s\le n+2$  or $s\ge n+3$ and  condition \eqref{gg equations Xr large s} is satisfied, then $D$ is effective and \emph{only linearly obstructed}, see \cite[Theorem 5.3]{bdp1}. In this case a log resolution of the corresponding pair is given by the iterated blow-up of the linear base locus, see Corollary \ref{log resolutions}.

The case $s=n+3$ is the first case where non-linear obstructions appear for divisors violating \eqref{gg equations Xr large s}. The base locus of effective divisors $D$ was studied in \cite{bdp3} (cfr. Lemma \ref{base locus lemma II}). In this case a log resolution of any pair given satisfying \eqref{abundance condition},
 will be constructed in Section \ref{construction log resolution} by means of the iterated blow-up along the subvarieties, linear and non-linear, contained in the base locus.

\end{remark}

\subsection{Proof of Theorem \ref{abundance Q}, only linearly obstructed case}
\label{section abundance n+2}

Let $s$ be an arbitrary integer.
Set $X:=X_{s,(0)}$ and $Y:=X_{s,(n-2)}$. Let
$$D=dH-\sum_{i=1}^{s}m_i E_i\ge0$$
be a divisor as in Notation \ref{generality definition}.

Assume moreover that $D$ is only linearly obstructed,
namely that condition \eqref{gg equations Xr large s} is verified.

\begin{remark}\label{movability}
Notice that under the assumption \eqref{abundance condition} the divisor $D$ is irreducible. We recall that $D$ represents a general member of the linear system $|D|$ and the assumptions above force $D$ to have no divisorial components.
Indeed, observe that no (strict transform of) hyperplane spanned by $n$ points is contained in the base locus of $D$. In fact, we can see first of all that \eqref{abundance condition}
implies that for every $i=1 \dots s$,
\begin{equation}\label{extra nef condition}
\epsilon (m_i-d) \le -2.
\end{equation}
Now,  if $I=I(n-1)$ is an index set of cardinality $n$, we can compute that the multiplicity of containment of the corresponding hyperplane is zero as follows:
\begin{align*}
\sum_{i\in I}m_i-(n-1)d&=\left(\sum_{i\in I\setminus\{i_1,i_2\}}m_i-(n-2)d\right)
+\left(m_{i_1}+m_{i_2}-d\right)\\
& \le -2(n-2)+(n-3)\\ & \le0.
\end{align*}
\end{remark}

Using the notation introduced in Subsection \ref{higher blow ups},
let $\pi:Y\to X$ be the composition of blow-ups  of $X$ along lines,
then planes etc., up to codimension-$2$ linear cycles $L_I$, $I\subset\{1,\dots,s\}$
 for which $k_I>0$, see \eqref{mult k} for the definition of $k_I$.
As in \eqref{strict transform}, the strict transform of $D$, is given by
$$
\tilde{D}=dH-\sum_{i}m_i E_i-\sum_{r=1}^{n-2}\sum_{I(r)}k_{I(r)}E_{I(r)}.
$$

\begin{proposition}\label{log resolution n+2}
In the above notation, the map $\pi:Y\to X$ is a log resolution of $(X, \Delta)$, for every
$\epsilon\ge0$.
\end{proposition}
\begin{proof} It follows from Corollary \ref{log resolutions}.

\end{proof}

Recall the classes of the canonical divisors on $Y$:
$$
K_Y=-(n+1)H+(n-1)\sum E_i+\sum_{r=1}^{n-2}(n-r-1)\sum_{I(r)}E_{I(r)}.
$$
Here we abuse notation by denoting by $H$ the hyperplane class in both $X$ and $Y$;
similarly by abuse of notation we denote by $E_i$ an exceptional divisor in $X$
and its strict transform in $Y$.

\vskip.3cm

For $0\le \epsilon<1$, $\epsilon\in\mathbb{Q}$,
consider the pairs $(X,\Delta)=(X_{s,(0)},\epsilon D)$ and  $(Y,\tilde{\Delta})=(X_{s,(n-2)},\epsilon\tilde{D})$ and write
$$
K_Y+\tilde{\Delta}=\pi^*(K_X+\Delta)+
\sum_{1\le r\le n-2}(n-r-1-\epsilon k_{I(r)})E_{I(r)}.
$$

 We have $$a_i(E_{I(r)},X,\Delta)=n-r-1-\epsilon k_{I(r)},$$
for any $I(r)$ such that $1\le r\le n-2$ (cfr. \cite[Lemma 2.29]{KM}).
Therefore $\textrm{discrep}(X,\Delta)\ge-1$ if
\begin{equation}\label{lc condition}
\epsilon k_{I(r)} \le n-r, \  \forall I(r), \ 1\le r\le n-2.
\end{equation}

\medskip
We are now ready to prove the first main result of this section.

\begin{proof}[Proof of Theorem \ref{abundance Q}, only linearly obstructed case]

We are now ready to prove Theorem \ref{abundance Q} for divisors with only linear obstructions.
By Proposition \ref{log resolution n+2}, $(Y,\tilde{\Delta})$
is log smooth and $\pi:Y\to X$ is a log resolution of $(X,\Delta)$.
We are going to prove that the pair is $(X,\Delta)$ is lc.

We  prove that \eqref{lc condition}
is implied by  \eqref{abundance condition}, second line,
and by \eqref{extra nef condition}.
Indeed, take for instance $r=2$. If $k_{I(2)}=0$ the statement
is obvious. Assume that $k_{I(2)}>0$.
Write $I(2)=\{i_1,i_2,i_3\}$.
We have $\epsilon k_{I(2)}=\epsilon((m_{i_1}-d)+(m_{i_2}+m_{i_3}-d))\le
-2+(n-3)\le n-2$. The same holds for $r\ge 3$.
\end{proof}

\subsection{Proof of Theorem \ref{abundance Q}, case $s=n+3$}
\label{section abundance n+3}

In this section we conclude the proof of Theorem \ref{abundance Q}.
In particular we consider all effective divisors with $s=n+3$ with
no restriction on the obstructions contained in the base locus
and we construct a log resolution of the corresponding pairs.

