\begin{document}

\title[Nakamaye's theorem on  log canonical pairs]{Nakamaye's theorem on log canonical pairs} 

\dedicatory{\normalsize \dag \ Dedicated to the memory of Fassi} 

\author[S. Cacciola and A.F. Lopez]{Salvatore Cacciola* and Angelo Felice Lopez*}

\thanks{* Research partially supported by the MIUR national project ``Geometria delle variet\`a 
algebriche" PRIN 2010-2011.}

\address{\hspace{.1in}kip -.43cm Dipartimento di Matematica e Fisica, Universit\`a di Roma 
Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. e-mail {\tt cacciola@mat.uniroma3.it, lopez@mat.uniroma3.it}}

\thanks{{^{-1}t 2010 Mathematics Subject Classification} : Primary 14C20, 14F18. Secondary 14E15, 14B05}

\begin{abstract} 
We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension $\leq 1$. We also generalize Ein-Lazarsfeld-Musta{\c{t}}{\u{a}}-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities. 
\end{abstract}

\maketitle

\section{Introduction}
\label{intro}

Let $X$ be a normal complex projective variety and let $D$ be a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. The stable base locus
\[ \mathbf{B}(D)= \bigcap_{E \geq 0 : E \sim _{{\mathbb Q}} D} {\rm Supp}(E) \]
is an important closed subset associated to $D$, but it is often difficult to handle. On the other hand, there are two, perhaps even more important, base loci associated to $D$. 

One of them is the augmented base locus (\cite{n}, \cite[Def. 1.2]{elmnp1}) 
\[ \mathbf{B}_+(D)= \bigcap_{E \geq 0 :  D - E \mbox{\ ample}} {\rm Supp}(E) \]
where $E$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor. Since this locus measures the failure of $D$ to be ample, it has proved to be a key tool in several recent important results
in birational geometry, such as Takayama \cite{t}, Hacon and McKernan's \cite{hm} effective birationality of pluricanonical maps or Birkar, Cascini, Hacon and McKernan's \cite{bchm} finite generation of the canonical ring, just to mention a few.

One way to compute $\mathbf{B}_+(D)$ is to pick a sufficiently small ample ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $A$ on $X$, because then one knows that $\mathbf{B}_+(D) = \mathbf{B}(D-A)$ by  \cite[Prop. 1.5]{elmnp1}. 

In the case when $D$ is also nef, for every subvariety $V \subseteqset X$ of dimension $d \geq 1$ such that $D^d \cdot V = 0$, we have that $D_{|V}$ is not big, whence $s(D-A)_{|V}$ cannot be effective for any $s ^{-1}n {\mathbb N}$ and therefore $V \subseteqseteq \mathbf{B}(D-A) = \mathbf{B}_+(D)$. Now define
\[ {\rm Null}(D) = \bigcup_{V \subseteqset X: D^d \cdot V = 0} V \]
so that, by what we just said,
\begin{equation}
\label{nak}
{\rm Null}(D) \subseteqseteq \mathbf{B}_+(D).
\end{equation}
A somewhat surprising result of Nakamaye \cite[Thm. 0.3]{n} (see also \cite[\S 10.3]{laz}) asserts that, if $X$ is {^{-1}t smooth} and $D$ is big and nef, then in fact equality holds in \eqref{nak}.
 
As is well-known, in birational geometry, one must work with normal varieties with some kind of (controlled) singularities. In the light of this, it becomes apparent that it would be nice to have a generalization of Nakamaye's Theorem to normal varieties. While in positive characteristic the latter has been recently proved to hold, on any projective scheme, by Cascini, McKernan and Musta{\c{t}}{\u{a}} \cite[Thm. 1.1]{cmm}, we will show in this article a generalization to normal complex varieties with log canonical singularities. This partially answers a question in \cite{cmm}. 

More precisely let us define

\begin{defn}
{\rm Let $X$ be a normal projective variety. The {\bf non-lc locus} of $X$ is
\[ X_{\mathrm{nlc}} =\bigcap_{{\Delta}} \mathrm{Nlc}(X,{\Delta}) \]
where ${\Delta}$ runs among all effective Weil ${\mathbb Q}$-divisors such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})$ is the locus of points $x ^{-1}n X$ such that $(X, {\Delta})$ is not log canonical at $x$.}
\end{defn}

Using Ambro's and Fujino's theory of non-lc ideal sheaves \cite{a}, \cite{f} and a modification of some results of de Fernex and Hacon \cite{dh}, we prove

^{\vee}skip .3cm

\begin{bthm}
\label{main}
Let $X$ be a normal projective variety such that $\dim X_{\mathrm{nlc}} \leq 1$. 
Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then 
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bthm}

^{\vee}skip .2cm

This easily gives the following

\begin{bcor}
\label{dim}
Let $X$ be a normal projective variety such that $\dim \operatorname{Sing}(X) \leq 1$ or $\dim X \leq 3$ or there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ such that $(X, {\Delta})$ is log canonical. 

Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then 
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bcor}

^{\vee}skip .2cm

Moreover, using a striking result of Gibney, Keel and Morrison \cite[Thm. 0.9]{gkm}, we can give a very quick application to the moduli space of stable pointed curves.
 
\begin{bcor}
\label{mod}
Let $g \geq 1$ and let  $D$ be a big and nef  ${\mathbb Q}$-divisor on $\overline{M}_{g,n}$. Then 
\[ \mathbf{B}_+(D) \subseteqseteq \partial \overline{M}_{g,n}. \]
\end{bcor}
Thus, for example, one gets new compactifications of $M_{g,n}$ by taking rational maps associated to such divisors.
 
