text:
\section{Introduction}
Let $S=\mathbb{C}[x,y,z]$ be the polynomial ring in three variables $x,y,z$ with complex coefficients. We denote by $ \partial_x, \partial_y, \partial_z$ the partial derivations with respect to $x,y,z$ respectively.
Let $Der(S)=\{\partial =a \partial_x + b  \partial_y+ c \partial_z \ : \ a,b,c \in S \}$ be the free $S$-module of $\mathbb{C}$-derivations of the polynomial ring $S$.
\begin{definition}
\label{def2} Let $g \in S$ be a polynomial.
The $S$-module $D(g)$ of derivations of $S$ {\it preserving the principal ideal} $(g) \subset S$ is by definition
$$D(g)=\{\partial \in Der(S) \ : \ \partial g \in (g) \}.$$
Moreover, the $S$-module $D_0(g)$ of derivations of $S$ {\it  killing the polynomial} $g$  is by definition
$$D_0(g)=\{\partial \in Der(R) \ : \ \partial g =0 \}.$$
\end{definition}
When $g$ is a homogeneous polynomial, then both modules $D(g)$ and $D_0(g)$ are graded $S$-modules and one has
$$D(g)=D_0(g) \oplus S(-1) \cdot E,$$
 where $E=x\partial_x+y\partial_y+z\partial_z$ denotes the {\it Euler derivation.} The curve $C_g: g=0$ in $\mathbb{P}^2$ is said to be {\it  free} if
 $D(g)$ or, equivalently, $D_0(g)$ is a free graded $S$-module.

 Note that $D_0(g)$ can be identified with the $S$-module of all Jacobian relations for $g$, namely to
$$AR(g)=\{(a,b,c) \in S^3 \ : \ ag_x+bg_y+cg_z=0 \},$$
where $g_u=\partial_u g$, for $u=x,y,z$. An important numerical invariant associated to a reduced curve $C:f=0$ in the projective plane $\mathbb{P}^2$ is the {\it minimal degree of a derivation killing $f$} or,
equivalently, the {\it minimal degree of a Jacobian relation (syzygy) for $f$}. This is defined by
$$mdr(f)= \min \{s \in \mathbb{N}: \ D_0(f)_s  \ne 0 \}=\min \{s \in \mathbb{N}: \ AR(f)_s  \ne 0 \}.$$
It can be used for instance to characterize the free or the nearly free curves, see \eqref{F1} and \eqref{F2} below.
In this note we study the relations of this invariant with a decomposition of $C$ as a union of two curves $C_1:f_1=0$ and $C_2:f_2=0$, without common irreducible components. In particular, we would like to relate $r=mdr(f)=mdr(f_1f_2)$ to $r_j=mdr(f_j)$ for $j=1,2$. The case when $C_1$ is a  a line arrangement and $C_2$ is a line was studied in detail in \cite{ADS}.

In section 2 we recall some basic notations and facts, for instance the definition of the Jacobian module $N(f)$ and of free, nearly free and plus one generated curves which play a key role in this paper.

Then we consider in section 3 the case when $C_2$ is a line and $C_1$ is  any reduced curve, not having $C_2$ as a component. We study in Theorem \ref{thm1} the behavior of our invariant $mdr(f)$ when $C_2$ is a member of a pencil of lines in $\mathbb{P}^2$, under the assumption that we know not only $r_1$, but also a non trivial derivation in $D_0(f_1)_{r_1}$. Several examples are given in section 4.

We study in section 5 the general case of two curves $C_1$ and $C_2$, and get bounds for $r=mdr(f)$ in terms of the degrees $d_j=\deg(f_j)$  and of the invariants $r_j$ for $j=1,2$, see Theorem \ref{thm10}. As an example, we discuss in Proposition \ref{prop10} all the possibilities when both $C_1$ and $C_2$ are smooth conics.

Finally, in section 6, {\it we assume that all the singularities of $C_1$ and $C$ are quasihomogeneous and that $C_2$ is a smooth curve (most of the time $C_2$ is also supposed to be rational).}
Under this assumption, we may use a key result by Schenck, Terao and Yoshinaga, see \cite{STY}, to get finer information on $r$. Our Theorem \ref{thm20} gives a description of the cohomology exact sequence associated to the short sheaf exact sequence obtained in \cite{STY}, paying special attention to the description of the morphisms between the corresponding cohomology groups.

This approach was already used in  \cite{STY} to relate the freeness of $C_1$ to the freeness of $C$. Here we show that even when the curve $C_1$ is not free, one can obtain valuable information on $r$ using this approach.
This idea works best when the Jacobian module $N(f_1)$ is small, and this explains why we consider mostly free and nearly free curves $C_1$.
Sometimes the determination of $r$ is rather easy, using just the knowledge of the numerical invariant $r_1$, as in most examples in section 6.
 In  Example \ref{ex21E} we present a situation where one needs to use the morphisms in the exact sequence given by Theorem \ref{thm20}, namely the multiplication by $f_2^2$ between the two Jacobian modules $N(f_1)$ and $N(f)$.
As a by-product, under the assumption for this final section,
we get lower bounds on the number of points in the intersection $C_1 \cap C_2$ in terms of $r_1$ when the curve $C_1$ is  free or nearly free and $C_2$ is either a line or a smooth conic, see Corollary \ref{corF} and  Corollary \ref{cor21NF}.

Finally we give some geometrical criteria, {\it the first ones of this type in the existing literature as far as we know}, for a line $L$  in $\mathbb{P}^2$ to be a jumping line for the rank 2 vector bundle  $T\langle C \rangle $  of logarithmic vector fields along a reduced curve $C$, see Theorem \ref{thmJ} and Example \ref{ex1.1} where this is applied to Thom-Sebastiani curves.

\medskip

We would like to thank Laurent Bus\'e, Piotr Pokora, \c Stefan Toh\u aneanu and Masahiko Yoshinaga for useful discussions related to this paper.

\section{Prerequisites}

In this section we recall some basic facts, see for instance  \cite{CDI,DSt3syz}. For any degree $e$ reduced homogeneous polynomial $g \in S_e$,  let $N(g)= \widehat{J}_g/J_g$ be the Jacobian module of $g$, with $J_g$ the Jacobian ideal of $g$ in $S$, spanned by the partial derivatives $g_x,g_y$, $g_z$ of $g$, and $\widehat{J}_g$ the saturation of the ideal $J_g$ with respect to the maximal ideal
${\bf m}=(x,y,z)$ in $S$.
We set   $n(g)_j=\dim N(g)_j$,  $T_g=3(e-2)$ and recall that  we have
\begin{equation}\label{LPN}
n(g)_0\leq n(g)_1\leq\ldots\leq n(g)_{\lfloor\frac{T_g}{2}\rfloor-1}\leq n(g)_{\lfloor\frac{T_g}{2}\rfloor}\geq n(g)_{\lfloor\frac{T_g}{2}\rfloor+1}\geq\ldots\geq n(g)_T.\end{equation}
For a reduced curve $C_g:g=0$, we consider the following  invariants   $$\sigma(C_g)=\min\left\{j : n(g)_j\neq 0\right\}=\indeg(N(f))
\text{ and }
\nu(C_g)=\max\left\{n(g)_j\right\}_j.$$
The self duality of the graded $S-$module $N(g)$ implies
$
 n(g)_j=n(g)_{T_g-j},
$
for any integer $j$,  see \cite{Se}. In particular $n(g)_k>0$ exactly when $\sigma(C_g) \leq k \leq T_g-\sigma(C_g).$\\
The  form of the minimal graded free resolution for the Milnor algebra $M(g)=S/J_g$
is
\begin{equation}\label{RMM}0\to \displaystyle{\oplus_{i=1}^{m-2}S(-e_i)}\to \displaystyle{\oplus_{i=1}^{m}S(1-e-d'_i)}\to S^3(1-e)\to S,\end{equation}
with $e_1\leq e_2\leq\ldots\leq e_{m-2}$ and $1 \leq d'_1\leq d'_2\leq\cdots\leq d'_m$. In this case the curve $C_g$ is said to be an $m$-{\it syzygy curve}. The first degree $r_g=d_1'$ is denoted by $mdr(g)$ and is the {\it minimal degree of a Jacobian relation (syzygy) for $g$}.
It follows from \cite[Lemma 1.1]{HS12} that one has $$e_j=e+d'_{j+2}-1+\epsilon_j,$$ for $j=1,\ldots, m-2$ and some integers $\epsilon_j\geq 1.$
The minimal resolution of $N(g)$ obtained from (\ref{RMM}), by \cite[Proposition 1.3]{HS12}, is
 $$0\to \displaystyle{\oplus_{i=1}^{m-2}S(-e_i)}\to\displaystyle{\oplus_{i=1}^mS(-\ell_i)}\to\displaystyle{\oplus_{i=1}^m}S(d'_i-2(e-1))\to \displaystyle{\oplus_{i=1}^{m-2}S(e_i-3(e-1))},$$
 where $\ell_i=e+d'_i-1$. It follows that
\begin{equation}\label{sigma}
 \sigma(C_g)=3(e-1)-e_{m-2}=2(e-1)-d'_m-\epsilon_{m-2}.
\end{equation}
The following are important special cases, see \cite{Abe18,DStRIMS,DSt3syz}.
Here $\tau(C_g)$ is the total Tjurina number of the curve $C_g$, which is the same as the degree of the Jacobian ideal $J_g$.
\begin{enumerate}

\item $C_g$ is a free curve if and only if $m=2$ and $d_1'+d'_2=e-1$. In this case $\nu(C_g)=0$ and  $N(g)=0$. The degrees $(d_1',d_2')$ are the {\it exponents} of the free curve $C_g$.
Moreover, a reduced curve $C_g$ is free if and only if
\begin{equation}\label{F1}
 \tau(C_g)=(e-1)^2-r_g(e-r_g-1),
\end{equation}
see \cite{Dmax,dPW}.