Take $X:=X_{n+3,(0)}$ and a divisor $D$ on $X$ of the form
$$D=dH-\sum_{i=1}^{n+3}m_iE_i\ge0.$$
 Write $\Delta=\epsilon D$, for $0\le\epsilon\ll1$ and assume it satisfies condition \eqref{abundance condition} of Theorem \ref{abundance Q}.

\begin{remark}\label{movable'}
Note that,  \eqref{abundance condition}  implies that $D$ is big (cfr. Theorem \ref{big n+3 X0}.).
In fact we can use \eqref{low mult of joins} that gives
$$
\epsilon k_{I(n-2t),\sigma_t}\le -n-1<0,
$$
for all $I(n-2t),\ t\ge 1$.
Moreover one can similarly check that
$$\epsilon k_{I(n-2t-1),\sigma_t} \le0,$$
for all $I(n-2t-1),\ t\ge 1$, hence $D$ is movable, see Theorem \ref{effectivity theorem}.
In particular the general element of $|D|$ is irreducible.
\end{remark}

\begin{remark}
The argument developed in Section \ref{section abundance n+2}
applies also to the case $s=n+3$ and $k_C:=k_C(D)=0$ (see \eqref{expanded}
for the definition of this number), namely to divisors $D$ on $X_{n+3,(0)}$
 that have only linear base locus and for which \eqref{gg equations Xr large s} is satisfied. Indeed under this condition and \eqref{abundance condition}, $D$ is movable (cfr. Theorem \ref{effectivity condition n+3}) and its strict transform on $X_{n+3,(n-2)}$,  the iterated blow-up along the linear cycles, is globally generated by Theorem \ref{gg theorem Xr}.
Moreover  the lc condition on the pair $(X,\epsilon\Delta)$
 can be verified in the same manner as in the case with \eqref{gg equations Xr large s},
as in  Section
\ref{section abundance n+2}.
\end{remark}

From now on we will assume that $k_C(D)\ge1$.

\subsubsection{Constructing a log resolution of $(X,\epsilon D)$}\label{construction log resolution}

The following result computes the  pairwise intersections of the joins $\J(L_I, {\sigma}_{t})$.

\begin{proposition}\label{center of blow-up is smooth}
\label{snc}
Let $\J(L_{I_1},\sigma_{t_1})$ and $\J(L_{I_2},\sigma_{t_2})$ be any join varieties of the same dimension $r_{I_1,\sigma_{t_1}}=r_{I_2,\sigma_{t_2}}$.
 Then,
 $$\J(L_{I_1},\sigma_{t_1})\cap\J(L_{I_2},\sigma_{t_2})=\bigcup_{J}\J(L_J,\sigma_{t_J}),$$
for some subsets $J\subseteq I_1\cup I_2.$
\end{proposition}

\begin{proof}
Assume first of all that $I_1\cap I_2=\emptyset$. We have
$$
0\le 2(n-1)-2\left(r_{I_1,\sigma_{t_1}}+r_{I_2,\sigma_{t_2}}\right),
$$
which gives

$$2r_{I_i,\sigma_{t_i}}\le 2n-(r_{I_1,\sigma_{t_1}}+1)-(r_{I_2,\sigma_{t_2}}+1),\ \forall i \in\{1,2\}.$$
Therefore, since $|I_i|\le   r_{I_i,\sigma_{t_i}}+1$, we
have $2r_{I_i,\sigma_{t_i}}\le 2n-(|I_1|+|I_2|)$, $i=1,2$, hence we are in the hypotheses of Proposition \ref{intersection of joins}. This concludes the proof of the statement in the case when $I_1,I_2$ are disjoint. Indeed at each step of blow-up, the intersection of any two subvarieties that are being blown-up is a union of smaller subvarieties that have been previously blown-up.

\vskip.3cm

Assume now that $I_1\cap I_2=:I_{12}\ne \emptyset$. Set $I'_i:=I_i\setminus I_{12}$, for $i=1,2$. Notice that
$$r_{I'_i,\sigma_{t_i}}=r_{I_i,\sigma_{t_i}}-|I_{12}|, \ \forall i\in\{1,2\}.$$
Moreover,
\begin{equation}\label{join of join}
\J(\J(L_{I'_i},\sigma_{t_i}),L_{I_{12}})=\J(L_{I_i},\sigma_{t_i}), \ \forall i\in\{1,2\}.
\end{equation}
Also,
$$\J(\J(L_{I'_1},\sigma_{t_1}),L_{I_{12}})\cap\J(\J(L_{I'_2},\sigma_{t_2}),L_{I_{12}})=
\J(\J(L_{I'_1},\sigma_{t_1})\cap\J(L_{I'_2},\sigma_{t_2}),L_{I_{12}}).$$
One proves the two last equalities of subvarieties using induction on $|I_{12}|$
 based on the case $|I_{12}|=0$ for which the statement is obvious.
With a similar argument as the one employed in the first case, we obtain that
\begin{align*}
2r_{I'_i,\sigma_{t_i}}&\le 2n-(r_{I'_1,\sigma_{t_1}}+1)-(r_{I'_2,\sigma_{t_2}}+1)-4|I_{12}|,\\
& \le 2n-(|I'_1|+|I'_2|)),
\end{align*}
for $i=1,2$.
Therefore,
\begin{equation}\label{intersection of joins extended}
\J(L_{I_1},\sigma_{t_1})\cap\J(L_{I_2},\sigma_{t_2})=
\J\left(\bigcup_J\J(L_J,\sigma_{t_J}),L_{I_{12}}\right)=\bigcup_J\J(L_{J\cup I_{12}},\sigma_{t_J}),
\end{equation}
where the union is taken over all subsets $J\subseteq I'_1\cup I'_2$ satisfying
$2|(I'_i\cup J)\setminus(I_i\cap J)|=|I'_1|+|I'_2|,\ \forall i\in\{1,2\},$
and for every such $J$, $t_J$ is the integer defined by the following equation
$2r_{J,\sigma_{t_J}}=2r_{I'_i, \sigma_{t_i}}-(|I'_1|+|I'_2|).$
The first equality of \eqref{intersection of joins extended} follows from Proposition
 \ref{intersection of joins}, the second inequality is an application of \eqref{join of join}.
\end{proof}

\begin{notation}\label{even odd notation}
For every $n$, let $\mathcal{T}:=\{(I,t):1\le r_{I,\sigma_t}\le n-2 \}$ be the set parametrizing all subvarieties  $\J(L_I, {\sigma}_{t})$ of ${\mathbb P}^n$ of dimension between $1$ and $n-2$.
\begin{enumerate}
\item
If $n$ is even, say $n=2\nu$, with $\nu\ge2$,
we consider the following set
$$\mathcal{C}^{\textrm{even}}:=\{(I,t)\in\mathcal{T}: 2r_{I,\sigma_t}\le 2\nu-1 \textrm{ if } I\neq\emptyset\}.$$
In other terms, $(I,t)\in\mathcal{C}^{\textrm{even}}$ if $I=\emptyset$ and $1\le t<\nu$ or if  $I\neq\emptyset$ and $|I|+2t\le\nu$.
\item
If $n$ is odd, say
$n=2\nu+1$, with $\nu\ge2$, we
consider the following set
$$\mathcal{C}^{\textrm{odd}}:=\{(I,t)\in\mathcal{T}: 2r_{I,\sigma_t}\le 2\nu \textrm{ if } I\neq\emptyset,\{1\}\}.$$
In other terms, $(I,t)\in\mathcal{C}^{\textrm{even}}$ if $I=\emptyset,\{1\}$ and $1\le t<\nu$ or if  $I\neq\emptyset,\{1\}$ and $|I|+2t\le\nu+1$.
\end{enumerate}

Using the same notation as in Subsection \ref{section vanishing cohomology n+3},
we consider $$\pi^\sigma:Y^\sigma\to X,$$   the iterated blow-up of $X=X_{2\nu+3,(0)}$
along all subvarieties $\J(L_I,\sigma_t)$ such that $(I,t)\in\mathcal{C}^{\textrm{even}}$
if $n$ is even, and such that $(I,t)\in\mathcal{C}^{\textrm{odd}}$ if $n$ is odd, in increasing dimension.
We denote by $E_{I,\sigma_t}$ the corresponding  exceptional divisors.