The second base locus associated to any pseudoeffective  ${\mathbb R}$-Cartier ${\mathbb R}$-divisor $D$, measuring how far $D$ is from being nef, is the restricted base locus \cite[Def. 1.12]{elmnp1}.

\begin{defn}
{\rm Let $X$ be a normal projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. The {\bf restricted base locus} of $D$ is
\[ \mathbf{B}_-(D)= \bigcup_{A {\ \rm ample}} \mathbf{B}(D+A) \]
where $A$ runs among all ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors such that $D+A$ is a ${\mathbb Q}$-divisor.}
\end{defn}
Restricted base loci are countable unions of subvarieties by \cite[Prop. 1.19]{elmnp1}, but not always closed \cite[Thm. 1.1]{les}.

For a big ${\mathbb Q}$-divisor $D$ on a {^{-1}t smooth} variety $X$, the subvarieties of $\mathbf{B}_-(D)$ are precisely described in \cite[Prop. 2.8]{elmnp1} (also in positive characteristic in \cite[Thm. 6.2]{mus}) in terms of asymptotic valuations. 

\begin{defn} 
\label{as1}
{\rm (\cite[Def. III.2.1]{nak}, \cite[Lemma 3.3]{elmnp1}, \cite[\S 1.3]{bbp}, \cite[\S 2]{dh})
Let $X$ be a normal projective variety, let  $D$ be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$, that is $v$ is a positive integer multiple of the valuation associated to a prime divisor $\Gamma$ lying on a birational model $f : Y \to X$. The center of $v$ on $X$ is $c_X(v)=f(\Gamma)$. 

If $D$ is big, we set
\[ v(\|D\|) = ^{-1}nf \{v(E), E \ \mbox{effective} \ {\mathbb R}\mbox{-Cartier} \ {\mathbb R}\mbox{-divisor on X such that} \ E \equiv D\}; \]
if $D$ is pseudoeffective, we pick an ample divisor $A$ and set
\[ v(\|D\|) = \lim_{^{\vee}arepsilon \to 0^+} v(\|D + ^{\vee}arepsilon A\|). \]
If  $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and $b ^{-1}n {\mathbb N}$ is such that $bD$ is Cartier and $|bD| \neq \emptyset$, we set
(see \cite[Def. 2.14]{cd} or \cite[Def. 2.2]{elmnp1} for the case $D$ big)
\[ v(\langle D \rangle) = \lim_{m \to + ^{-1}nfty}\frac{v(|mbD|)}{mb} \]
where, if $g$ is an equation, at the generic point of $c_X(v)$, of a general element in $|mbD|$, then $v(|mbD|) = v(g)$.}
\end{defn}

Now the main content of \cite[Prop. 2.8]{elmnp1} is that, given a discrete valuation $v$ on a {^{-1}t smooth} $X$ with center $c_X(v)$ and a big divisor $D$, then $c_X(v) \subseteqseteq \mathbf{B}_-(D)$ if and only if $v(\|D\|) > 0$. Using the main result of \cite{cd} we give a generalization to normal pairs with klt singularities.

\begin{bthm}
\label{main2}
Let  $X$ be a normal projective variety such that there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ with $(X,{\Delta})$ a klt pair. Let $v$ be a divisorial valuation on $X$.
Then
\begin{itemize}
^{-1}tem [(i)] If $D$ is a big Cartier divisor on $X$ we have 
\[ v(\langle D \rangle) > 0 \ \mbox{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D) \ \mbox{if and only if} \ \limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty. \]
^{-1}tem [(ii)] If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
\[ v(\|D\|) > 0 \ \textrm{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D). \]
\end{itemize}
\end{bthm}

^{\vee}skip .2cm

\noindent {^{-1}t Acknowledgments}. We wish to thank Lorenzo Di Biagio for some helpful discussions.

\section{Non-lc ideal sheaves}
\label{basic}

\begin{notn}
{\rm Throughout the article we work over the complex numbers. Given a variety $X$ and a coherent sheaf of ideals $\mathcal{J} \subseteqset  \mathcal{O}_X$, we denote by $\mathcal{Z}(\mathcal{J})$ the closed subscheme of $X$ defined by $\mathcal{J}$. If $X$ is a normal projective variety and ${\Delta}$ is a Weil ${\mathbb Q}$-divisor on $X$,  we call $(X, {\Delta})$ a {\bf pair} if $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier. We refer to \cite[Def. 2.34]{km} for the various notions of singularities of pairs.}
\end{notn} 

\begin{defn}
{\rm Let $X$ be a normal projective variety and let ${\Delta} = \sum\limits_{i=1}^s d_i D_i$ be a Weil ${\mathbb Q}$-divisor on $X$, where the ${D_i}'s$ are distinct prime divisors. 

Given $a ^{-1}n {\mathbb R}$ we set ${\Delta}^{>a}=\sum\limits_{1 \leq i \leq s : d_i>a} d_i D_i$, ${\Delta}^+={\Delta}^{>0}$, ${\Delta}^- =(-{\Delta})^+$ and ${\Delta}^{<a}=-((-{\Delta})^{>-a})$. The {\bf round up} of ${\Delta}$ is $\lceil {\Delta} \rceil = \sum\limits_{i=1}^s \lceil d_i \rceil D_i$ and the {\bf round down} is $\lfloor {\Delta} \rfloor = \sum\limits_{i=1}^s \lfloor d_i \rfloor D_i$. We also set ${\Delta}^{\#} =  {\Delta}^{<-1} + {\Delta}^{>-1}$.}
\end{defn} 

The following is easily proved.