\item $C_g$ is a nearly free curve if and only if $m=3$ and $d_1'+d'_2=e$, $d_3'=d_2'$. In this case $\nu(C_g)=1$ and  $\sigma(C_g)=e+d_1'-3$. The degrees $(d_1',d_2')$ are the {\it exponents} of the nearly free curve $C_g$.
Moreover, $C_g$ is nearly free if and only if
\begin{equation}\label{F2}
 \tau(C_g)=(e-1)^2-r_g(e-r_g-1)-1,
\end{equation}
see \cite{Dmax}.

\item $C_g$ is a plus one generated curve if and only if $m=3$ and $d_1'+d'_2=e$, $d_3'>d_2'$, see \cite{Abe18} for the case $C_g$ a line arrangement and \cite{DSt3syz} for the general case. In this case $\nu(C_g)=d_3'-d_2'+1$ and   $\sigma(C_g)=2e-d_3'-3$.

\end{enumerate}

\section{Adding a line to a reduced curve}

Consider a reduced plane curve $C_1:f_1=0$  of degree $d_1$ in $\mathbb{P}^2$ such that $mdr(f_1)=r_1$. Let $L$ be a line in $\mathbb{P}^2$, which is not an irreducible component of $C_1$ and consider the curve $C=C_1 \cup L: f=0$.
Then $C$ has degree $d=d_1+1$, and we denote $r=mdr(f)$.
In this section we analyze the relation between $r$ and $r_1$, starting with the following result.
\begin{prop}
\label{prop1}
With the above notation, one has $r_1 \leq r \leq r_1+1$.
\end{prop}
\proof
Choose a coordinate system on $\mathbb{P}^2$ such that the line $L$ is given by $z=0$, and hence $f=zf_1$. Let
\begin{equation}
\label{syzf}
af_x+bf_y+cf_z=0
\end{equation}
be a Jacobian syzygy of minimal degree $r$ for $f$, and
\begin{equation}
\label{syzf'}
a_1f_{1x}+b_1f_{1y}+c_1f_{1z}=0
\end{equation}
 a Jacobian syzygy of minimal degree $r_1$ for $f_1$.
Note that one has
\begin{equation}
\label{e1}
f_x=zf_{1x}, \  f_y=zf_{1y} \text{ and } f_z=zf_{1z}+f_1=\frac{1}{d_1}xf_{1x}+\frac{1}{d_1}yf_{1y} +\frac{d}{d_1}zf_{1z}.
\end{equation}
Using \eqref{syzf} we get
$$azf_{1x}+bzf_{1y}+c(zf_{1z}+f_1)=0,$$
and hence the polynomial $c$ is divisible by $z$, so we can write $c=zc'$. Indeed, note that $f_1$ is not divisible by $z$ by our assumptions. With this notation, and using \eqref{e1}, we get after division by $z$ the following equation.
\begin{equation}
\label{e2}
(a+\frac{1}{d_1}c'x)f_{1x}+(b+\frac{1}{d_1}c'y)f_{1y}+\frac{d}{d_1}c'zf_{1z}=0.
\end{equation}
This implies $r_1 \leq r$.
Similarly, using \eqref{syzf'} and \eqref{e1} we get
\begin{equation}
\label{e3}
(d_1a_1z-c_1x)f_x+(d_1b_1z-c_1y)f_y+c_1d_1zf_z=0.
\end{equation}
Note that this is a non trivial syzygy, namely one cannot have
$$d_1a_1z-c_1x= d_1b_1z-c_1y=c_1d_1z= 0.$$
This implies  $r \leq r_1+1$.
\endproof
\begin{rk}
\label{rk01}
With the above notation, if $z$ divides  $c_1$, the coefficient of $f_{1z}$ in \eqref{syzf'}, then all the coefficients in \eqref{e3} are divisible by $z$,
and hence after simplification by $z$ we get $r=r_1$ in this case. When $\dim D_0(f_1)_{r_1}>1$, there is a choice of the syzygy \eqref{syzf'} within a linear system, and some choices may be better than others, i.e. for the good ones $z$ divides  $c_1$, see Example \ref{exPOG} below for such a situation.
\end{rk}
To say more about
the value of $r$, it is convenient to look not only at a single line $L$, but at all the lines in a pencil.
The pencil we consider is formed by all the lines in $\mathbb{P}^2$ passing through a point $p \in \mathbb{P}^2$, which may or may not be on the curve $C_1$. We choose a coordinate system on $\mathbb{P}^2$ such that $p=(1:0:0)$,
hence a line in the pencil has the equation $L_u: sy+tz=0$ for some
$u=(s:t) \in \mathbb{P}^1$. Assume that \eqref{syzf} and
\eqref{syzf'} are minimal degree Jacobian syzygies for $f=(sy+tz)f_1$ and
respectively for $f_1$, with respect to this coordinate system.
Note that the coefficients $a_1,b_1,c_1$ are known and independent of $u$, since they depend only on $C_1$ and the choice of the coordinate system.
Let
$$r=d'_1(f) \leq d'_2(f) \leq \dots \leq d'_m(f)$$
 be the degrees of a minimal set of generators for $AR(f)$ coming from the resolution \eqref{RMM} of the Milnor algebra $M(f)$, which depend in general on $u$, see Example \ref{ex2} below. Elementary computations similar to those done above yield the following syzygy
\begin{equation}
\label{e4}
A_uf_x+B_uf_y+C_uf_z=0,
\end{equation}
where $A_u=d(sy+tz)a_1-x(sb_1+tc_1)$, $B_u=d(sy+tz)b_1-y(sb_1+tc_1)$ and finally
$C_u=d(sy+tz)c_1-z(sb_1+tc_1)$.
Using this syzygy, we can prove the following result.
\begin{thm}
\label{thm1}
With the above notation, if $sy+tz$ is a factor of $sb_1+tc_1$, then $r=r_1$.
If $sy+tz$ is not a factor of $sb_1+tc_1$, then either
\begin{enumerate}
\item $r=r_1+1$, or

\item $r=r_1$ and $d'_2(f) \leq r+1$.
\end{enumerate}
Moreover, the case $(2)$ is impossible when $2r_1<d_1-1$, or when $2r_1=d_1-1$ and $C$ is not free.

\end{thm}

\proof
The first claim is obvious. Indeed, when $sy+tz$ is a factor of $sb_1+tc_1$, the coefficients $A_u$, $B_u$ and $C_u$ can be divided by $sy+tz$, and the syzygy \eqref{e4} yields a syzygy of degree $r_1$. Since $r \geq r_1$ by Proposition \ref{prop1}, we get $r=r_1$.
Assume now that $sy+tz$ is not a factor of $sb_1+tc_1$. Then we claim that
the syzygy \eqref{e4} is primitive, i.e. it is not a multiple of a syzygy of strictly lower degree. In other words, we have to show that $A_u$, $B_u$ and $C_u$ have no common factor in this case.
Note that $yA_u-xB_u=d(sy+tz)(ya_1-xb_1)$, $zA_u-xC_u=d(sy+tz)(za_1-xc_1)$ and
$zB_u-yC_u=d(sy+tz)(zb_1-yc'_1$.
Let $D$ be a common irreducible factor of $A_u$, $B_u$ and $C_u$, supposed to be a homogeneous polynomial of degree $>0$. It it clear that $D$ cannot be $sy+tz$, since $sy+tz$ is not a factor of $sb_1+tc_1$. Hence $D$ has to divide the polynomials $m_{12}=ya_1-xb_1$, $m_{13}=za_1-xc_1$ and
$m_{23}=zb_1-yc_1$.
Recall now the construction of the Bourbaki ideal $B(C_1, \rho'_1)$ associated to the curve $C_1$ and to the minimal degree syzygy $\rho'_1$
given by \eqref{syzf'}, as described in \cite[Section 5]{DStJump}.
It follows that the Bourbaki ideal $B(C_1, \rho'_1)$ is contained in the principal ideal generated by $D$. This is a contradiction, since
the Bourbaki ideal $B(C_1, \rho'_1)$ defines a subscheme which is either empty (when $C_1$ is a free curve), or zero-dimensional, see
\cite[Theorem 5.1]{DStJump}.

Therefore the syzygy \eqref{e4} is indeed primitive.
It follows that either $r=r_1+1$, or $r=r_1$ and $d_2(f) \leq r+1$.
Note that in this latter case we have
$$d_1=d-1 \leq d'_1(f) +  d'_2(f) \leq r_1+r_1+1=2r_1+1.$$
Indeed, recall that
$d-1=d'_1(f)+d'_2(f)$
exactly when $C$ is free, and
$d-1<d'_1(f)+d'_2(f)$ otherwise, see for instance \cite{ST}.
\endproof
\begin{prop}
\label{prop01}
With the notation from Theorem \ref{thm1}, we have the following equivalent properties.
\begin{enumerate}
\item $sy+tz$ is a factor of $sb_1+tc_1$ for infinitely many $u=(s:t) \in \mathbb{P}^1$;

\item $sy+tz$ is a factor of $sb_1+tc_1$ for all $u=(s:t) \in \mathbb{P}^1$;

\item the reduced curve $C_1:f_1=0$ is the union of the curve $h=0$ with a pencil of lines $g=0$ passing through the point $p=(1:0:0)$.