Let us denote  $\Exc(\pi^\sigma)=\sum_{(I,t)\in\mathcal{C}^{\textrm{even}}}E_{I,\sigma_t}$
if $n$ is even and $\Exc(\pi^\sigma)=\sum_{(I,t)\in\mathcal{C}^{\textrm{odd}}}E_{I,\sigma_t}$ if $n$ is odd.
\end{notation}

\begin{proposition}\label{Ysigma smooth even}
For any $n$, the blown-up space $Y^\sigma$ is smooth. Moreover $\Exc(\pi^\sigma)$
is simple normal crossing.
\end{proposition}
\begin{proof}
At each level of the blow-up, the center is a disjoint union of smooth subvarieties.
Indeed, if $\J(L_{I_1},\sigma_{t_1})$ and $\J(L_{I_2},\sigma_{t_2})$ are two
subvarieties such that  $2r_{I_1,\sigma_{t_1}}=2r_{I_2,\sigma_{t_2}}\le n-1$,
 then we conclude by Proposition \ref{snc}.
Otherwise, if $2r_{I,\sigma_{t}}\ge n$, the only subvarieties that we blow-up
 are the strict transforms of the secant varieties $\sigma_t$'s, with
$\frac{\nu+1}{2}\le t\le\nu$, in the case $n=2\nu$, and the strict transforms
of the $\sigma_t$'s and of the pointed cones $\J(L_{\{1\}},\sigma_t)$, with $\frac{\nu}{2}+1\le t\le\nu$, if $n=2\nu+1$. It is easy to see that such pointed cones intersect along smaller joins of the same form, that have been previously blown-up.
\end{proof}

\subsubsection{Properties of the strict transform of $|\epsilon D|$}

Denote by $\tilde{D}^\sigma$ the strict transform of $D$, as in \eqref{definition D sigma}:
\begin{equation}\label{D tilde sigma}
\tilde{D}^{\sigma}:=dH-\sum_{i}m_i E_i-\sum_{r=1}^{n-2}\sum_{\substack{I,t:\\ r_{I,\sigma_t}=r}}k_{I,\sigma_t}E_{I,\sigma_t}.
\end{equation}

\begin{remark}
Notice that, as in the only linearly obstructed case (handled in Section \ref{section abundance n+2}), the integer $r$ in the above summation of exceptional divisors ranges up to $n-2$, as $D$
is movable (cfr. Theorem \ref{effectivity theorem}).
\end{remark}

We use the sets $\mathcal{C}^{\textrm{even}}$ and $\mathcal{C}^{\textrm{odd}}$ introduced in Notation \ref{even odd notation} in the following lemma.

\begin{lemma}\label{tildeD intersect exc}
For any $n$, $s=n+3$ and $D$ as in Theorem \ref{abundance Q}, if
 $k_{I,\sigma_t}>0$, then $2r_{I,\sigma_t}< n-1$. In particular such a pair
$(I,t)\in\mathcal{T}$ belongs to $\mathcal{C}^{\textrm{even}}$ if $n$ is even, and to $\mathcal{C}^{\textrm{odd}}$, if $n$ is odd.
\end{lemma}
 \begin{proof}
Let us  compute
\begin{align*}
\epsilon k_C& =\epsilon\max\left\{0, \left(\sum_{i=1}^{n-3}m_i-(n-3)d\right)
+(m_{n-2}+m_{n-1}-d)\right.+\\
& \ \  \ + (m_{n}+m_{n+1}-d)+(m_{n+2}+m_{n+3}-d)\Bigg\}\\
& \le-2(n-3)+3(n-2)
\\ & =n-3,
\end{align*}
where $k_C:=k_{\sigma_1}$ is the multiplicity of containment of the rational normal curve
of degree $n$. If $k_C>0$, we also compute
\begin{align*}
\epsilon (k_C-d)&=\epsilon\left(\sum_{i=1}^{n-1}m_i-(n-1)d
+(m_{n}+m_{n+1}-d)+(m_{n+2}+m_{n+3}-d)\right)\\
& \le-2(n-1)+2(n-2)\\ & =-4.
\end{align*}
Both of the above inequalities follow from  \eqref{abundance condition}, second line, and the inequality \eqref{extra nef condition}, computed
in the proof of Theorem  \ref{abundance Q}.
Hence, for every $t\ge0$, we obtain
\begin{align*}
\epsilon k_{\sigma_t}& =\epsilon\max\left\{0, k_C+(t-1)(k_C-d)\right\}\\
& \le \max\{0, n-3-4(t-1)\}.
\end{align*}
Furthermore, for every $I$ with $|I|\ge0$ and $t\ge0$, we obtain
\begin{align*}
\epsilon k_{I,\sigma_t}&=\epsilon\max\left\{0,k_{\sigma_t}+\sum_{i\in I}m_i-(|I|)d\right\}\\
& \le\max\{0,
(n-3-4(t-1))-2|I|\}\\ & =\max\{0,(n-|I|-2t+1)-(|I|+2t)\}.
\end{align*}

Hence, for every variety $\J(L_I,\sigma_t)\subset{\mathbb P}^n$
with $r_{I,\sigma_t}=|I|+2t-1\le n$, we obtain
\begin{equation}\label{low mult of joins}
\epsilon k_{I,\sigma_t}\le \max\{0,n-1-2r_{I,\sigma_t}\}.
\end{equation}
\end{proof}

\begin{corollary}\label{tildesigma transversal intersection}
In the same notation as Lemma \ref{tildeD intersect exc},
the divisor $\tilde{D}^\sigma$ intersects transversally each exceptional divisors.
\end{corollary}

\begin{theorem}\label{base point free tilde-sigma-D}
For any $n$, $s=n+3$ and $D$ as in Theorem \ref{abundance Q}. If the divisor $D$
is general, then $\tilde{D}^\sigma$ is base point free.
\end{theorem}

We will show Theorem \ref{base point free tilde-sigma-D}  in Section
\ref{proof base point freeness D case even} (if $n$ is even) and in Section
\ref{proof base point freeness D case odd} (is $n$ is odd).
The next corollary follows from  Bertini's Theorem.

\begin{corollary}\label{general tildesigma smooth}
In the notation of Theorem \ref{base point free tilde-sigma-D}, the general element of $|\tilde{D}^\sigma|$ is smooth.
\end{corollary}

\subsubsection{Proof of Theorem \ref{base point free tilde-sigma-D}, case $n$ even}
\label{proof base point freeness D case even}