\begin{rem}
\label{sha}
{\rm Let $X$ be a normal projective variety and let ${\Delta}, {\Delta}'$ be Weil ${\mathbb Q}$-divisors on $X$. Then
\begin{itemize}
^{-1}tem[(i)] $\lceil  (-{\Delta})^{\#} \rceil = \lceil -({\Delta}^{<1}) \rceil - \lfloor {\Delta}^{>1} \rfloor$; 
^{-1}tem[(ii)] If ${\Delta} \leq {\Delta}'$, then $\lceil {\Delta}^{\#} \rceil \leq \lceil  ({\Delta}')^{\#} \rceil$.
\end{itemize}}
\end{rem}

We recall the definition of non-lc ideal sheaves \cite[Def. 4.1]{a}, \cite[Def. 2.1]{f}.

\begin{defn}
\label{inlc}
{\rm Let $(X, {\Delta})$ be a pair and let $f: Y \to X$ be a resolution  of $X$ such that ${\Delta}_Y :=f^{\ast}(K_X+{\Delta})-K_Y$ has simple normal crossing support.
The {\bf non-lc ideal sheaf associated to} $(X,{\Delta})$ is
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil -  \lfloor {\Delta}_Y^{>1}  \rfloor). \]}
\end{defn}

\begin{rem}
\label{nlc}
{\rm Non-lc ideal sheaves are well-defined by \cite[Prop. 2.6]{f}, \cite[Rmk. 4.2(iv)]{a}. Moreover, when ${\Delta}$ is effective and $f : Y \to X$ is a log-resolution of $(X, {\Delta})$, we have that the non-lc locus of $(X, {\Delta})$ is, set-theoretically, $\operatorname{Nlc}(X,{\Delta})= f(\operatorname{Supp}({\Delta}_Y^{>1})) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))$ \cite[Lemma 2.2]{f}.}
\end{rem}

\begin{rem}
\label{ic}
{\rm The non-lc ideal sheaf of a pair $(X,{\Delta})$ with ${\Delta}$ effective is an integrally closed ideal.}
\begin{proof}
With notation as in Definition \ref{inlc}, set $G =  \lceil - ({\Delta}_Y^{<1})  \rceil$ and $N=  \lfloor {\Delta}_Y^{>1}  \rfloor$, so that $G$ and $N$ are effective divisors without common components, $G$ is $f$-exceptional and $\mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(G-N) = f_{\ast}\mathcal{O}_Y(-N)$ by Fujita's lemma \cite[Lemma 2.2]{fta}, \cite[Lemma 1-3-2]{kmm}, \cite[Lemma 4.5]{dh}. Therefore $\mathcal{J}_{NLC}(X,{\Delta})$ is an ideal sheaf and it is integrally closed by \cite[Prop. 9.6.11]{laz}.
\end{proof}
\end{rem}

Our next goal is to prove, using techniques and results in de Fernex-Hacon \cite{dh}, that non-lc ideal sheaves have a unique maximal element. To this end we will use some results of Fujino \cite{f} and de Fernex-Hacon \cite{dh} that we wish to recall for the reader's convenience.

\begin{lemma} \cite[Lemma 2.7]{f} 
\label{fuj}
Let $g : Y' \to Y$ be a proper birational morphism between smooth varieties and let $B_Y$ be an ${\mathbb R}$-divisor on $Y$ having simple normal crossing support. Assume that  $B_{Y'} := g^{\ast}(K_Y+B_Y)-K_{Y'}$ also has simple normal crossing support. Then 
\[ g_{\ast} {\mathcal O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) \cong {\mathcal O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ). \]
\end{lemma}

\begin{defn} 
\label{dfh1}
{\rm (\cite[Def. 3.1]{dh})
Let $f: Y \to X$ be a proper birational morphism between normal varieties and let  $K_Y$ be a canonical divisor on $Y$ and $K_X = f_{\ast} K_Y$. For every $m \geq 1$ define $K_{m, Y/X} = K_Y - \frac{1}{m}f^{\natural}(mK_X)$, where $f^{\natural}(mK_X)$ is the divisor on $Y$ such that ${\mathcal O}_Y(-f^{\natural}(mK_X)) = ({\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y)^{^{\vee}ee ^{\vee}ee}$.}
\end{defn}

\begin{lemma}
\label{dh2}
Let $m \geq 1$. In (i)-(iv) below let $f: Y \to X$ be a proper birational morphism between normal varieties. Then
\begin{itemize}
^{-1}tem [(i)]  If $X$ is Gorenstein then $K_{m, Y/X} = K_{Y/X} := K_Y + f^{\ast}(-K_X)$;
^{-1}tem [(ii)]  \cite[Rmk 3.3]{dh} For all $q \geq 1$ we have $K_{m, Y/X} \leq K_{mq, Y/X}$;
^{-1}tem [(iii)] \cite[Lemma 3.5]{dh} Assume that $mK_Y$ is Cartier and ${\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y$ is invertible. Let $Y'$ be a normal variety and let $g : Y' \to Y$ be a proper birational morphism. Then $K_{m, Y'/X} = K_{m, Y'/Y} + g^{\ast} K_{m, Y/X}$;
^{-1}tem [(iv)] \cite[Rmk 3.9]{dh} Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and assume that  $m(K_X + {\Delta}elta)$ is Cartier. Then $K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{m, Y/X}$;
^{-1}tem [(v)] \cite[Thm. 5.4 and its proof]{dh} For every $m \geq 2$ there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ such that $m{\Delta}_m$ is integral, $\lfloor {\Delta}_m \rfloor = 0$, $({\Delta}_m)_Y$ has simple normal crossing support, $f$ is a log-resolution for the log-pair $((X, {\Delta}_m), \mathcal{O}_X(-mK_X))$, $K_X + {\Delta}_m$ is ${\mathbb Q}$-Cartier and $K_{m, Y/X} = K_Y + f_{\ast}^{-1}({\Delta}_m) - f^{\ast}(K_X+{\Delta}_m)$.
\end{itemize}
\end{lemma}
In (iv) and (v) above $f_{\ast}^{-1}({\Delta})$ is the proper transform of ${\Delta}$. Note that our ${\Delta}_Y$ (Definition \ref{inlc}) is different from the one in \cite[Def. 3.8]{dh}.