\end{enumerate}

\end{prop}

\proof
The fact that (2) implies (1) is clear. First we show that (1) implies (3).
Note that $sy+tz$ is a factor of $sb_1+tc_1$ for infinitely many $u=(s:t) \in \mathbb{P}^1$ if and only if there is a polynomial $h$ of degree $r_1-1$ such that
$b_1=yh$ and $c_1=zh$. Replacing these values in \eqref{syzf'} we conclude that $f_{1x}$ is divisible by $h$, say $f_{1x}=hg$, with $\deg g=d_1-r_1\geq 1$. If we divide the syzygy \eqref{syzf'} by $h$, we get
\begin{equation}
\label{e5}
a_1g+yf_{1y}+zf_{1z}=0
\end{equation}
or, equivalently,
\begin{equation}
\label{e6}
a_1g+d_1f_1-xf_{1x}=0.
\end{equation}
It follows that $g$ is a common factor of $f_1$ and $f_{1x}$. To conclude the proof of the implication $(1) \implies (3)$ we use the following result, communicated to us by Laurent Bus\' e.

\begin{lem}
\label{lempropBL}
With the above notation, assume that $g=G.C.D.(f_1,f_{1x})$ has degree $\geq 1$. Then $g$ is a homogeneous polynomial in $y$ and $z$ only, and
$$f_1(x,y,z)=g(y,z)h(x,y,z),$$
for some homogeneous polynomial $h \in S$. In geometric terms, the reduced curve $C_1:f_1=0$ is the union of the curve $h=0$ with a pencil of lines $g=0$ passing through the point $p=(1:0:0)$.
\end{lem}

\proof

Let $A$ be an irreducible common factor of $f_1$ and $f_{1x}$, such that
$f_1=A U$ for $U \in S$. This implies
$f_{1x}=A_xU+AU_x$, and hence, if $A_x \ne 0$, then $A$ has to divide
$U$. Indeed, $A$ cannot divide $A_x$ since $\deg A_x <\deg A$.
But this contradicts the fact that $C_1:f_1=0$ is a reduced curve.
Hence $A_x=0$, in other words $A$ is a homogeneous polynomial in $y$ and $z$ only. Since $g$ is a product of such polynomials, the claim is proved.
\endproof

Finally we show that (3) implies (2). Assume that $g=G.C.D.(f_1,f_{1x})$ has degree $\geq 1$, then
one can define $a_1$ using the above equation \eqref{e6}. Then, if we multiply the equation  \eqref{e5} by $h=f_{1x}g^{-1}$, we get a primitive syzygy of the form \eqref{syzf'}, where $b_1=yh$ and $c_1=zh$.
\endproof

\section{Examples}

\begin{ex}
\label{ex1}
Assume $C_1$ is an irreducible nodal curve and $L_u:sy+tz=0$ is a line such that $C=C_1 \cup L_u$ is nodal.
Then it is known that $r_1=d_1-1$ and $r=d-2=d_1-1$, see \cite{DStEdin, E}. Note that one has in this case $d_2(f)=r+1$,  see \cite[Theorem 4.1] {DStEdin}.
Hence  the case (2) of  Theorem \ref{thm1} might occur.
\end{ex}

\begin{ex}
\label{ex2}
Consider the rational cuspidal curve $C_1:f_1=xy^{d_1-1}+z^{d_1}=0$, $d_1 \geq3$, which is nearly free, and $L_u:sy+tz=0$ a line passing through the singular point $p=(1:0:0)$.
Then the syzygy \eqref{syzf'} becomes
$$(d_1-1)xf_{1x}-yf_{1y}=0.$$
Hence $sb_1+tc_1=-sy$ is divisible by $sy+tz$ only for $(s:t)=(1:0)$ and for $(s:t)=(0:1)$, and we get in these cases $r=r_1=1$ using Theorem \ref{thm1} as we see now.
The curve $C': f=xy^{d_1}+yz^{d_1}=0$ corresponding to $(s:t)=(1:0)$
is free, the two generating syzygies being
$$(d_1)^2xf_x-d_1yf_y+zf_z=0$$
and
$$d_1z^{d_1-1}f_x-y^{d_1-1}f_z=0$$
satisfying $d'_1(f)+d'_2(f)=1+ (d_1-1)=d-1$.
The curve $C'': f=xy^{d_1-1}z+z^{d_1+1}=0$ corresponding to $(s:t)=(0:1)$,
is nearly free with exponents $d'_1(f)=1$, $d'_2(f)=d'_3(f)=d-1$. Indeed, note that the curve $C''$ has two singularities, namely $p=(1:0:0)$ and
$q=(0:1:0)$. The singularity at $q$ is a simple node $A_1$, and the singularity at $p$ is given in local coordinates $y'=y/x$ and $z'=z/x$ by
$(y')^{d_1-1}z'+(z')^{d_1+1}=0$. This is a quasi homogeneous singularity,
with weights $wt(z')=d^{-1}$ and $wt(y')=d_1[(d_1-1)d]^{-1}$.
It follows that
$$\tau(C'',p)=\mu(C'',p)=d^2-3d+1$$
and hence the total Tjurina number of $C''$ is given by
$$\tau(C'')=\tau(C'',q)+ \tau(C'',p)=d^2-3d+2.$$
The fact that $C''$ is nearly free follows now from \eqref{F2}.

For $d_1 \geq 4$ and for $L_u:y+z=0$, we have $r=r'+1=2$
by Theorem \ref{thm1}, since $2r_1<d_1-1$ in this case.
The corresponding curves $C_u$ are again nearly free, but this time with exponents
$d'_1(f)=2$, $d'_2(f)=d'_3(f)=d-2$. To see this, one notes that a curve $C_u$ in this family has two singularities, a node and a semi quasi homogeneous singularity  $(C_u,p):g(y',z')=g_0(y',z')+g_+(y',z')=0$,
where $g_0$ is quasi homogeneous and $g_+(y',z')$ is the sum of two monomials of strictly higher degree. Working in the Milnor algebra $M(g_0)$, we see that the Tjurina algebra of $g$ is isomorphic to the quotient $M(g_0)/(y^{d_1})$. This implies that
$$\mu(C_u,p)=(d_1)^2-d_1-1 \text{ and } \tau(C_u,p)=(d_1-1)^2+1.$$
It follows that $\tau(C_u)=(d_1-1)^2+2=(d-2)^2+2=(d-1)^2-2(d-3)-1$,
showing that $C_u$ is nearly free by \eqref{F2}.

\end{ex}

\begin{ex}
\label{ex3}
Let $C_1:f_1=(y^2-2xy+z^2)(y^2+4xy+z^2)=0$, be the union of two smooth conics tangent at one point $p=(1:0:0)$ and meeting transversely at $q_{\pm}=(0:1:\pm i)$. Then using Singular we see that $r_1=2$ and a minimal degree derivation is given by
$$\partial'=xz\partial_x-yz\partial_y+y^2\partial_z.$$
Then the equation  $sy+tz$ of a line $L$ passing through the tangency point $p$ divides
$$sb_1+tc_1=y(ty-sz)$$
if and only if either $(s:t)=(1:0)$ or $(s:t)=(1:\pm i)$.

The case $(s:t)=(1:0)$ corresponds to a common tangent $y=0$ to the two conics at $p$. Using Singular, we see that the corresponding curve
$$C: f=yf_1=y(y^2-2xy+z^2)(y^2+4xy+z^2)=0$$
is free with exponents $(2,2)$, in particular $r=2=r_1$ as predicted by Theorem \ref{thm1}.

The case $(s:t)=(1:\pm i)$ corresponds to a line joining the tangency point $p$ to one of the two nodes $q_{\pm}$ of $C_1$.
Using Singular, we see that the corresponding curve
$$C: f=(y \pm iz)f_1=(y \pm iz)(y^2-2xy+z^2)(y^2+4xy+z^2)=0$$
is nearly free with exponents $(2,3)$, in particular, again $r=2=r_1$ as predicted by Theorem \ref{thm1}.

Finally, to see what happens when $sy+tz$ does not  divide
$sb_1+tc_1=y(ty-sz)$, namely when the line through $p$ is general, we consider the special case $(s:t)=(1:1)$. Using Singular, we see that the corresponding curve
$$C: f=(y +z)f_1=(y +z)(y^2-2xy+z^2)(y^2+4xy+z^2)=0$$
is a maximal Tjurina curve of type $(d,r)=(5,3)$, see \cite{DStmax} for the definition and the properties of such curves, and in particular $C$ has exponents  $(3,3,3,3)$. Hence $r=3=r_1+1$. Note that we can show that for any line  $L:sy+tz=0$ with $t \ne 0$, the singularity of $C$ at $p$ is of type $D_6$. Indeed, it follows easily that this singularity is semi weighted homogeneous of type
$(2,1;5)$, where $wt(y)=2$ and $wt(z)=1$. The claim follows using \cite[Corollary (7.39)]{RCS}. In particular, when $sy+tz$ does not  divide
$sb_1+tc_1=y(ty-sz)$, we always have $\tau(C)=10$, since there are 4 nodes $A_1$  on $C$ in addition to the $D_6$ singularity.

\end{ex}

\begin{ex}
\label{exPOG}
Let $C_1: f_1=(y^2-xz)^2+y^2z^2+z^4=0$ be the curve considered in
 \cite[Example 4.1]{DSt3syz}. This curve is plus one generated with exponents $(d_1',d_2',d_3')=(2,2,3)$, in particular $\dim D_0(f_1)_2=2$.
 If we choose the right element in $D_0(f_1)_2$, namely
 $$ \partial'=(2xy+3yz)\partial_x+(xz+2z^2)\partial_y-yz\partial_z,$$
 then $z$ divides the coefficient of $\partial_z$, and
it follows that $r=r_1=2$ by Theorem \ref{thm1}.