Write $n=2\nu$, $\nu\ge2$.
Recall that the strict transform on $X_{2\nu+3,(0)}$ of the secant variety
$\sigma_\nu\subset{\mathbb P}^{2\nu}$ is the fixed divisor
$$
\Sigma:=(\nu+1)H-\nu\sum_{i=1}^{2\nu+3}E_i.
$$

\begin{proposition}
In the above notation, the strict transform $\tilde{\Sigma}^\sigma$ of $\Sigma$ on $Y^\sigma$ is smooth.
\end{proposition}
\begin{proof}
Recall that the singular locus of  $\sigma_\nu\subset{\mathbb P}^{2\nu}$ is $\sigma_{\nu-1}\subset\sigma_\nu$ and, more precisely, the non-reduced union
of all joins $\J(L_I,\sigma_t)$ such that $k_{I,\sigma_t}(\Sigma)>1$.
In particular we compute the following multiplicities.
For every $I$ and $t$ such that $|I|,t\ge0$,  we have
\begin{equation}\label{mult Sigma}
k_{I,\sigma_t}(\Sigma)=\max\{0,\nu-|I|-t+1\}.
\end{equation}
All subvarieties $\J(L_I,\sigma_t)$ such that $|I|+t\le \nu$ have been blown-up,
hence we conclude that $\pi^\sigma$ is a resolution of the singularities of $\Sigma$.
\end{proof}

Choose $\alpha\in\mathbb{N}$ such that
\begin{equation}\label{definition alpha}
\frac{k_C(D)}{\nu}
\le \alpha\le \min_{1\le i\le 2\nu+3}\left\{\frac{m_i}{\nu},d-m_i\right\}.
\end{equation}
\begin{lemma}
Under the assumptions of Theorem \ref{abundance Q}, such an integer $\alpha$ exists.
\end{lemma}
\begin{proof}
It is enough to prove that $\frac{k_C(D)}{\nu}\le \frac{m_i}{\nu}-1$ and that
$\frac{k_C(D)}{\nu}\le d-m_i-1$ for all $i\in\{1\dots,2\nu+3\}$.
The first statement follows from the following computation:
$$
\epsilon k_C(D)\le 2\nu-3\le 2\nu-1-\epsilon\nu\le \epsilon (m_i-\nu).
$$
The first inequality follows from \eqref{low mult of joins}. The second inequality follows from
the assumption
 $\epsilon\ll1$; in fact it is enough to take $\epsilon\le\frac{4}{n}=\frac{2}{\nu}$. The last inequality follows from
\eqref{abundance condition}. Similarly, one proves the second statement by observing that
$$
\epsilon k_C(D)\le 2\nu-3\le 2\nu-\epsilon\nu\le \epsilon \nu(d-m_i-1).
$$
The last inequality follows from \eqref{extra nef condition}.
\end{proof}

Consider the linear system
$$
|D'|:=|D-\alpha \Sigma|=|d'H-\sum_{i=1}^{2\nu+3}m'_iE_i|,
$$
with $d':=d-\alpha(\nu+1)$ and $m'_i:=m_i-\alpha\nu$, for all $i\in\{1\dots,2\nu+3\}$.
We have
$$
|D'|+\alpha\Sigma\subseteq|D|.
$$

\begin{proposition}\label{Delta' smooth even}
In the notation of above, the  linear system $|D'|$ is non-empty.
Moreover $|D'|$ has only linear base locus and
 the strict transform in $Y^\sigma$ of $|D'|$  is base point free.
\end{proposition}
\begin{proof}
For every $i\in\{1\dots,2\nu+3\}$,
since by \eqref{definition alpha} we have
$\alpha\le \frac{m_i}{\nu}$ and $\alpha\le d-m_i$,
then $m'_i\ge0$ and $d\ge m'_i$ respectively.
Moreover, let us compute
\begin{align*}
k_C(D')&=\max\left\{0,\sum_{i=1}^{2\nu+3}(m_i-\alpha\nu)-2\nu(d-\alpha(\nu+1))\right\}\\
& =\max\left\{0,k_C(D)-\alpha\nu\right\}\\
&=0.
\end{align*}
The last equality follows from \eqref{definition alpha}: $\alpha\ge \frac{k_C(D)}{\nu}$.
This proves the first statement, namely that $\Delta'$ is effective, see Theorem \ref{effectivity theorem}.

To prove the second statement,
notice that if $|I|\le\nu$, then the pair $(I,0)\in\mathcal{C}^{\textrm{even}}$, hence
the corresponding linear subspace $L_I\subset{\mathbb P}^{2\nu}$ has been blown-up.
Otherwise, if $|I|\ge\nu+1$, we claim that $k_I(D')=0$.  Theorem \ref{gg theorem Xr}
imply the second statement, namely that $\tilde{D}^\sigma$ is a globally generated divisor.
To prove the claim,  for every $I$ we choose $J\subset I$ with $|J|=\nu$ and
 we compute
\begin{align*}
\epsilon k_{I}(D)&=\max\left\{0, \epsilon\left(\sum_{i\in I}m_i-(|I|-1)d\right)+
\epsilon\alpha(|I|-\nu-1)\right\}\\
&=\max\left\{0,\epsilon\left(\sum_{i\in J}m_i-(\nu-1)d\right)+
\epsilon\left(\sum_{i\in I\setminus J}(m_i-d)\right)+\epsilon \alpha(|I|-\nu-1)\right\}\\
&\le \max\{0,-\epsilon \alpha\}\\ &=0.
\end{align*}
The inequality holds because
$\epsilon \left(\sum_{i\in J}m_i-(\nu-1)d\right)\le0$ by \eqref{low mult of joins} and
$\alpha(|I|-\nu)\le \sum_{i\in I\setminus J}(d-m_i)$ by \eqref{definition alpha}.
\end{proof}

\begin{proposition}\label{linear series of strict transform even}
In the notation of above, there exist non-negative numbers
 $\alpha_{I,\sigma_t}\in\mathbb{Z}$ such that
$$
|\tilde{D'}^\sigma|+\alpha\tilde{\Sigma}^\sigma+\sum_{(I,t)\in\mathcal{C}^{\textrm{even}}}
\alpha_{I,\sigma_t} E_{I,\sigma_t}\subseteq
|\tilde{D}^\sigma|.
$$
\end{proposition}
\begin{proof}
In order to prove the statement, we compare  the strict transform,
$|\tilde{D'}^\sigma|+\alpha\tilde{\Sigma}^\sigma$  with that the strict transform of
$D$.