Now we have

\begin{prop} 
\label{max}
Let $X$ be a normal projective variety. Then there exists a Weil ${\mathbb Q}$-divisor ${\Delta}_0$ on $X$ such that $\lfloor {\Delta}_0 \rfloor = 0$,
$K_X + {\Delta}_0$ is ${\mathbb Q}$-Cartier and 
\[ \mathcal{J}_{NLC}(X,{\Delta}) \subseteqseteq \mathcal{J}_{NLC}(X,{\Delta}_0) \]
for every pair $(X,{\Delta})$ with ${\Delta}$ effective.
\end{prop}
\begin{proof}
Fix a canonical divisor $K_X$ on $X$ and an integer $m \geq 2$. By Lemma \ref{dh2}(v) there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ with the properties in (v). In particular $K_{m, Y/X}$ is $f$-exceptional. Now set
\[ \mathfrak{a}_m(X) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil).\]
As in the proof of Remark \ref{ic} we get that $\mathfrak{a}_m(X)$ is a coherent ideal sheaf. Let us check that its definition is independent of the choice of $f$. Let $f': Y' \to X$ be another log-resolution of $(X, \mathcal{O}_X(-mK_X))$ and assume, as we may, that $f'$ factors through $f$ and a morphism $g : Y' \to Y$. By Lemma \ref{dh2}(iii) and (i) we have
$K_{m, Y'/X} = K_{Y'/Y} + g^{\ast} K_{m, Y/X} = K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X})$, whence
\begin{equation}
\label{ben}
(fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast}(g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil)).
\end{equation}
Now set $B_Y = - K_{m, Y/X}$ and $B_{Y'} = g^{\ast}(K_Y+B_Y) - K_{Y'}$ so that, using Remark \ref{sha}(i) and Lemma \ref{fuj}, we have
\[ g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil (-B_{Y'})^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) = \]
\[ = \mathcal{O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ) = \mathcal{O}_Y (\lceil (-B_Y)^{\#} \rceil) = \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]

and by \eqref{ben} we get
\[ (fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast} \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]
that is $\mathfrak{a}_m(X)$ is well defined.

We now claim that the set $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element. In fact, given $m, q \geq 2$, let $f: Y \to X$ be a log-resolution of $(X, \mathcal{O}_X(-mK_X)) + \mathcal{O}_X(-mqK_X))$. By Lemma \ref{dh2}(ii) and Remark \ref{sha}(ii) we have $\lceil (K_{m, Y/X})^{\#} \rceil \leq \lceil (K_{mq, Y/X})^{\#} \rceil$ and therefore $\mathfrak{a}_m(X) \subseteqseteq \mathfrak{a}_{mq}(X)$. Using the ascending chain condition on ideals we conclude that $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element, which we will denote by $\mathfrak{a}_{\rm max}(X)$. 

Next let us show that all the ideal sheaves $\mathfrak{a}_m(X)$, for $m \geq 2$ (whence in particular also $\mathfrak{a}_{\rm max}(X)$), are in fact non-lc ideal sheaves of a suitable pair. 

Let  ${\Delta}_m$ be as above, so that, by Remark \ref{sha}(i) and using $\lceil  (-f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = 0$, we have
\[ \lceil - (({\Delta}_m)_Y^{<1}) \rceil -  \lfloor ({\Delta}_m)_Y^{>1}  \rfloor = \lceil  (-({\Delta}_m)_Y)^{\#} \rceil = \lceil  (K_{m, Y/X} - f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \]
\[  = \lceil  (K_{m, Y/X})^{\#}  \rceil + \lceil (- f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \lceil  (K_{m, Y/X})^{\#} \rceil \]
whence
\[ \mathcal{J}_{NLC}(X,{\Delta}_m)=f_{\ast}\mathcal{O}_Y(\lceil - (({\Delta}_m)_Y^{<1}) \rceil -  \lfloor ({\Delta}_m)_Y^{>1}  \rfloor) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) = \mathfrak{a}_m(X). \]
To finish the proof, let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $q ^{-1}n {\mathbb N}$ be such that $q(K_X + {\Delta})$ is Cartier. Let $m_0 \geq 2$ be such that $\mathfrak{a}_{\rm max}(X) = \mathfrak{a}_{m_0}(X) = \mathfrak{a}_{qm_0}(X)$. By what we proved above, there exists ${\Delta}_0 := {\Delta}_{qm_0}$ such that $\mathcal{J}_{NLC}(X,{\Delta}_0) =  \mathfrak{a}_{\rm max}(X)$. By Lemma \ref{dh2}(iv)  we have that  $- {\Delta}_Y \leq K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{qm_0, Y/X}$, whence also, by Remark \ref{sha} (i) and (ii),
\[ \lceil - ({\Delta}_Y^{<1}) \rceil -  \lfloor {\Delta}_Y^{>1}  \rfloor = \lceil (- {\Delta}_Y)^{\#} \rceil \leq \lceil (K_{qm_0, Y/X})^{\#} \rceil \]
and therefore
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil -  \lfloor {\Delta}_Y^{>1}  \rfloor) \subseteqseteq f_{\ast}\mathcal{O}_Y(\lceil (K_{qm_0, Y/X})^{\#} \rceil) = \mathfrak{a}_{\rm max}(X) = \mathcal{J}_{NLC}(X,{\Delta}_0). \]
\end{proof}