\end{ex}

\section{The general case: the union of two curves}

Let $C_1:f_1=0$ and $C_2:f_2=0$ be two reduced curves in $\mathbb{P}^2$, without common irreducible components. We denote $d_j=\deg f_j$ and $r_j=mdr(f_j)$ for $j=1,2$. Consider now the union of the two curves
$C:f=f_1f_2=0$, and let $d=d_1+d_2=\deg f$ and $r=mdr(f)$.
\begin{thm}
\label{thm10}
With the above notation, one has the following.
\begin{enumerate}
\item If $\delta_1 \in D_0(f_1)$, then
$$\delta=f_2\delta_1-\frac{\delta_1(f_2)}{d}E \in D_0(f),$$
where $E=x\partial_x+y\partial_y+z\partial_z$ denotes the Euler derivation. In particular
$$r\leq \min\{r_1+d_2, r_2+d_1\}.$$

\item  $D_0(f) \subset D(f_1)\cap D(f_2)$. More precisely,  for $\delta \ne 0$, one has $\delta \in D_0(f)$  if and only if $\delta$ can be written in a unique way in the form
$$ \delta=\frac{h}{d_1}E+ \delta_1 = -\frac{h}{d_2}E+ \delta_2, $$
where $h \in S$ and $\delta_j \in D_0(f_j)$ are non-zero derivations.
In particular
$$r \geq \max \{r_1, r_2\}.$$
\end{enumerate}

\end{thm}
\proof
To prove $(1)$, first we check that $\delta( f)=0$. Then we note that
$\delta \ne 0$ if $\delta_1 \ne 0$. Indeed, if $\delta_1(f_2)=0$, then clearly $\delta =f_2\delta_1\ne 0$. When $\delta_1(f_2)\ne 0$, note that
$$\delta (f_1)=\frac{d_1f_1\delta_1(f_2)}{d} \ne 0.$$
The last claim follows by noting that if $\delta_1$ is a homogeneous derivation then also $\delta$ is a homogeneous derivation. Moreover, the roles played by $f_1$ and $f_2$ are symmetric.

To prove $(2)$,
start with $\delta \in D_0(f)$ and hence
$$\delta (f)=f_2\delta (f_1)+f_1\delta (f_2)=0.$$
If $\delta (f_1)=0$, then $\delta (f_2)=0$ and hence $\delta \in D_0(f_1)\cap D_0(f_2)$.
If $\delta( f_1)\ne 0$, then $f_2$ divides the product $f_1\delta (f_2)$. Since
$f_1$ and $f_2$ have no common factor by our assumptions, it follows that $f_2$ divides $\delta (f_2)$, hence $\delta \in D(f_2)$. This is possible only if   $\delta \in D(f_1)$ as well. It follows that we  can write $\delta \in D_0(f)$  in the form
$$\delta=h_jE+\delta_j$$
where $h_j \in S$ and $\delta_j \in D_0(f_j)$. Clearly $\delta_j \ne 0$, since otherwise $\delta (f) \ne 0$.
Then $\delta (f_1)= d_1h_1f_1$ and $\delta (f_2)=d_2h_2f_2$.
It follows that
$$0=\delta (f)=\delta (f_1)f_2+f_1\delta (f_2)=f_1f_2(d_1h_1+d_2h_2).$$
Then one implication in the claim follows by taking $h=d_1h_1=-d_2h_2$. The other implication is obvious.
\endproof

\begin{rk}
\label{rk10}The inequality $r \geq \max \{r_1, r_2\}$ was already noticed in \cite[Proposition 3.2. (ii)]{BST}, where the $S$-module $D_0(g)=AR(g)$ is denoted by $\Syz (J_g)$ and $mdr(g)$ is denoted by
$\indeg (\Syz (J_g))$. Note also that in \cite{BST} one works over the polynomial ring in $n$-variables with coefficients in an arbitrary infinite field.
The corresponding result for a product $f=f_1f_2 \cdot \ldots \cdot f_m$ of $m\geq 2$ forms in $n$-variables is considered in \cite[Proposition 3.5]{BST}. Interesting information on the invariant  $\indeg (\Syz (J_f))$ when  $C:f=0$ is the union of several smooth plane curves meeting transversally is given in \cite[Proposition 3.6]{T}.

\end{rk}

\begin{cor}
\label{cor10}
With the above notations, $r=mdr(f)$ is the minimal integer $s$ such that
either $D_0(f_1)_s\cap D_0(f_2)_s \ne 0$, or $D_0(f_1)_s+D_0(f_2)_s$ contains a non-zero multiple of the Euler derivation $E$.
\end{cor}
\proof
The first case corresponds to $h=0$ in Theorem \ref{thm10} (2), while the second case corresponds to $h \ne 0$.
\endproof

\begin{ex}
\label{ex10}
Let $C_1:f_1=x^2 + y^2 -z^2=0$ and $C_2:f_1=x^2 + y^2 -4z^2=0$ be  two smooth conics with 2 tacnodes as in Proposition \ref{prop10} $(3)$. Hence $d_1=d_2=2$, $r_1=r_2=1$.
Note that  $y\partial_x-x\partial_y \in D_0(f_1)_1 \cap D_0(f_2)_1$. Therefore, according to Corollary \ref{cor10} we have $r=1$, see also Proposition \ref{prop10}, (3).\\
Consider next the case  $C_1:f_1=xyz=0$ and $C_2:f_2=xy+yz+xz=0$.
Then $C_2$ is a smooth conic circumscribed in the triangle $C_1$.
Using Singular, we see that $r=2$ and $D_0(f)_2$ is spanned by
$$\delta=2x(y-z)\partial _x-y(3y+2z)\partial_y+z(2y+3z)\partial_z$$
and
$$\delta'=x(3x+4y-2z)\partial_x-y(2x+6y+2z)\partial_y+z(-2x+4y+3z)\partial_z.$$
Then $\delta (f_1)= xyz(y+3z)=d_2h_2f_1$, which implies $h=-d_2h_2=-(y+3z)$.
Similarly $\delta' (f_1)= xyz(-x+2y-z)=d_2h_2f_1$, which implies that in this case $h=-d_2h_2=x-2y+z$. It follows that in this case
$D_0(f_1)_2 \cap D_0(f_2)_2=0$. Therefore, both situations may occur in Corollary \ref{cor10}. The fact that $r=2$ in this case is discussed from another view-point, without the use of Singular, in Example \ref{ex10bis}.
The curve $C$ has three $D_4$ singularities and hence $\tau(C)=12$.
Using the characterization of free curves in \eqref{F1}, it follows that $C$ is a free curve.
\end{ex}

Let $C_1$ and $C_2$ be smooth conics, hence $d_1=d_2=2$ and $r_1=r_2=1$. For $C=C_1 \cup C_2$, Theorem \ref{thm10} gives us $1 \leq r \leq 3$. We have the following precise result.

\begin{prop}
\label{prop10}
The two conics $C_1$ and $C_2$  can be in one of the following four situations.
\begin{enumerate}

\item $|C_1 \cap C_2|=4$, and then all the intersection points are nodes for $C$.
 In this case  $r=2$.

\item  $|C_1 \cap C_2|=3$, and then one intersection point is a tacnode  and the other two intersection points are nodes for $C$. In this case $r=2$.

\item  $|C_1 \cap C_2|=2$. Then the two intersection points are either two tacnodes for $C$,  and in this case $r=1$ and the curve $C$ is nearly free with exponents $(1,3)$, or a node $A_1$ and a singularity $A_5$ for $C$,
and in this case $r=2$ and the curve $C$ is nearly free with exponents $(2,2)$.

\item  $|C_1 \cap C_2|=1$, and then the intersection point is a singularity $A_7$ for $C$,  $r=1$ and $C$ is a free curve.

\end{enumerate}
\end{prop}

Computations with Singular suggest that in case $(1)$ the curve  $C=C_1 \cup C_2$ is a 4-syzygy curve with
exponents $(2,3,3,3)$, and in case $(2)$ the curve $C=C_1 \cup C_2$ is a plus one generated curve with
exponents $(2,2,3)$.

\proof
The claim $(1)$ follows from \cite[Theorem 4.1]{DStEdin}.
For the claim $(2)$ we use the inequalities involving $r$ and the Tjurina number $\tau(C)$ due to du Plessis and Wall, see \cite{dPW}.
In case $(2)$ we have $\tau(C)=5$. We know that
$$5=\tau(C) \geq (d-1)(d-1-r)=3(3-r).$$
This implies $r \geq 2$. For $r=3$ we also have
$$5=\tau(C) \leq (d-1)(d-1-r)+r^2-{2r-d+2 \choose 2}= 3,$$
a contradiction. So the only possibility is $r=2$.
In case $(3)$, when the contact between $C_1$ and $C_2$ consists of two tacnodes, using the results in  \cite{Meg}, we see that a pair of conics in this situation is projectively equivalent to a pair of conics of the form
\begin{equation} \label{M1}
C_1:f_1=x^2 - y^2 -z^2=0 \text{ and } C_2:f_1=x^2 - y^2 -k z^2=0,
\end{equation}
with $k \in \mathbb{C}^*$, $k \ne 0$.
 We have $\tau(C)=6$, and the same approach as in Example \ref{ex10} gives $r=1$ in this case. Since we have $\tau(C)=6$ in this case, the equality
$$\tau(C) = (d-1)(d-1-r)+r^2-1,$$
holds, and
it follows from \eqref{F2} that $C$ is nearly free.
Assume now that the smooth conics $C_1$ and $C_2$ have a contact of type $A_5$  at $(0:0:1)$. Choosing the coordinates, we may assume that $C_1: f_1=yz-x^2=0$. Then it is easy to see that the other conic has an equation of the form $C_2=f_2=yz-x^2+a\cdot xy+b \cdot y^2=0$ for some $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$. Since $\tau(C) =6$, we get as above $r \leq 2$. To show that $r>1$ one can use Theorem \ref{thm10} (2), since we have a simple description of $D(f_1)_1$. We see that $\delta_1 (f_2) \notin \mathbb{C} \cdot f_2$, for any $\delta_1 \in D_0(f_1)_1$. It follows that $r=2$ and we get the nearly freeness of $C$ as above.