Recall that, for any $1\le t<\nu$ and $I$ such that $|I|\ge 0$, we have $k_{I,\sigma_t}(D')=0$
by Proposition \ref{Delta' smooth even}. Moreover for all $I$ and $t$,
  \eqref{low mult of joins} implies that
\begin{equation}\label{Sigma is more singular than Delta}
k_{I,\sigma_t}(\alpha \Sigma)=\alpha k_{I,\sigma_t}(\Sigma)\ge k_{I,\sigma_t}(D).
\end{equation}

This, together with \eqref{mult Sigma} and the computation of $k_I(D)$ made in the
proof of Proposition \ref{Delta' smooth even}, implies the following
linear equivalence of divisors
$$\tilde{D}^\sigma\sim \tilde{D'}^\sigma+\alpha\tilde{\Sigma}^\sigma+
\sum_{(I,t)\in\mathcal{C}^{\textrm{even}}}
\alpha_{I,\sigma_t}E_{I,\sigma_t},$$
where
 $\alpha_{I}:=k_{I}(\alpha\Sigma)+k_{I}(D')
-k_{I}(D)$
if  $1\le |I|\le \nu$, while
 $\alpha_{I,\sigma_t}:=k_{I,\sigma_t}(\alpha\Sigma)-k_{I,\sigma_t}(D)$ for
$1\le t\le\nu$, $\nu+1\le |I|\le 2\nu$.
Finally, from \eqref{Sigma is more singular than Delta} we obtain $a_{I,\sigma_t}\ge0$,
for all $I$ and $t$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{base point free tilde-sigma-D}, case $n$ even]
We recall that a general member of $|D|$ vanishes at every point of $J(L_I,\sigma_t)$
with multiplicity equal to $k_{I,\sigma_t}$ and that $|\tilde{D}^\sigma|$ is the
 strict transform of $D$ under the blow-up of its secant base locus. Assume that the base locus, ${\rm{Bs}}|\tilde{D}^\sigma|$, is non-empty.
By Proposition \ref{linear series of strict transform even}, we have
$$
{\rm{Bs}}|\tilde{D}^\sigma| \subset {\rm{Bs}}|\tilde{D'}^\sigma|\cup \tilde{\Sigma}\cup
\bigcup_{(I,t)\in\mathcal{C}^{\textrm{even}}}E_{I,\sigma_t}.
$$

Note that the divisor $|\tilde{D'}^\sigma|$ is base point free by Theorem \ref{gg theorem Xr}, therefore  ${\rm{Bs}}|\tilde{D'}^\sigma|$ is empty. Hence
$$
{\rm{Bs}}|\tilde{D}^\sigma| \subset  \tilde{\Sigma}\cup
\bigcup_{(I,t)\in\mathcal{C}^{\textrm{even}}}E_{I,\sigma_t}.
$$

 Assume there is a base point for $|\tilde{D}^\sigma|$ on one of the exceptional divisors $E_{I,\sigma_t}$, $(I,t)\in\mathcal{C}^{\textrm{even}}$, or on $\tilde{\Sigma}$.
This implies the existence of a point in a join
$J(L_I,\sigma_t)$, or in $\Sigma$, on which the general divisor $D$ has infinitesimal tangencies. The proof of Proposition \ref{infinitesimal base locus} implies that no point of $J(L_I,\sigma_t)$,
nor of $\Sigma$, carries any infinitesimal information. This leads to a contradiction.

\end{proof}

\subsubsection{Proof of Theorem \ref{base point free tilde-sigma-D}, case $n$ odd}
\label{proof base point freeness D case odd}

Recall that the strict transform on $X_{2\nu+3,(0)}$ of the cone $\J(L_{\{1\}},\sigma_\nu)\subset{\mathbb P}^{2\nu}$ is the divisor
$$
\Gamma:=(\nu+1)H-(\nu+1)E_1-\nu\sum_{i=2}^{2\nu+4}E_i.
$$

\begin{proposition}
In the above notation, the strict transform $\tilde{\Gamma}^\sigma$ of $\Gamma$ on $Y^\sigma$ is smooth.
\end{proposition}
\begin{proof}
Recall that the singular locus of  $\J(L_{\{1\}},\sigma_\nu)\subset{\mathbb P}^{2\nu}$ is $\J(L_{\{1\}},\sigma_{\nu-1})\subset \J(L_{\{1\}},\sigma_\nu)$ and, more precisely, the non-reduced union
of all joins $\J(L_I,\sigma_t)$ such that $k_{I,\sigma_t}(\Sigma)>1$.
In particular we compute the following multiplicities.
For every $I$ and $t$ such that $|I|,t\ge0$,  we have
\begin{equation}\label{mult Sigma odd}
k_{I,\sigma_t}(\Sigma)=\max\left\{0, \nu-|I|-t+1+\delta_I\right\},
\end{equation}
where $\delta_I$ is defined as
\begin{equation}\label{Kronecker}
\delta_I:=\delta_{1,I}=\left\{\begin{array}{ll}
1 & \textrm{if } 1\in I\\
0 & \textrm{if } 1\notin I.\\
\end{array}\right.
\end{equation}
All subvarieties $\J(L_I,\sigma_t)$ such that $|I|+t\le \nu+\delta_I$ have been blown-up,
hence we conclude that $\pi^\sigma$ is a resolution of the singularities of $\Sigma$.
\end{proof}

Choose $\beta\in\mathbb{N}$ such that
\begin{equation}\label{definition beta}
\frac{k_C(D)}{\nu}
\le \beta\le \min_{1\le i\le 2\nu+4}\left\{\frac{m_1}{\nu+\delta_{\{i\}}},d-m_i\right\}.
\end{equation}
where $\delta_{\{i\}}$ is the  Kronecker delta defined in \eqref{Kronecker}
\begin{lemma}
Under the assumptions of Theorem \ref{abundance Q}, such an integer $\beta$ exists.
\end{lemma}
\begin{proof}
For every $i\in\{1,\dots, 2\nu+4\}$, we compute
$$
\epsilon k_C(D)\le 2\nu-2\le 2\frac{\nu^2}{\nu+\delta_{\{i\}}}-\epsilon\nu\le
\epsilon\nu\left(\frac{m_1}{\nu+\delta_{\{i\}}}-1\right).
$$
In the above expression,
the first inequality follows from \eqref{low mult of joins}. The second inequality follows from
the assumption    $\epsilon\ll1$; in fact it is enough to take
 $\epsilon\le\frac{8}{n^2-1}=\frac{2}{\nu^2+\nu}$.
The last inequality follows from
\eqref{abundance condition}. Furthermore we have
$$
\epsilon k_C(D)\le 2\nu-2\le 2\nu-\epsilon\nu\le \epsilon \nu(d-m_i-1).
$$
The last inequality follows from \eqref{extra nef condition}.
\end{proof}

Consider the linear system
$$
|D'|:=|D-\beta \Gamma|=|d'H-\sum_{i=1}^{2\nu+4}m'_iE_i|,
$$
with $d':=d-\beta(\nu+1)$,   $m'_1:=m_1-\beta(\nu+1)$ and $m'_i:=m_i-\beta\nu$,
 for all $i\in\{2\dots,2\nu+4\}$.
We have
$$
|D'|+\beta\Gamma\subseteq|D|.
$$

\begin{proposition}\label{Delta' smooth odd}
In the notation of above, the  linear system $|D'|$ is non-empty.
Moreover $|D'|$ has only linear base locus and
 the strict transform in $Y^\sigma$ of $|D'|$  is base point free.
\end{proposition}
\begin{proof}
By \eqref{definition beta}, we have $0\le m'_i\le d$ for all $i\in\{1,\dots, 2\nu+4\}$.
Moreover, as in the proof of Proposition \ref{Delta' smooth even}, we compute
$$
k_C(D')=\max\left\{0, \epsilon\left(\sum_{i=1}^{2\nu+4}m'_i-(2\nu+1)d'\right)
\right\}=\max\left\{0,k_C(D)-\beta\nu\right\}=0.
$$
This proves the first statement.