\section{Proof of Theorem \ref{main}}
\label{b+}

We record the following lemma, which is also of independent interest.

\begin{lemma} 
\label{bs}
Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $D$ be an effective Cartier divisor on $X$. Then there exists $c = c(X,{\Delta}, D) ^{-1}n {\mathbb N}$ such that the set-theoretic equality
\[ \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E_1+\dots+E_c)) \]
holds for some $E_1,\dots, E_c ^{-1}n |D|$.
\end{lemma}

\begin{proof}
Let $f:Y \to X$ be a log-resolution of $(X,{\Delta})$ and of the linear series $|D|$ such that $f_{\ast}^{-1} {\Delta} + \mathbf{B}s|f^{\ast}D| + \operatorname{Exc}(f)$ has simple normal crossing support.
Write ${\Delta}_Y = {\Delta}_Y^+ - {\Delta}_Y^-$, where ${\Delta}_Y^+$ and ${\Delta}_Y^-$ are effective simple normal crossing support ${\mathbb Q}$-divisors without common components. Then ${\Delta}_Y^-= \sum_{i=1}^s \delta_i D_i$, for some non-negative $\delta_i ^{-1}n {\mathbb Q}$ and distinct prime divisors $D_i$'s and define 
\[ c =  \lceil \operatorname{max}\{\delta_i, 1 \leq i \leq s \}  \rceil +2. \]
Moreover we have that $|f^{\ast}D|=|M|+F$, where $|M|$ is base-point free and $\operatorname{Supp}(F)= \mathbf{B}s|f^{\ast}D|$. By Bertini's Theorem and \cite[Lemma 9.1.9]{laz}, we can choose $M_1,\dots, M_c ^{-1}n |M|$ general divisors such that, for all $j=1,\dots,c$, $M_j$ is smooth, every component of $M_j$ is not a component of ${\Delta}_Y, M_1,\dots, M_{j-1}$ and ${\Delta}_Y+M_1+\dots+M_c+F$ has simple normal crossing support. Now, for all $j = 1, \ldots, c$, $M_j+F ^{-1}n |f^{\ast}D|$, so that there exists $E_j ^{-1}n |D|$ such that $M_j+F =f^{\ast}E_j$. Set $E=E_1+\dots +E_c$ and notice that $f$ is also a log-resolution of $(X,{\Delta}+E)$.

By Remark \ref{nlc} we have $\operatorname{Nlc}(X,{\Delta})=\mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))\subseteqseteq \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$, the latter inclusion following by Remark \ref{sha}(i) and (ii), because $E$ is effective. Also, for every prime divisor $\Gamma$ in the support of $F$ we get for the discrepancies
\[ a(\Gamma,X,{\Delta}+E)=a(\Gamma,X,{\Delta})-\operatorname{ord}_{\Gamma}(f^{\ast} E) =-\operatorname{ord}_{\Gamma} ({\Delta}_Y) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq \]
\[ \leq \operatorname{ord}_{\Gamma} ({\Delta}_Y^-) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq  \operatorname{max}\{\delta_i, 1 \leq i \leq s \} -\operatorname{ord}_{\Gamma}(M_1+\dots+M_c+cF)\leq -2 \]
whence $f(\Gamma) \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E)$. As $\mathbf{B}s|D|$ is the union of such $f(\Gamma)$'s, using Remark \ref{nlc}, we get the inclusion $\mathbf{B}s |D| \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$.
 
On the other hand notice that $({\Delta}+E)_Y =f^{\ast}(K_X+{\Delta}+E)-K_Y= {\Delta}_Y +f^{\ast}E$. Also ${\Delta}_Y + f^{\ast} E = {\Delta}_Y + M_1+\dots+M_c+cF$, so that
\[ \operatorname{Supp}(({\Delta}+E)_Y^{>1}) = \operatorname{Supp}(({\Delta}_Y + f^{\ast} E)^{>1}) \subseteqseteq \operatorname{Supp}(F) \cup \operatorname{Supp} ({\Delta}_Y^{>1}) \]
whence
\[ f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq f(\operatorname{Supp}(F)) \cup f(\operatorname{Supp} ({\Delta}_Y^{>1})) = \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
Therefore, by Remark \ref{nlc}, 
\[ \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) = \operatorname{Nlc}(X, {\Delta}+E) = f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
\end{proof}

Now we essentially follow the proof of Nakamaye's Theorem as in \cite[\S 10.3]{laz} and \cite[Thm. 0.3]{n}.
 
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main}}}  
\begin{proof}
We can assume that $D$ is a Cartier divisor. The issue is of course to prove that $\mathbf{B}_+(D) \subseteqseteq \operatorname{Null}(D)$, since the opposite inclusion holds on any normal projective variety, as explained in the introduction.

By Proposition \ref{max} and Remark \ref{nlc} there is an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ on $X$ such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})= X_{\mathrm{nlc}}$, so that $\dim \mathrm{Nlc}(X,{\Delta}) \leq 1$.

Let $A$ be an ample Cartier divisor such that $A-(K_X+{\Delta})$ is ample. As in \cite[Proof of Thm. 10.3.5]{laz}) we can choose $a, p ^{-1}n {\mathbb N}$ sufficiently large such that
\[ \mathbf{B}_+(D)= \mathbf{B}(aD-2A)= \mathbf{B}s|paD-2pA|. \]
By Lemma \ref{bs} there exist $c ^{-1}n {\mathbb N}$ and a Cartier divisor $E$ on $X$ such that
\[ \mathbf{B}_+(D)\cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) \]
and $E \equiv c(paD-2pA)=qaD-2qA$, where $q:= cp ^{-1}n {\mathbb N}$. 