In the case (4), it follows from \cite[Proposition 1.3]{Bruce}, that the equations of the two conics can be chosen as follows
$$C_1: f_1=x(x+y)+yz-a \cdot y^2=0 \text{ and } C_2: f_2=x(x+y)+yz+a \cdot y^2=0$$
for some $a \in \mathbb{C}^*$. It follows that $(f_1)_x=(f_2)_x$, $(f_1)_z=(f_2)_z$, $\delta=(f_1)_z\partial_x-(f_1)_x\partial_z \in
D_0(f_1)_1 \cap D_0(f_1)_1 \ne 0$, and hence $r=1$. Moreover, the freeness of $C$ follows from
\cite{Dmax,dPW}.
\endproof

\begin{ex}
\label{ex11}
Let $C_1:f_1=(x^2 + y^2 -z^2)(x^2+y^2-4z^2)=0$ be the union of two smooth conics with 2 tacnodes as in Proposition \ref{prop10} $(3)$. Hence $d_1=4$, $r_1=1$.
Let $C_2:f_2= (x-z)(3y^2-(x+2z)^2)=0$ be the union of 3 lines, forming a triangle which is inscribed in the conic $C_2$ and circumscribed to the conic $C_1$. Then $d_2=3$, $r_2=1$.  For $C=C_1 \cup C_2$, Theorem \ref{thm10} gives us $1 \leq r \leq 3$.
Using Singular we see that $r=3$, see also Example \ref{ex11bis} below for a different approach. In fact, $C$ is a free curve with exponents $(3,3)$ as follows from \eqref{F1}, see also \cite{DPo}.
\end{ex}

\section{The case of quasihomogeneous singularities}

Consider the sheafification
$$E_C:=
\widetilde{AR(f)}= \widetilde{D_0(f)} $$
of the graded $S$-module $AR(f)=D_0(f)$, which is a rank two vector bundle on $\mathbb{P}^2$, see \cite{Se} for details. Moreover, recall that
\begin{equation} \label{equa1}
E_C=T\langle C \rangle (-1),
\end{equation}
where $T\langle C \rangle $ is the sheaf of logarithmic vector fields along $C$ as considered for instance in \cite{FV1,MaVa,DS14}.
One has,  for any integer $k$,
\begin{equation}
\label{e7}
H^0(\mathbb{P}^2, E_C(k))=D_0(f)_k \text{ and }  H^1(\mathbb{P}^2, E_C(k))=N(f)_{k+d-1},
\end{equation}
where $d=\deg(f)$, for which we refer
to \cite[Proposition 2.1]{Se}.
Return now to the setting of the previous section, where $C=C_1 \cup C_2$ and $f=f_1f_2$, and recall the following  result, see \cite[Theorem 1.6 and Remark 1.8]{STY}.

\begin{thm}
\label{thm20STY}
With the above notation, assume that $C_2$ is an irreducible curve, and that all singularities of $C_1$, $C_2$ and $C$ are quasihomogeneous.
If $C_1 \cap C_2$ is contained in the smooth part of $C_2$, then there is an exact  sequence of sheaves on $ \mathbb{P}^2$ given by
$$ 0 \to E_{C_1}(1-d_2) \stackrel{f_2} \longrightarrow  E_C(1) \to i_{2*}{\mathcal F} \to 0$$
 where  $i_2: C_2  \to \mathbb{P}^2$ is the inclusion and ${\mathcal F}$ a torsion free sheaf  on ${C_2}$.
Moreover,
when $C_2$ is smooth, then one has ${\mathcal F}= {\mathcal O}_{C_2}(-K_{C_2}-R)$, where $K_{C_2}$ is the canonical divisor on ${C_2}$ and $R$ is the reduced scheme of $C_1 \cap C_2$.
\end{thm}

{\it For simplicity, in this note we consider only the case $C_2$ smooth.}
If we set
$${\mathcal O}_{C_2}(1)=i_2^*{\mathcal O}_{\mathbb{P}^2}(1),$$
then one can write ${\mathcal O}_{C_2}(1)={\mathcal O}_{C_2}(D)$, where the divisor $D$ corresponds to the intersection of a line in $\mathbb{P}^2$ with the curve $C_2$, and hence $\deg D=d_2$. With this notation, by tensoring the above exact sequence with ${\mathcal O}_{\mathbb{P}^2}(k-1)$, for any integer $k$, we get the exact sequence
\begin{equation}
\label{e8}
0 \to E_{C_1}(k-d_2) \stackrel{f_2} \longrightarrow  E_C(k) \to i_{2*}{\mathcal O}_{C_2}(-K_{C_2}-R+(k-1)D) \to 0.
\end{equation}
By taking the corresponding long cohomology sequence and using \eqref{e7}, we get the following result.
\begin{thm}
\label{thm20}
With the above notation, assume that $C_2$ is a smooth curve, and that all singularities of $C_1$ and $C$ are quasihomogeneous. Then there is an exact sequence  for any integer $k$ given by
$$ 0 \to D_0(f_1)_{k-d_2} \stackrel{\phi_k} \longrightarrow  D_0(f)_k \to H^0(C_2,{\mathcal O}_{C_2}(-K_{C_2}-R+(k-1)D)) \to $$
$$ \to N(f_1)_{k-d_2+d_1-1} \stackrel{\psi_k} \longrightarrow N(f)_{k+d-1}   \to H^1(C_2,{\mathcal O}_{C_2}(-K_{C_2}-R+(k-1)D)), $$
where the morphism $\phi_k: D_0(f_1)_{k-d_2} \to  D_0(f)_k$ is given by
$$\phi_k(\delta_1)=f_2\delta_1-\frac{\delta_1(f_2)}{d}E$$
for $\delta_1 \in D_0(f_1)$ and $\psi_k$ is induced by the multiplication by $f_2^2$.
 In particular, if
$$(k+2)d_2 <d_2^2+ |R|,$$
then the morphism $\phi_k$ is an isomorphism and $\psi_k$ is a monomorphism.
\end{thm}
\proof
The exact sequence above is part of the long cohomology exact sequence associated to the exact sequence of sheaves \eqref{e8}. It remains to explain the claims about the morphisms $\phi_k$ and $\psi_k$.
Using the identification $D_0(g)=D(g)/SE$, valid for any homogeneous polynomial $g \in S$, it is shown in \cite{STY} that the morphism $E_{C_1}(1-d_2) \to E_C(1)$ is induced by the multiplication by $f_2$.
In terms of the modules $D_0(g)$, this is precisely the mapping
$D_0(f_1)(-d_2) \to D_0(f)$ given by
$$\phi: \delta_1 \mapsto \delta=f_2\delta_1-\frac{\delta_1(f_2)}{d}E,$$
as constructed in Theorem \ref{thm10}. To explain the formula for $\psi_k$, consider the diagram of graded $S$-modules
$$
\xymatrix{0 \ar[r] & D_0(f_1)(-d_2) \ar[d]_{\phi}  \ar[r]^{\iota} & S^3(-d_2) \ar[d]_{\phi}  \ar[r]^{\nabla f_1} &
                            J_{f_1}(d_1-d_2-1) \ar[d]_{f_2^2}  \ar[r] & 0 \\
 0 \ar[r] & D_0(f)   \ar[r]^{\iota}  & S^3  \ar[r]^{\nabla f}  &    J_f(d-1)   \ar[r] & 0
}$$
Here $\iota$ are the obvious inclusions, $\phi$ is the morphism defined above and its extension to a map $Der(S)(-d_2)=S^3(-d_2) \to Der(S)=S^3$ given by the same formula,
$\nabla f :S^3 \to J_f$ is the map $(a,b,c) \mapsto af_x+bf_y+cf_z$ and similarly for $\nabla f_1$, while
$$f_2^2: J_{f_1}(d_1-d_2-1) \to J_f(d-1)$$
is the multiplication by $f_2^2$. A simple computation shows that this diagram is commutative. Since $N(f_1)= \widehat{J}_{f_1}/J_{f_1}$ and $N(f)= \widehat{J}_f/J_f$, it follows that the morphism $N(f_1)_{k-d_2+d_1-1} \stackrel{\psi_k} \longrightarrow N(f)_{k+d-1} $ induced by the long cohomology exact sequence, and hence coming from $\phi$, is nothing else but multiplication by $f_2^2$.
The final inequality says that
$$\deg( -K_{C_2}-R+(k-1)D) <0,$$
and so the claim follows from the exact sequence. To see this, recall that
$$\deg K_{C_2}=2g_{C_2}-2=d_2^2-3d_2,$$
where $g_{C_2}$ is the genus of the smooth curve $C_2$.
\endproof

\begin{rk}
\label{rk20} The formula for $\psi_k$ given in Theorem \ref{thm20} implies the following fact: for any $h \in \widehat{J}_{f_1}$, one has $f_2^2h \in
 \widehat{J}_{f}$. When $C_1$ is a smooth curve, then $ \widehat{J}_{f_1}=S$ and this situation occurs already in
\cite[Proposition 3.2. (i)]{BST}, where $S$ is the  polynomial ring in $n$-variables with coefficients in an arbitrary infinite field.