To prove the second statement,
we notice that if $|I|\le\nu+1$ the pair $(I,0)\in\mathcal{C}^{\textrm{odd}}$, while
 we claim that $k_I(D')=0$ if $|I|\ge\nu+2$. This and Theorem \ref{gg theorem Xr}
imply the second statement.
We prove the claim for $1\in I$; the case $1\notin I$ is similar and we leave the details to the reader.
 Choose $J\subset I$ with $|J|=\nu+1$ and $1\in J$. Let us
 compute

\begin{align*}
\epsilon k_{I}(D)&=\max\left\{0, \epsilon\left(\sum_{i\in I}m_i-(|I|-1)d\right)+
\epsilon\beta(|I|-\nu-2)\right\}\\
&=\max\left\{0,\epsilon\left(\sum_{i\in J}m_i-(\nu)d\right)+
\epsilon\left(\sum_{i\in i\setminus J}(m_i-d)\right)+\epsilon \beta(|I|-\nu-2)\right\}\\
&\le \max\{0,-\epsilon \beta\}\\ &=0.
\end{align*}
The inequality holds thanks to \eqref{low mult of joins} and
 \eqref{definition beta}.
\end{proof}

\begin{proposition}\label{linear series of strict transform odd}
In the notation of above, there exist non-negative numbers
 $\beta_{I,\sigma_t}\in\mathbb{Z}$ such that
$$
|\tilde{D'}^\sigma|+\beta\tilde{\Gamma}^\sigma+\sum_{(I,t)\in\mathcal{C}^{\textrm{odd}}}
\beta_{I,\sigma_t} E_{I,\sigma_t}\subseteq
|\tilde{D}^\sigma|.
$$
\end{proposition}
\begin{proof}
The proof follows the same lines as that of Proposition
 \ref{linear series of strict transform even} and it uses Proposition \ref{Delta' smooth odd}. We leave the details to the reader.
\end{proof}

\begin{proof}[Proof of Theorem \ref{base point free tilde-sigma-D}, case $n$ odd]
The proof follows the same idea as that of the case $n$ even, at the end of Section
\ref{proof base point freeness D case even}.

\end{proof}

\subsubsection{The pair $(X,\epsilon D)$ is lc, for $D$ general}

Notice that  the canonical divisor of $Y^\sigma$ is
$$
K_{Y^\sigma}=-(n+1)H+(n-1)\sum E_i+\sum_{r=1}^{n-2}(n-r-1)
\sum_{\substack{I,t:\\ r_{I,\sigma_t}=r}}E_{I,\sigma_t}.
$$

We are now ready to prove Theorem \ref{abundance Q} for $s=n+3$.

\begin{proof}[Proof of Theorem \ref{abundance Q}, case $s=n+3$]
By Corollary \ref{general tildesigma smooth},
$(Y^\sigma,\tilde{\Delta})$
is log smooth and $\pi:Y\to X$ is a log resolution of $(X,\Delta)$.

To complete the proof,
similarly to the case of only linearly obstructed divisors,
we are going to show that \eqref{abundance condition} implies

\begin{equation}\label{lc' condition}
  \epsilon k_{I,\sigma_t}  \le n-|I|-2t+1,
\quad  \forall I(r), \ 2\le r_{I,\sigma_t}\le n-2,
\end{equation}

that in turns implies that $\textrm{discrep}(X,\Delta)\ge-1$.

This follows from the inequalities \eqref{low mult of joins} computed in the proof of Proposition
 Proposition \ref{tildeD intersect exc}.
\end{proof}

\section{On the F-conjecture for $\overline{\mathcal{M}}_{0,n}$} \label{fulton}

Let $\overline{\mathcal{M}}_{0,n}$ be the \textit{moduli space of stable rational
curves with $n$ marked points}.
For $n= 5$, $\overline{\mathcal{M}}_{0,n}$ is a del Pezzo surface and it has the property of being a
Mori dream space. Hu and Keel in \cite{HuKeel} showed that $\overline{\mathcal{M}}_{0,6}$
 is a log Fano threefold, hence a Mori Dream Space;  Castravet computed its Cox ring in \cite{CA}. For $n\geq 13$,  $\overline{\mathcal{M}}_{0,n}$ is known to not be a Mori Dream Space, see \cite{CT2,karu}.

We recall here the \emph{F-conjecture} on the nef cone of $\overline{\mathcal{M}}_{0,n}$ due to Fulton.
The elements of the $1$-dimensional boundary strata on  $\overline{\mathcal{M}}_{0,n}$ are called \textit{F-curves}.
A divisor intersecting non-negatively all F-curves is said to be \textit{F-nef}.
The F-Conjecture states that a divisor on $\overline{\mathcal{M}}_{0,n}$ is nef if and only if it is F-nef. This conjecture was proved for $n\leq 7$ in by Keel and McKernan \cite{KeMc}.

\subsection{Preliminaries and notation}
Let $\mathcal{I}$ be a subset of $\{1, \ldots, n+3\}$ with cardinality
$2\leq |\mathcal{I}|\leq n+1$ and let $\Delta_{\mathcal{I}}$ denote a
 boundary divisor on $\overline{\mathcal{M}}_{0,n+3}$. Here, $\Delta_{\mathcal{I}}$
 is the divisor parametrizing curves with one component marked by the elements
of $\mathcal{I}$ and the other component marked by elements of its complement,
${\mathcal{I}}^c$, in $\{1, \ldots, n+3\}$. Obviously $\Delta_{\mathcal{I}}=\Delta_{{\mathcal{I}}^c}$.

We recall that, for any $1\leq i\leq n+3$,  the tautological class $\psi_i$ is defined as the first Chern class of
the cotangent bundle, $c_1(\mathbb{L}_i)$, where $\mathbb{L}_i$ is the line bundle on $\overline{\mathcal{M}}_{0,n+3}$
such that over a moduli point $(C, x_1, \ldots, x_{n+3})$ the fiber is the cotangent space to $C$ at $x_i$, $T_{x_i}^{*}C.$

 In the Kapranov's model given by $\psi_{n+3}$ , denote by $\mathcal{S}$ the collection of
 $n+2$ points in general position in ${\mathbb P}^n$
 obtained by contraction of sections
 $\sigma_i$ of the forgetful morphism of the $n+3$ marked point. Denote by $S$ the
set of indices parametrizing $\mathcal{S}$.