Set $Z =  \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$. For $m \geq qa$, we get that 
\[ mD-qA-(K_X+{\Delta}+E)\equiv (m-qa)D+qA-(K_X+{\Delta}) \] 
is ample, whence $H^1(X, \mathcal{J}_{NLC}(X,{\Delta}+E) \otimes \mathcal{O}_X(mD-qA))=0$, for $m\geq qa$
by  \cite[Thm. 3.2]{f},  \cite[Thm. 4.4]{a}, so that the restriction map
\begin{equation} 
\label{uno}
H^0(X,\mathcal{O}_X(mD-qA))\to H^0(Z, \mathcal{O}_Z(mD-qA)) \mbox{\ is surjective for} \ m \geq qa.
\end{equation} 
By contradiction let us assume that there exists an irreducible component $V$ of $\mathbf{B}_+(D)$, such that $V \not \subseteqseteq \operatorname{Null}(D)$.
Now $V \subseteqseteq \mathbf{B}_+(D) \subseteqseteq \mathbf{B}(D-\frac{q}{m}A)\subseteqseteq \mathbf{B}s |mD-qA|$ for $m ^{-1}n {\mathbb N}$, whence the restriction map
\[ H^0(X,\mathcal{O}_X(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m  ^{-1}n {\mathbb N} \]
and therefore, by \eqref{uno}, also
\begin{equation} 
\label{due}
H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m \geq qa.
\end{equation} 
On the other hand $\dim V\geq 1$, as $\mathbf{B}_+(D)$ does not contain isolated points by \cite[Proposition 1.1]{elmnp2}(which holds on $X$ normal).
As $\dim \operatorname{Nlc}(X,{\Delta}) \leq 1$, this implies that $V$ is an irreducible component of $Z$. Moreover, as $V\not \subseteqseteq \operatorname{Null}(D)$, we have that $D_{|_V}$ is big.

Now,  by Remark \ref{ic},  $\mathcal{J}_{NLC}(X,{\Delta}+E)$ is integrally closed, and exactly as in  \cite[Proof of Thm. 10.3.5]{laz} (the proof of this part holds on any normal projective variety) it follows that, for $m \gg 0$, $H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA))$ is not zero, thus contradicting \eqref{due}.
This concludes the proof.
\end{proof}
\renewcommand{Proof}{Proof}

\renewcommand{Proof}{Proof of Corollary {\rm \ref{dim}}}  
\begin{proof}
Note that, on any normal projective variety $X$, we have $X_{\mathrm{nlc}} \subseteqseteq \operatorname{Sing}(X)$ (see for example \cite[Rmk 4.8]{cd}) and  if $\dim X\leq 3$, then $\dim \operatorname{Sing}(X) \leq 1$. Then just apply Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof}

\renewcommand{Proof}{Proof of Corollary {\rm \ref{mod}}}
\begin{proof}
By \cite[Thm. 0.9]{gkm} we know that $\operatorname{Null}(D) \subseteqseteq \partial \overline{M}_{g,n}$. On the other hand it is well-known
(see for example \cite[Lemma 10.1]{bchm}) that $(\overline{M}_{g,n}, 0)$ is klt, whence the conclusion follows by Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof}

\section{Restricted base loci on klt pairs}
\label{rests}

We first recall that, associated to a pseudoeffective divisor $D$, there are two more loci,  one that also measures how far $D$ is from being nef and another one that measures how far $D$ is from being nef and abundant.

\begin{defn}
\label{as2}
{\rm Let $X$ be a normal projective variety and let  $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. As in \cite[Def. 1.7]{bbp}, we define the {\bf non-nef locus}
\[ \operatorname{Nnef}(D) = \bigcup_{v : v(\|D\|) > 0} c_X(v) \]
where $v$ runs among all divisorial valuations on $X$, $c_X(v)$ is its center and  $v(\|D\|)$ is as in Definition \ref{as1}.

Let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$. As in \cite[Def. 2.18]{cd}, we define the {\bf non nef-abundant locus}
\[ \operatorname{Nna}(D) = \bigcup_{v : v(\langle D \rangle) > 0} c_X(v) \]
where again $v$ runs among all divisorial valuations on $X$ and  $v(\langle D \rangle)$ is as in Definition \ref{as1}.}
\end{defn}

In the sequel we will use the fact that, for $D$ big (\cite[Lemma 3.3]{elmnp1}) or even abundant (\cite[Prop. 6.4]{leh}), we have $v(\|D\|) = v(\langle D \rangle)$, while in general they are different when $D$ is only pseudoeffective (\cite[Rmk 2.16]{cd}).

We will also use (see \cite[page 2]{bfj} and references therein) 

\noindent {\bf Izumi's Theorem}
\label{iz}
{^{-1}t Let $X$ be a normal variety over an algebraically closed field $k$ and let $0 ^{-1}n X$ be a closed point. Let $m_0$ be the maximal ideal of the local ring ${\mathcal O}_{X,0}$ and set, for any $f  ^{-1}n {\mathcal O}_{X,0}$, $\operatorname{ord}_0 (f) = \operatorname{max}\{j \geq 0 :  f ^{-1}n m_0^j \}$.
For any divisorial valuation $v$ of $k(X)$ centered at $0$, there exists a constant $C = C(v) > 0$ such that
\[ C^{-1} \operatorname{ord}_0(f) \leq v(f) \leq C \operatorname{ord}_0(f).\]}
We start by proving an analogue of \cite[Prop. 2.8]{elmnp1} for $\operatorname{Nna}(D)$.

\begin{thm}
\label{nna}
Let $X$ be a normal projective variety, let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and let $v$ be a divisorial valuation on $X$.
Then  
\[ c_X(v)\subseteqseteq \operatorname{Nna}(D) \ \textrm{if and only if} \ v(\langle D \rangle)>0. \]
\end{thm}
\begin{proof}
We can assume that $D$ is Cartier and effective. By definition of $\operatorname{Nna}(D)$, we just need to prove that if $c_X(v) \subseteqseteq \operatorname{Nna}(D)$, then $v(\langle D \rangle)>0$. 