\end{rk}

\begin{cor}
\label{cor20}
With the above notation, assume that $C_2$ is a smooth curve, and that all singularities of $C_1$ and $C$ are quasihomogeneous.  Let $R$ be the reduced scheme of $C_1 \cap C_2$. If
$$|R|>(r_1+1)d_2,$$
then $r=r_1+d_2$.
This applies in particular when $C_2$ is a generic  curve and $r_1\ne d_1-1$.
\end{cor}
\proof
The hypothesis $|R|>(r_1+1)d_2$ implies that the inequality
$$(k+2)d_2 <d_2^2+ |R|$$
holds for all $k \leq r_1+d_2-1$. Using Theorem \ref{thm20} and the definition of $r_1$, it follows that $D_0(f)_k=0$ for all $k \leq r_1+d_2-1$.
The exact sequence in Theorem \ref{thm20} also implies that
$D_0(f)_{r_1+d_2} \ne 0$, which proves our claim. When $C_2$ is a generic  curve, then $C_1 \cap C_2$ consists of $d_1d_2$ nodes for $C$ and the claim is clear.
\endproof
\begin{ex}
\label{ex20}
Let $C_1$ be a reduced curve satisfying $r_1 \ne d_1-1$ and such that all the singularities of $C_1$ are quasihomogeneous. Then for any point $p \notin C_1$  and  any  line $C_2$ through $p$ such that $C_2$ meets transversally $C_1$ at smooth points, one has $r=r_1+1$. The claim follows from Corollary \ref{cor20}, since $d_2=1$ and $C_1 \cap C_2$ consists of $d_1$ nodes for $C$ in this case.
This result should be compared to Theorem \ref{thm1}. Moreover, Example \ref{ex1} shows that the restriction $r_1 \ne d_1-1$
is necessary.
If we take $p \in C_1$, then the condition that $(C,p)$ is quasihomogeneous {\it limits drastically the choices} for the line $C_2$ passing through $p$.
Consider the rational cuspidal curve $C_1:f_1=xy^{d_1-1}+z^{d_1}=0$, with $d_1>2$. We have seen in  Example \ref{ex2} that, if we take $C_2$ to be the line through the  singular point $p=(1:0:0)$  given by
$y=0$ or $z=0$, then $(C,p)$ is quasihomogeneous, and $r=r_1$ in these two cases. In fact, in these cases $|R|=1$ and Corollary \ref{cor20} does not apply. When $C_2$ is given by $sy+tz=0$ with $st \ne 0$, then
the singularity $(C,p)$ is not quasihomogeneous, as we have seen in
Example \ref{ex2} for the case $s=t=1$.
\end{ex}
Assume from now on that $|R|\leq (r_1+1)d_2$, or equivalently $(k+2)d_2 \geq d_2^2+ |R|$ and set
$$k_0=d_2-2+ \left \lceil \frac{ |R|}{d_2}\right \rceil.$$
To simplify the discussion, we also assume that $C_2$ is a smooth rational curve, hence $d_2 \in \{1,2\}$. It follows that $k_0$ is the smallest integer $k$ such that $H^0(C_2,{\mathcal O}_{C_2}(-K_{C_2}-R+(k-1)D))\ne 0$. If we assume in addition that $C_1$ is a free curve, then $N(f_1)=0$ and Theorem \ref{thm20}  implies the following.
\begin{cor}
\label{corF}
With the above notation and assumptions, if in addition $C_1$ is a free curve and $C_2$ is rational, then $|R|\leq (r_1+1)d_2$ implies $k_0\geq r_1$ and
$$r_1 \leq r= k_0 \leq r_1+d_2-1.$$
In particular, $|R| > (r_1+1)d_2-d_2^2$, that is we have the following cases.
\begin{enumerate}

\item Let $C_1:f_1=0$ be a free curve and $L$ be a line such that $C_1$ and $C_1 \cup L$ have only quasihomogeneous singularities. Then
$$|C_1 \cap L| > r_1=mdr(f_1).$$

\item  Let $C_1:f_1=0$ be a free curve and $Q$ be a smooth conic such that $C_1$ and $C_1 \cup Q$ have only quasihomogeneous singularities. Then
$$|C_1 \cap Q| > 2r_1-2, \text{ where } r_1=mdr(f_1).$$

\end{enumerate}

\end{cor}
\proof
Note that $r \geq r_1$  implies $k_0 \geq r_1$, which yields in particular the last claim.
\endproof

When $C_2$ is a line, then $r=r_1$ and $|R|=r_1+1$ in these conditions, a known result when $C_1$ is a line arrangement, see for instance \cite[Theorem 3.6 (2)]{ADS}.
\begin{ex}
\label{exF}
Consider $C_1: xyz(x-y)(y-z)(x-z)=0$, which is free with $d_1=6$ and $r_1=2$. Let $C_2$ be a general conic passing through 2 triple points and 2 double points of $C_1$, for instance
$C_2:x^2+z^2-xy-yz=0$. Then $d_2=2$, $r_2=1$ and $|R|=6$. Corollary \ref{corF} implies
$r=k_0=3.$ It follows that the curve $C$ is free with exponents $(3,4)$ by \eqref{F1}.
Indeed, this curve $C$ has 3 nodes, 4 ordinary triple points and 2 ordinary quadruple points, hence $\tau(C)=37$.
\end{ex}
The application to the exact sequence \eqref{e8} to study free curves goes back to \cite{STY}. Now we show that this sequence gives valuable information even when $C_1$ is not a free curve. We start with the case $C_2$ is a line, hence we have to decide by Proposition \ref{prop1} or by Theorem \ref{thm1} whether $r=r_1$ or $r=r_1+1$.
\begin{cor}
\label{cor21}
With the above notation, assume that  $C_2$ is a line  and that all singularities of $C_1$ and $C$ are quasihomogeneous.  Let $R$ be the reduced scheme of $C_1 \cap C_2$. If
$$|R|\leq r_1+1,$$
then there is the following exact sequence
$$ 0 \to   D_0(f)_{r_1} \to H^0(C_2,{\mathcal O}_{C_2}(r_1+1-|R|))
 \to N(f_1)_{r_1+d_1-2} \to N(f)_{r_1+d_2+d-2}   \to 0. $$
 \end{cor}
\proof
The proof is as above, using Theorem \ref{thm20} for $k =r_1$
and the fact that $H^1(C_2,{\mathcal O}_{C_2}(\ell)) =0$ if $\ell \geq 0$.
\endproof

\begin{cor}
\label{cor21NF}
\begin{enumerate}

\item Let $C_1:f_1=0$ be a nearly free curve and $L$ be a line such that $C_1$ and $C_1 \cup L$ have only quasihomogeneous singularities. Then
$$|C_1 \cap L| \geq  r_1=mdr(f_1).$$

\item  Let $C_1:f_1=0$ be a nearly free curve and $Q$ be a smooth conic such that $C_1$ and $C_1 \cup Q$ have only quasihomogeneous singularities. Then
$$|C_1 \cap Q| \geq 2r_1-1, \text{ where } r_1=mdr(f_1).$$

\end{enumerate}

 \end{cor}
\proof
The proof is as above, using Theorem \ref{thm20} for $k =r_1-1$
and the fact that $\dim H^(C_2,{\mathcal O}_{C_2}(\ell)) \geq 2> \nu(C_1)=1$ if $\ell \geq 1$. For $d_2=2$, we use the stronger fact that $\sigma(C_1)=d_1+r_1-3$.

\endproof

\begin{ex}
\label{ex21}
Let $C_1$ be a nearly free curve having only quasihomogeneous singularities.   Let $C_2$ be a line such that
$|R| =r_1$, the minimal possible value, and $C=C_1 \cup C_2$ has again only quasihomogeneous singularities. Then in the exact sequence of Corollary \ref{cor21} we have
$$ \dim H^0(C_2,{\mathcal O}_{C_2}(r_1+1-|R|))\geq 2=\dim H^0(C_2,{\mathcal O}_{C_2}(1)) >\dim N(f_1)_{r_1+d_1-2}=1.$$
The last equality follows from the equality $\sigma(C_1)=d_1+r_1-3$, see \cite[Corollary 2.17]{DStRIMS}. This implies $r=r_1$ in this situation.\\
A first explicit example of such a situation is provided by the curves $C_1$ discussed in Example \ref{ex2} with the line $C_2$ given by $y=0$ or $z=0$, when $r_1=1$. \\
A second  example  is provided by
the quartic with 3 cusps
$$C_1: x^2y^2+y^2z^2+x^2z^2-2xyz(x+y+z)=0,$$
which is nearly free with  $r_1=2$, see \cite[Example 2.13]{DStRIMS} and $C_2:z=0$, a line joining 2 cusps.
The curve $C$ has in this case one cusp $A_2$ and two $D_5$ singularities, hence has  only quasihomogeneous singularities.
Since $|R|=2=r_1$, the above discussion applies and
it follows that $r=r_1=2$.
 Using \cite{Dmax,dPW}, it follows that  the obtained quintic curve
$$C: x^2y^2z+y^2z^3+x^2z^3-2xyz^2(x+y+z)=0,$$
 is free with exponents $(2,2)$.\\
 As a third example, consider $C_1$ to be the union of two smooth conics tangent to each other in two points, as in Proposition \ref{prop10}.
 Let $C_2$ be the line joining these two points. Then $C$ has two $D_6$ singularities, $2=|R|>r_1=1$. Hence {\it Corollary \ref{cor21} cannot be used to conclude}. Note that using the equation \eqref{M1}, we see that
 $y\partial_x+x\partial_y \in D(f_1)_1 \cap D(f_2)_1 \ne 0$, and hence $r=1$.
  Using \cite{Dmax} it follows that this curve $C$ is nearly free with exponents $(1,4)$.
\end{ex}