Further, denote by $\mathcal{X}_{n+2,(n-2)}$ the iterated
 blow-up of ${\mathbb P}^n$ along the strict transforms of \textit{all} linear subspaces $L_I$
of dimension at most $n-2$ spanned by sets of points $I\subset\mathcal{S}$ with
$|I|\le n-1$, ordered by increasing dimension. Notice that, in the notation of Section \ref{higher blow ups}, for an effective divisor $D$, the iterated blown-up
space $X_{n+2, (n-2)}$ along the linear subspaces that are in the base locus of $|D|$ is a resolution of singularities of $D$ (see Corollary \ref{log resolutions}), so it depends on the divisor we start with. However, $\mathcal{X}_{n+2,(n-2)}$ depends only on the original set of $n+2$ points in ${\mathbb P}^n$.

In \cite{Ka} Kapranov identifies the moduli space $\overline{\mathcal{M}}_{0,n+3}$ with the
projective variety  $\mathcal{X}_{n+2, (n-2)}$, in the notation of Section
 \ref{notations}, by constructing birational maps from $\overline{\mathcal{M}}_{0,n+3}$
to ${\mathbb P}^{n}$ induced by the divisors $\psi_i$, for any choice of $i$ with $1\leq i\leq n+3$.

 We recall that the Picard group of $\mathcal{X}_{n+2, (n-2)}$ is spanned by a general
hyperplane class and exceptional divisors, $\Pic(\mathcal{X}_{n+2, (n-2)})= \langle  H, E_{J} \rangle$,
where $J$ is any non-empty subset of $S$ with $1\leq |J|\leq n-1$.

\begin{remark} In the Kapranov's model given by $\psi_{n+3}$, $\mathcal{X}_{n+2, (n-2)}$, there are $n+2$ Cremona transformations that are based on any subset of $n+1$ points of $S$, $S_i:=S \setminus \{p_i\}$. The $\psi_{i}$ classes with $i\neq n+3$ correspond to the image of the Cremona transformation of a general hyperplane class $H$, based on the set $S_i$, denoted by $\Cr_i(H)=\Cr(H)$ in \eqref{cremona}, while $\psi_{n+3}$ corresponds to $H$.
\end{remark}

Furthermore, we have the following identification:
\begin{equation}
 \label{boundary divisors}
\Delta_{\mathcal{I}}= \begin{cases}
 E_J, \qquad |\mathcal{I}|\leq n,\\
 H_J,  \qquad |\mathcal{I}|= n+1.
 \end{cases}
 \end{equation}
where $E_J$ is the strict transform of the exceptional divisor obtained by blowing-up
 the linear cycle spanned points of $J$, while $H_J$ is the strict transform of the
hyperplane passing thought the points of $J$, namely
$$H_J:=H-\sum_{\substack {I\subset J:\\ 1\leq |I|\leq n-2}}E_I.$$

The \textit{$F$-curves on $\overline{\mathcal{M}}_{0,n+3}$} correspond to
partitions of the index set
$$\mathcal{I}_1\sqcup \mathcal{I}_2\sqcup \mathcal{I}_3\sqcup \mathcal{I}_4=
\{1,\ldots, n+3\}.$$
We remark that by definition, all subsets $\mathcal{I}_i$ are non-empty.
We denote by $F_{\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3, \mathcal{I}_4}$ the class of the
corresponding F-curve.
We have the following intersection table (see \cite{KeMc}).
 \begin{equation}
 \label{intersection f-curves}
F_{\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3, \mathcal{I}_4}\cdot \Delta_{\mathcal{I}}= \begin{cases}
 1 \qquad  \ \  \mathcal{I}= \mathcal{I}_i\sqcup \mathcal{I}_j, \textrm{ for some } i\neq j,\\
 -1  \qquad \mathcal{I}=\mathcal{I}_i, \textrm{ for some } i,\\
 0  \qquad \  \textrm{ otherwise}.
 \end{cases}
 \end{equation}

We first describe the F-conjecture in a Kapranov's model using the coordinates of the
 N\'eron-Severi  group $N^1(\mathcal{X}_{n+2, (n-2)})$.
Consider a general divisor on $\mathcal{X}_{n+2, (n-2)}$ of the form
\begin{equation}\label{general div on M}
dH -\sum_{\substack {I\subset S:\\1\leq |I| \leq n-1}} m_{I}E_{I}.
\end{equation}
For a non-empty subset $I$ of the  points parametrized by  $S$ we define
 \begin{equation}
 \label{coefficient1}
a_{I}:= \begin{cases}
 0 \qquad |I|\geq n ,\\
 1  \qquad |I|\leq n-1.
 \end{cases}
 \end{equation}
For any partition of the set of $n+2$ points $G \sqcup J \sqcup L=S$, set
\begin{equation*}
\begin{split}
A_{G, J, L}:&=d-a_G\cdot m_G-a_J\cdot m_J - a_L\cdot m_L + a_{J \sqcup L}\cdot m_{J \sqcup L} \\
&+ a_{J \sqcup G}\cdot m_{J \sqcup G}+ a_{L \sqcup G}\cdot m_{L \sqcup G}.
\end{split}
\end{equation*}

\begin{example}\label{f curves1}
Consider $|G|=n$ then $J$ and $L$  consist of one element each, say $j$
 and respectively $l$. Then $A_{G,J,L}$ is independent of $G$ since
$A_{G,J,L}=d-m_j-m_l+m_{jl}$ for $n\geq 3$.
 Whenever $|G|\leq n-1$ then $A_{G,J,L}$ depends on all three subsets.
\end{example}

Moreover, for any two non-empty subsets of $S$, $I$ and $J$, set
 \begin{equation}
 \label{coefficient2}
b_{I\sqcup J}:= \begin{cases}
 0 \qquad |I|+|J|\geq n,\\
 1  \qquad |I|+|J|\leq n-1.
 \end{cases}
 \end{equation}
For any partition $I \sqcup G \sqcup J \sqcup L=S$ set
$b_{I \sqcup J \sqcup L}:=b_{I \sqcup (J \sqcup L)}$, as defined in \eqref{coefficient2}, and
\begin{equation*}
\begin{split}
B_{I, G, J, L}:&=m_{I}- b_{ I \sqcup G}\cdot m_{I \sqcup G} - b_{I \sqcup J}\cdot m_{I \sqcup J} - b_{I \sqcup L}\cdot m_{I \sqcup L} + \\
&+ b_{I \sqcup J \sqcup L}\cdot m_{I \sqcup J \sqcup L} + b_{I \sqcup G \sqcup J}\cdot m_{I \sqcup G \sqcup J}+ b_{I \sqcup G \sqcup L}\cdot m_{I \sqcup G \sqcup L}.
\end{split}
\end{equation*}

\begin{example}\label{f curves2} If $|I|+|G|=n$ then $J$ and $L$ consist each of one element, call $j$ and $l$ respectively. In this case
\begin{itemize}
\item
If $|I|=n-1$ then the subsets $G$, $J$ and $L$ consist of one element each and $B_{I, G, J, L}=m_{I}.$
\item
If $|I|=n-2$ then $B_{I, G, J, L}=m_{I}-m_{I\sqcup \{j\}}-m_{I\sqcup \{l\}}.$
\item
If $|I|\leq n-3$ then $B_{I, G, J, L}=m_{I}-m_{I\sqcup \{j\}}-m_{I\sqcup \{l\}}+m_{I\sqcup \{j\}\sqcup \{l\}}.$
\end{itemize}
Whenever $|I|+|G| \leq n-1$ then $B_{I, G, J, L}$ depends on the four subsets of the partition.
\end{example}
\begin{remark}
The number $A_{G, J, L}$ represents the intersection product between the divisor $D$ and the corresponding F-curve contained in a hyperplane divisor and $B_{I, G, J, L}$ represents the intersection product between the divisor $D$ and the F-curve contained in some exceptional divisor $E_I$.
\end{remark}
Using the identification of boundary divisors \eqref{boundary divisors} and the intersection table \eqref{intersection f-curves},
it  is easy to see that the following remark holds.