We first prove the theorem when $X$ is smooth.  For any $p ^{-1}n {\mathbb N}$ let $b(|pD|)$ be the base ideal of $|pD|$, $\mathcal{J}(X,\|pD\|))$ the asymptotic multiplier ideal and denote by $b_p$ and $j_p$ the corresponding images in $R_v$, the DVR associated to $v$. As in \cite[\S 2]{elmnp1}, we get
\begin{equation}
\label{j}
v(\langle D \rangle) = \lim_{p \to +^{-1}nfty} \frac{v(b_p)}{p} \geq \lim_{p \to +^{-1}nfty} \frac{v(j_p)}{p} = \sup_{p ^{-1}n {\mathbb N}} \{\frac{v(j_p)}{p}\}.
\end{equation}
By \cite[Cor. 5.2]{cd} we have the set-theoretic equality
\[ \operatorname{Nna}(D)=\bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}(X,\|pD\|)) \]
whence there exists $p_0 ^{-1}n {\mathbb N}$ such that $c_X(v) \subseteqseteq \mathcal{Z}(\mathcal{J}(X,\|p_0D\|))$, so that $v(j_{p_0})>0$ and \eqref{j} gives that $v(\langle D \rangle)>0$. 

We now prove the theorem for a divisorial valuation $\nu$ on $X$ such that $c_X(\nu) = \{x\}$ is a closed point.

As $c_X(\nu) \subseteqseteq \operatorname{Nna}(D)$, there exists a divisorial valuation $v_0$ on $X$ such that $v_0(\langle D \rangle)>0$ and $x ^{-1}n c_X(v_0)$.
Let $E_0$ be a prime divisor over $X$ such that  $v_0 = k \operatorname{ord}_{E_0}$ for some $k ^{-1}n {\mathbb N}$. We can assume that there is a birational morphism  $\mu:Y\to X$ from a smooth 
variety $Y$ such that $E_0 \subseteqset Y$. As $\mu(E_0)=c_X(\operatorname{ord}_{E_0}) = c_X(v_0 )$, there is a point $y ^{-1}n E_0$ such that $\mu(y) = x$. Let $\pi : Y' \to Y$ be the blow-up of $Y$ on $y$ with exceptional divisor $E_y$. For any $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$$\operatorname{ord}_{E_y}(G) =\operatorname{ord}_{E_y}(\pi^*(\mu^*G)) = \operatorname{ord}_{y}(\mu^*G) \geq \operatorname{ord}_{E_0}(\mu^*G) = \operatorname{ord}_{E_0}(G)$$
therefore $\operatorname{ord}_{E_y}(\langle D \rangle) \geq \operatorname{ord}_{E_0}(\langle D \rangle) = \frac{1}{k} v_0(\langle D \rangle)>0$.
Since $c_X(\operatorname{ord}_{E_y})=\{x\}$, by Izumi's Theorem applied twice, there exist $C > 0, C' > 0$ such that for all $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$\operatorname{ord}_{E_y}(G) \leq C' \operatorname{ord}_{x}(G) \leq C \nu(G)$.  Hence $\nu(\langle D \rangle) \geq \frac{1}{C} \operatorname{ord}_{E_y}(\langle D \rangle)>0$.

Finally let $v$ be any divisorial valuation on $X$ with $c_X(v)\subseteqseteq \operatorname{Nna}(D)$. As above there is a birational morphism $f : Z \to X$ from a smooth 
variety $Z$ and a prime divisor $E \subseteqset Z$ such that $v = h \operatorname{ord}_E$ for some $h ^{-1}n {\mathbb N}$. For every closed point $z ^{-1}n E$ we have that  $\nu:= \operatorname{ord}_z$ is a divisorial valuation with $c_X(\nu) \subseteqseteq c_X(\operatorname{ord}_E) \subseteqseteq \operatorname{Nna}(D)$ and $c_X(\nu)$ is a closed point. Thus, by what we proved above, we have that $\operatorname{ord}_z(\langle f^*(D) \rangle) = \operatorname{ord}_z(\langle D \rangle)>0$ for all $z ^{-1}n E$, so that $E \subseteqseteq \operatorname{Nna}(f^*(D))$. As $Z$ is smooth, we get $v(\langle D \rangle) = h\operatorname{ord}_E(\langle D \rangle) = h \operatorname{ord}_E(\langle f^*(D) \rangle)>0$.
\end{proof}

We next prove an analogous result for $\operatorname{Nnef}(D)$.

\begin{thm}
\label{nnef}
Let $X$ be a normal projective variety, let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$.
Then
$$c_X(v) \subseteqseteq \operatorname{Nnef}(D) \ \textrm{if and only if} \ v(\|D\|)>0.$$
\end{thm}

\begin{proof} Again we need to prove that $v(\|D\|) > 0$ if $c_X(v) \subseteqseteq \operatorname{Nnef}(D)$.
By \cite[Lemmas 2.13 and 2.12]{cd}, there exists a sequence of ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors $\{A_m\}_{m ^{-1}n {\mathbb N}}$
such that $\|A_m\| \to 0$, $D+A_m$ is a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor for all $m ^{-1}n {\mathbb N}$ and
$$\operatorname{Nnef}(D)=\bigcup_{m^{-1}n {\mathbb N}} \operatorname{Nnef}(D+A_m).$$
Then there is $m_0 ^{-1}n {\mathbb N}$ such that $c_X(v)\subseteqseteq \operatorname{Nnef}(D+A_{m_0})$. As $D+A_{m_0}$ is big, we have $\operatorname{Nnef}(D+A_{m_0}) = \operatorname{Nna}(D+A_{m_0})$, whence $v(\|D+A_{m_0}\|) = v(\langle D+A_{m_0}\rangle) >0$ by Theorem \ref{nna}.
Therefore $0 < v(\|D+A_{m_0}\|) \leq v(\|D\|) + v(\|A_{m_0}\|) = v(\|D\|)$.
\end{proof}

\begin{rem}
{\rm Note that, given a normal projective variety $X$, Theorems {\rm \ref{nna}} and {\rm \ref{nnef}} can be rewritten as follows (where $x$ is a closed point).