Here is the version of Corollary \ref{cor21} when $C_2$ is a smooth conic. Here we know already that $r_1 \leq r \leq r_1+2$ by Theorem \ref{thm10}.
\begin{cor}
\label{cor21C}
With the above notation, assume that  $C_2$ is a smooth conic and that all singularities of $C_1$ and $C$ are quasihomogeneous.  Let $R$ be the reduced scheme of $C_1 \cap C_2$. If
$$|R|\leq 2(r_1+1),$$
then there is the following exact sequences.
$$ 0 \to   D_0(f)_{r_1} \to H^0(C_2,{\mathcal O}_{C_2}(2r_1-|R|))
 \to N(f_1)_{r_1+d_1-3} \to N(f)_{r_1+d_2+d-3}  $$
 and
 $$ 0 \to   D_0(f)_{r_1+1} \to H^0(C_2,{\mathcal O}_{C_2}(2r_1+2-|R|))
 \to N(f_1)_{r_1+d_1-2} \to N(f)_{r_1+d_2+d-2}   \to 0.$$
 \end{cor}
 \proof
Use Theorem \ref{thm20} for $k =r_1$ and for $k=r_1+1$.
\endproof

\begin{ex}
\label{ex10bis}
Consider next the case  $C_1:f_1=xyz=0$ and $C_2:f_2=xy+yz+xz=0$.
Then $C_2$ is a smooth conic circumscribed in the triangle $C_1$, as in the second part of Example \ref{ex10}. In this case $r_1=1$ and $|R|=3$,
hence we can apply Corollary \ref{cor21C}. The first exact sequence implies that $D_0(f)_1=0$, and the second exact sequence implies that
$$\dim D_0(f)_2=2=\dim H^0(C_2,{\mathcal O}_{C_2}(1)) $$
since $C_1$ is a free curve, and hence $N(f_1)=0$.
\end{ex}
\begin{ex}
\label{ex11bis}
Let $C_1:f_1=(x-z)(3y^2-(x+2z)^2) (x^2+y^2-4z^2)=0$ be a smooth conic $Q$ circumscribed in a triangle $\Delta$ as in Example \ref{ex10bis}.
 Let $C_2:f_2= (x^2 + y^2 -z^2)=0$ be a conic inscribed in the triangle $\Delta$ and tangent to the conic $Q$ in two points. Then $d_2=2$, $r_2=1$. In this case $r_1=2$ and $|R|=5$,
hence we can apply Corollary \ref{cor21C}. The first exact sequence implies that $D_0(f)_2=0$, and the second exact sequence implies that
$$\dim D_0(f)_3=3=\dim H^0(C_2,{\mathcal O}_{C_2}(1)) $$
since $C_1$ is a free curve, and hence $N(f_1)=0$.
\end{ex}

\begin{ex}
\label{ex21C}
Let $C_1$ be a nearly free curve having only quasihomogeneous singularities.   Let $C_2$ be a smooth conic such that either $|R| \leq 2r_1-1$ or
$|R| =2r_1+1$ and $C=C_1 \cup C_2$ has again only quasihomogeneous singularities. Then, when  $|R| = 2r_1-1$, we get
exactly as in Example \ref{ex21} that $r=r_1$. Assume now that $|R| =2r_1+1$.
In the first exact sequence of Corollary \ref{cor21C} we have
$$H^0(C_2,{\mathcal O}_{C_2}(2r_1-|R|))=H^0(C_2,{\mathcal O}_{C_2}(-1))=0,$$
and hence $ D_0(f)_{r_1} =0$.
 In the second exact sequence of Corollary \ref{cor21C} we have
$$2 = \dim H^0(C_2,{\mathcal O}_{C_2}(2r_1+2-|R|)) =H^0(C_2,{\mathcal O}_{C_2}(1)) >\dim N(f_1)_{r_1+d_1-3}=1.$$
The last equality follows from the equality $\sigma(C_1)=d_1+r_1-3$, see \cite[Corollary 2.17]{DStRIMS}. This implies $r=r_1+1$ in this situation.\\
To have an explicit example, we consider again the quartic $C_1$ with 3 cusps from Example \ref{ex21}, and take now $C_2$ to be a smooth generic conic passing through the 3 cusps, then the resulting curve $C$ will have 3 $D_5$ singularities and 2 nodes $A_1$.
 It follows that $|R|=5=2r_1+1$. It follows that in this case
 $r=r_1+1=3$. As an explicit example, one can take
$$C: (xy+yz+xz)(x^2y^2+y^2z^2+x^2z^2-2xyz(x+y+z))=0,$$
which is a plus one generated curve with exponents $(3,3,4)$.
\end{ex}
\begin{ex}
\label{ex21D}
Let $C_1$ be a nearly free curve having only quasihomogeneous singularities.   Let $C_2$ be a smooth conic such that  $|R|= 2r_1+2$.  In the first exact sequence of Corollary \ref{cor21C} we have
$$\dim H^0(C_2,{\mathcal O}_{C_2}(2r_1-|R|))=\dim H^0(C_2,{\mathcal O}_{C_2}(-2))=0,$$
and in the second
exact sequence of Corollary \ref{cor21C} we have
$$\dim H^0(C_2,{\mathcal O}_{C_2}(2r_1+2-|R|))=\dim H^0(C_2,{\mathcal O}_{C_2})=1.$$
If  $N(f_1)_{r_1+d_1-2}=0$, then we have $r=r_1+1$.
We consider the following explicit situation. Let $C_1: f_1=x^2y^2+z^4-xz^3-2xyz^3=0$, which has an $A_4$-singularity at $p=(0:1:0)$ and an $A_2$-singularity at $q=(1:0:0)$. Then $C_1$ is a nearly free curve with
$d_1=4$ and $r_1=2$, see \cite[Example 2.13]{DStRIMS}. Let $C_2: xy+yz+xz=0$ be a smooth conic, passing transversely through $p$ and $q$ and meeting $C_1$ transversally at another 4 points. It follows that
$|R|=2+4=6=2r_1+2$ and $C$ has 4 nodes $A_1$, one $D_5$ singularity and one $D_7$-singularity in all. Note that
$$\sigma(C_1)=d_1+r_1-3=T_{f_1}/2=3.$$
This implies that $N(f_1)_{4}=0$, exactly what we need to conclude that $r=r_1+1=3$.
\end{ex}

\begin{ex}
\label{ex21E}
We end with an example in order to conclude via Corollary \ref{cor21C} we need to analyze the morphism
$$N(f_1)_{r_1+d_1-3} \stackrel{f_2^2} \longrightarrow N(f)_{r_1+d-1}.$$
Let $C_1: f_1=(x^2-2xz+y^2)(x^2+2xz+y^2)=0$, hence $C_1$ is a pair of smooth conics tangent at one point, as in Proposition \ref{prop10} (2).
This curve $C_1$ is a plus one generated curve with exponents $(2,2,3)$. Hence $d_1=4$ and $r_1=2$. Let $C_2: x^2+y^2-4z^2=0$, a circle tangent to each circle in $C_1$ at one point and passing through the 2 nodes of $C_1$. Hence $C$ has 3 singularities $A_3$ and 2 singularities $D_4$. It follows that $|R|=4=2r_1$. In the first exact sequence of Corollary \ref{cor21C} we have
$$\dim H^0(C_2,{\mathcal O}_{C_2}(2r_1-|R|))=\dim H^0(C_2,{\mathcal O}_{C_2})=1,$$
and $\dim N(f_1)_{r_1+d_1-3}=2$. More precisely, a basis of $N(f_1)_{r_1+d_1-3}$ is given by the monomials $xyz$ and $xz^2$.
A computation using Singular shows that the kernel of the morphism
$$N(f_1)_{r_1+d_1-3} \stackrel{f_2^2} \longrightarrow N(f)_{r_1+d-1}$$
is 1-dimensional, generated by $xyz-xz^2$. As a result $D_0(f)_2=0$.
 In the second
exact sequence of Corollary \ref{cor21C} we have
$$\dim H^0(C_2,{\mathcal O}_{C_2}(2r_1+2-|R|))=\dim H^0(C_2,{\mathcal O}_{C_2}(2))=3$$
and $\dim N(f_1)_{r_1+d_1-2}=1$. Therefore we have $r=r_1+1=3$.
A computation with Singular shows that $C$ is a plus one generated curve with exponents $(3,3,4)$.
\end{ex}

\subsection{An application to the jumping lines of the rank 2 vector bundle $E_{C_1}$}