\begin{remark}[The cone of F-nef divisors]\label{f-nef}
 A divisor on $\mathcal{X}_{n+2, (n-2)}$ of the form \eqref{general div on M} is \textit{F-nef} if the following sets of inequalities hold:
\begin{equation}\label{f-nef ineq}
\begin{split}
A_{G, J, L}\geq 0, & \quad \textrm{ for any partition }G \sqcup J \sqcup L=S,\\
B_{I, G, J, L} \geq 0, &\quad \textrm{ for any partition } I \sqcup G \sqcup J \sqcup L=S.
\end{split}
\end{equation}
 This cone is often referred to as the \emph{Faber cone} in the literature, see e.g. \cite{GKM}.
\end{remark}

\begin{conjecture}[F-conjecture]\label{f-conjecture} A divisor on $\mathcal{X}_{n+2, (n-2)}$
of the form \eqref{general div on M}
 is nef if and only if \eqref{f-nef ineq} holds.
\end{conjecture}

Take a general divisor with degree and multiplicities labeled as in
\eqref{general div on M}. We will now describe general properties of the $F$-nef divisors in a Kapranov's model that are useful in computations.

\begin{lemma}\label{degree of a f-nef divisor}
Any F-nef divisor satisfies $d\geq m_I\geq 0$, for every $I\subset S$, and $m_I\geq m_J$,
for every $I,J\subset S$ with $I\subset J$.
\end{lemma}

\begin{proof}
We claim that these inequalities follow from \eqref{f-nef ineq}. For $n=2$ the claim is obvious, hence we assume $n\geq 3$.
In fact, the following inequalities hold:
\begin{enumerate}
\item $m_{I}\geq 0$, for every non-empty set $I$ with $|I|=n-1$,
\item $m_{I}\geq m_{J}$, for every non-empty sets $I,J$ with $I\subset J$.
\end{enumerate}
Claim (1) follows from Example \ref{f curves2} and  Remark \ref{f-nef}.
To prove claim (2) we apply induction on $|I|$. For $i\neq j$ and $i,j \notin I$
 we introduce the following notations: $I_i:=I \sqcup \{i\}$ and $I_{ij}:=I \sqcup \{i,j\}$.
For the first step of induction consider the sets $I$ and $G$ with $|I|=n-2$ and $|G|=2$. For any $i\neq j$  one has, by \eqref{f-nef ineq},
that $$m_I - m_{I_i} -m_{I_j} \geq 0.$$
Therefore claim (2) follows from claim (1) for any $I$ with $|I|=n-2$.
If $|I|\leq n-3$ the claim follows using backward induction on $|I|$. Indeed, by Example \ref{f curves2} we have
$$m_I- m_{I_i} - m_{I_j} + m_{I_{ij}}\geq 0,$$
therefore
$$m_I- m_{I_i} \geq m_{I_j} - m_{I_{ij}}\geq 0.$$
Since $|I_i|=|I|+1$ and $I_i\subset I_{ij}$, the induction hypothesis holds for $I_i$, so the claim follows.

To see that $d\geq m_I$ we use Example \ref{f curves1} and claim (2) to obtain
$$d\geq m_i+(m_j-m_{ij})\geq m_I.$$
\end{proof}

\subsection{The F-conjecture holds for strict transforms on $\mathcal{X}_{n+2,(n-2)}$}
The main result of this section is Theorem \ref{fulton nef}, stating that the F-conjecture holds
for all divisors on $\mathcal{X}_{n+2, (n-2)}$ that are strict transforms of an effective divisor on $X_{n+2, (0)}$.

In the notation of Section \ref{notations}, $D$ be any effective divisor on $X_{n+2, (0)}$ and let $\tilde{D}$ denote its strict transform on $X_{n+2, (n-2)}$.
A general divisor on $\mathcal{X}_{n+2, (n-2)}$, for arbitrary coefficients $d$
and $m_I$, is of the form \eqref{general div on M} while the divisors $\tilde{D}$
have arbitrary coefficients $d$ and $m_i$ while $m_I:=k_I$ defined in \eqref{mult k}
for any index $I$ with $|I|\geq 2$. We can consider $\tilde{D}$ a divisor on
$\mathcal{X}_{n+2, (n-2)}$.

\begin{theorem}\label{fulton nef}
Assume $m_i\geq 0$, for all $i\in S$, then Conjecture \ref{f-conjecture} holds for $\tilde{D}$ on $\mathcal{X}_{n+2, (n-2)}$.
\end{theorem}

\begin{proof}
To prove the claim, notice first  that the effectivity of $D$ implies $\sum_{i\in S}m_i\le nd$ and
 $\sum_{i\in I}m_i\le nd$, for all $I\subset S$ such that $|I|=n+1$. Moreover, since $m_i\ge 0$, we
conclude by Theorem \ref{gg theorem Xr} and Remark \ref{gg for Dn-1}.

We proved that $\tilde{D}$ is globally generated. Therefore
 $\tilde{D}$ is nef and in particular F-nef.
\end{proof}

\begin{corollary}
For divisors of the form $\tilde{D}\ge 0$ on $\mathcal{X}_{n+2, (n-2)}$, the three properties of being
$F$-nef, nef and globally generated are equivalent.
\end{corollary}

\begin{remark}
The divisors in \eqref{general div on M} for which $m_I<k_I$ are not nef, as they
 intersect negatively the class of a general line on the exceptional divisor $E_I$, for $2\leq|I|\leq n-1.$

The divisors $\tilde{D}$ with $k_{I}\geq 1$, for some set $I$ with $|I|\geq 2$ are not globally generated since they contract the exceptional divisors $E_{I}$.

\end{remark}

\begin{remark}
Studying divisors interpolating higher dimensional linear cycles, $L_{I}$ for  $|I|\ge2$,
$m_I>k_I$, is a possible approach to the F-conjecture. Indeed,
once the vanishing theorems are established by techniques developed in \cite{bdp1} and \cite{dp} they could be used for describing globally generated divisors or
 ample and nef cones of $\overline{\mathcal{M}}_{0,n}$. The description of ample divisors on $\overline{\mathcal{M}}_{g,n}$ is an important question originally asked by Mumford and conjectured by Fulton for $g=0$.
\end{remark}

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