If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$ such that $\kappa(D)\geq 0$, then
$$\operatorname{Nna}(D)=\bigcup_{x ^{-1}n X}  \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\langle D\rangle)>0\}.$$

If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, then
$$\operatorname{Nnef}(D)=\bigcup_{x ^{-1}n X}  \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\|D\|)>0\}.$$}
\end{rem}

Next we will prove Theorem {\rm \ref{main2}. We will use a singular version (see for example \cite[Def. 2.2]{cd}) of standard asymptotic multiplier ideal sheaves \cite[Ch. 11]{laz}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main2}}}
\begin{proof}
In both cases we have that $\operatorname{Nnef}(D) = \mathbf{B}_-(D)$ by \cite[Thm. 1.2]{cd}, whence also $\operatorname{Nna}(D) = \mathbf{B}_-(D)$ in case (i). Then (ii) follows by Theorem \ref{nnef} and the first equivalence in (i) by Theorem \ref{nna}. To complete the proof of (i) we need to show that if $\limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty$ then $v(\langle D\rangle) > 0$, the reverse implication being obvious. We will proceed similarly to \cite[Proof of Prop. 2.8]{elmnp1} and \cite[Proof of Lemma 4.1]{cd}. If $v(\langle D\rangle) = 0$, by what we just proved, we have that $c_X(v) \not\subseteqseteq \mathbf{B}_-(D)$ and, by \cite[Cor. 5.2]{cd}, we have the set-theoretic equality
\[ \mathbf{B}_-(D) = \bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}((X,{\Delta});\|pD\|)) \]
where $\mathcal{J}((X,{\Delta});\|pD\|)$ is as in \cite[Def. 2.2]{cd}. Therefore $c_X(v) \not\subseteqseteq  \mathcal{Z}(\mathcal{J}((X,{\Delta}); \|pD\|))$ for any $p ^{-1}n {\mathbb N}$. Let $H$ be a very ample Cartier divisor such that $H - (K_X + {\Delta})$ is ample and let $n = \dim X$. By Nadel's vanishing theorem \cite[Thm. 9.4.17]{laz}, we deduce that $\mathcal{J}((X,{\Delta});\|pD\|) \otimes \mathcal{O}_X((n + 1)H + pD)$ is $0$-regular in the sense of Castelnuovo-Mumford, whence globally generated, for every $p ^{-1}n {\mathbb N}$, and therefore $c_X(v) \not\subseteqseteq \mathbf{B}s|(n+1)H + pD|$. On the other hand, as $D$ is big, there is $m_0 ^{-1}n {\mathbb N}$ such that $m_0 D \sim (n + 1)H + E$ for some effective Cartier divisor $E$. Hence, for any $m \geq m_0$, we get $v(|mD|) = v(|(m-m_0)D + (n + 1)H + E|) \leq  v(|(m-m_0)D + (n + 1)H|) + v(|E|) =  v(|E|)$ and the theorem follows. 
\end{proof}
\renewcommand{Proof}{Proof}

We end the section with an observation on the behavior of these loci under birational maps.

\begin{cor}
Let $f:Y \to X$ be a projective birational morphism between normal projective varieties. Then:

\begin{itemize}
^{-1}tem [(i)] For every ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $D$ on $X$ such that $\kappa(D)\geq 0$, we have
$$\operatorname{Nna}(f^*(D))=f^{-1}(\operatorname{Nna}(D));$$
^{-1}tem [(ii)] For every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have 
$$\operatorname{Nnef}(f^*(D))=f^{-1}(\operatorname{Nnef}(D));$$
^{-1}tem [(iii)] If there exist effective Weil ${\mathbb Q}$-divisors ${\Delta}_X$ on $X$ and ${\Delta}_Y$ on $Y$ such that $(X,{\Delta}_X)$ and $(Y,{\Delta}_Y)$ are klt pairs,
then, for every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have 
$$\mathbf{B}_-(f^*(D))=f^{-1}(\mathbf{B}_-(D))$$
\end{itemize}
\end{cor}
\begin{proof}
To see (i), for every closed point $y ^{-1}n Y$, let $v_y$ be a divisorial valuation such that $c_Y(v_y)=\{y\}$.
Then, by Theorem \ref{nna}, we have, 
\[ y^{-1}n f^{-1}(\operatorname{Nna}(D)){\mathcal L}eftrightarrow \{f(y)\}=c_X(v_y)\subseteqseteq \operatorname{Nna}(D){\mathcal L}eftrightarrow \]
\[ {\mathcal L}eftrightarrow v_y(\langle f^*(D) \rangle) = v_y(\langle D\rangle)>0{\mathcal L}eftrightarrow \{y\}=c_Y(v_y)\subseteqseteq \operatorname{Nna}(f^*(D)). \]
Now (ii) can be proved exactly in the same way by using Theorem  \ref{nnef}, while (iii) follows from (ii) and \cite[Thm. 1.2]{cd}.
\end{proof}

\end{document}