For a reduced plane curve $C$ and a line $L$ in $\mathbb{P}^2$, the pair of integers $( d_1^L(C), d_2^L(C))$ such that  $ d_1^L(C) \leq d_2^L(C)$  and  $ E_C|_L \simeq {\mathcal O}_L(-d_1^L(C)) \oplus {\mathcal O}_L(-d_2^L(C))$ is called the (ordered) splitting type  of $E_C$ along $L$, see for instance \cite{OSS}.
For a generic line $L_0$, the corresponding splitting type $( d_1^{L_0}(C), d_2^{L_0}(C))$ is known to be constant, see \cite[Definition 2.2.3 and Lemma 3.2.2]{OSS}.
A line $L$ in $\mathbb{P}^2$ is called a jumping line  for $E_C$ or, equivalently, for $T\langle C \rangle $, if
$$d_1^{L_0}(C)-d_1^{L}(C)>0.$$
The following result relates the splitting type of $E_C$ along a line $L: {\alpha}_L=0$, to the Lefschetz properties of the Jacobian module $N(f)$ with respect to the multiplication by ${\alpha}_L$, see \cite[Proposition 4.1]{DStJump}.
\begin{prop} \label{propthmA}
 For any reduced curve $C:f=0$ and any line $L: {\alpha}_L=0$ in $\mathbb{P}^2$, we have
 $d_1^L(C)=\min \{mdr(f), k(f,L)\},$
 where $$k(f,L)= \min \{ k \in \mathbb{N} \ : \ N(f)_{k+d-2} \stackrel{\cdot \alpha_L}{\to}
N(f)_{k+d-1} \text{ is not injective } \}.$$
 \end{prop}
 Using this result, we give now an easy geometric way to check that a line is a jumping line, under some conditions.
\begin{thm} \label{thmJ}
Let $C_1:f_1=0$ be a reduced curve and $C_2:f_2=0$ be a line in $\mathbb{P}^2$.
 Assume that  all the singularities of $C_1$ and of $C=C_1 \cup C_2$ are quasihomogeneous, and let $R$ be the reduced scheme of $C_1 \cap C_2$. If $|R| < r_1+1$, then the morphism
 $$ \psi'_k: N(f_1)_{k+d_1-2} \stackrel{f_2} \longrightarrow N(f_1)_{k+d_1-1} $$
 is not injective, for any $k$ satisfying $|R|-1 \leq k <r_1$ and one has
 $$d_1^{C_2}(C_1)=k(f_1,C_2) \leq |R|-1.$$
 Moreover, if one of the following two conditions holds
 \begin{enumerate}

\item  either $2r_1<d_1$, or

\item  $$ 2r_1 \geq d_1 \text{ and } |R|-1 < \left \lfloor \frac{d_1-1}{2} \right \rfloor,$$

\end{enumerate}
then $C_2$ is  a jumping line for  the rank two vector bundle $E_{C_1}$.

\end{thm}
\proof
Note that the condition $k \geq |R|-1$ implies that
$$H^0(C_2,{\mathcal O}_{C_2}(-K_{C_2}-R+(k-1)D))=H^0(C_2,{\mathcal O}_{C_2}(k+1-|R|))\ne 0$$
in the exact sequence from Theorem \ref{thm20}. On the other hand, the condition $k<r_1$ implies that $D_0(f)_k=0$.
Hence, using  Theorem \ref{thm20}, we see that the morphism
$$\psi_k: N(f_1)_{k+d_1-2} \stackrel{f_2^2} \longrightarrow N(f)_{k+d_1}.$$
is not injective. To prove our claim,  it is enough to show that
$$\ker  \psi_k \subset \ker \psi'_k.$$
Let $h \in \ker \psi_k$ be (the representative of ) some element in this kernel. This means that $f_2^2h \in J_f$, in other words there is a derivation $\delta \in Der(S)$ such that
$$f_2^2h=\delta (f)= f_2 \delta (f_1)+ f_1 \delta (f_2).$$
Since $f_1$ and $f_2$ have no common factor, this means that $\delta (f_2)$ is divisible by $f_2$, say $\delta (f_2)=f_2 g$ for some $g \in S$.
Dividing the above relation by $f_2$ we get
$$f_2h=  \delta (f_1)+ f_1g,$$
which implies $f_2h \in J_{f_1}$. Hence $h \in \ker \psi'_k$ as we claimed. Now we prove the final claims in Theorem \ref{thmJ}.
Since we know that $\psi'_k$ is not injective for $|R|-1 \leq k <r_1$,
it follows by Proposition \ref{propthmA} that
$$d_1^{C_2}(C_1)=k(f_1,C_2) \leq |R|-1.$$
Assume now $2r_1<d_1$. First note that the curve $C_1$ cannot be a free curve in view of Corollary \ref{corF} $(1)$ saying that in this case $|R|>r_1$. If $C_1$ is a nearly free curve, then the exponents $r=d_1'\leq d_2'$ verify $d_1'+d_2'=d_1$, and hence the condition $2r_1<d_1$ implies $d_1'<d_2'$. Using for instance \cite[Example 4.8]{DStJump}
we see that in this case $d_1^{L_0}(C_1)=r_1$. The same equality holds for all the other non free reduced plane curves $C_1$ satisfying $2r_1<d_1$, see \cite[Corollary 4.5]{DStJump}. This fact implies that
$C_2$ is a jumping line for $E_{C_1}$ in the case $(1)$. The claim in case (2) follows from  \cite[Corollary 4.6 and Example 4.8]{DStJump}.
Indeed, these results imply that we have
$$d_1^{L_0}(C_1)= \left \lfloor \frac{d_1-1}{2} \right \rfloor$$
when $2r_1 \geq d_1$.
\endproof

\begin{ex}
\label{ex1.1} Let $C_1: f_1=0$ be a Thom-Sebastiani plane curve, i.e. a curve such that $f_1(x,y,z)=g(x,y)+z^{d_1}$, where
$g$ is a homogeneous polynomial of degree $d_1$ in $S'=\mathbb{C}[x,y]$. Assume that
$$g=\ell_1^{k_1} \cdots \ell_m^{k_m},$$
where the linear forms $\ell_j \in S'$ are distinct and $m\geq 2$.
It follows that $C_1$ is a 3-syzygy curve with exponents $r_1=d'_1=m-1$ and $d'_2=d'_3=d_1-1$ for $m \geq 3$ and $C_1$ is nearly free with exponents $r_1=d'_1=1$ and $d'_2=d_1-1$ for $m=2$, see \cite[Example 4.5]{DSt3syz}.

Assume first that $2(m-1)<d_1$ and take the line $C_2$ to be given by one of the factors of $g$, say $C_2:f_2=\ell_1=0$. Then it is easy to check that all the singularities of $C_1$ and of $C=C_1 \cup C_2$ are quasihomogeneous. To do this, one can assume that $\ell_1=x$ and hence $R$ is the point $(0:1:0)$. It follows that $|R|=1<r_1+1$, and Theorem \ref{thmJ} (1) implies that the line $C_2$ is a jumping line for $E_{C_1}$ and moreover
$$d_1^{C_2}(C_1)=k(f_1,C_2)=0.$$
Note that the inequality $d_1^{C_2}(C_1)=k(f_1,C_2) \geq 0$ holds in general, see for instance \cite[Proposition 2.5]{DStJump}.

If we assume now that $2(m-1)\geq d_1\geq 3$, then we get the same result using Theorem \ref{thmJ} (2).
In this way we have found out $m$ jumping lines $\ell_j=0$ for $j=1, \ldots, m$ for the vector bundle $E_{C_1}$.

\end{ex}
Recall that when $C_1$ is a free curve, then all the lines $L$ in $\mathbb{P}^2$ are not jumping lines for the vector bundle $E_{C_1}$. With this in mind, the following result, which is a reformulation of Theorem \ref{thmJ}, can be regarded as a generalization of Corollary \ref{corF} (1).
\begin{cor}
\label{corR}
Let $C_1:f_1=0$ be a reduced curve and $L$ be a line in $\mathbb{P}^2$, which is not a jumping line for the vector bundle $E_{C_1}$. Let $d_1=\deg(f_1)$ and $r_1=mdr(f_1)$. If all the singularities of $C_1$ and of $C_1 \cup L$ are quasihomogeneous, then
$$|C_1 \cap L| >d_1^{L}(C_1)= \begin{cases} r_1 &\mbox{if }  \  2r_1<d_1, \\
 \left \lfloor \frac{d_1-1}{2} \right \rfloor & \mbox{if }  \  2r_1 \geq d_1. \end{cases} $$
\end{cor}

redpajama_set_name: RedPajamaArXiv


text:
13 August, 2015 - 00:52 lizleafloor
Ancient Ancestors Had More DNA Than We Do Now: Have we Devolved?
Adding to the number of unsolved mysteries regarding the ancient origins of humanity and the biology of our species, scientists have discovered that we have less DNA now than our ancient ancestors possessed. Are we de-evolving?
A study published this week in the journal Science has found that modern humans lost DNA as we evolved after our split from apes. Our ancient ancestors, early humans, possessed substantially more amounts of genetic data than we do now. This surprising discovery raises many questions, the most obvious one being: why did we lose all that genetic information? Also, what difference has the loss made?
The short answer is: we don't know yet.
According to news website Gizmodo, the team of researchers led by Professor Evan Eichler, geneticist at the Department of Genome Sciences , University of Washington, sequenced the genomes of 236 individuals from 125 distinct populations. They found that Homo sapiens have shed approximately 40.7 million base pairs of DNA after breaking from our closest living relatives, chimpanzees, around 13 million years ago.
Map of the 125 populations sampled in the study and their relation to each other. ( P. Sudmandt et al 2015 )
Calcified remains of Altamura Man yield oldest ever Neanderthal DNA sample
Jawbone found in Ethiopia set to rewrite history, push back origins of humans
Unusual 20,000-year-old human skull shows diversity of prehistoric humanity
White Skin Developed in Europe Only As Recently as 8,000 Years Ago Say Anthropologists
The genome of modern humans now contains 3 billion base pairs of DNA (complex molecules which contain all of the information necessary to build and maintain an organism, the building blocks of life), and even then scientists are unsure how much of that number is so-called "junk DNA"—genomic data whose function, if it has any, is not understood—but they do assert that at least 27.96 million of the base pairs lost were unique.
Common relationships were traced via DNA deletions among groups of humans. The longer lines show groups with more missing DNA. ( P. Sudmandt et al 2015 )
Have modern humans beneficially shed superfluous DNA, or have we lost something important over the generations?
Eichler proposes that a migration out of Africa reduced the human population in new areas and played a role in the loss of DNA.
Study authors write, "the breadth of the dataset allowed us to reconstruct the structure and content of the ancestral human genome prior to human migration and subsequent gene loss.'
'As expected, Africans were more likely to show evidence of these ancestral sequences compared to non-African populations, as the latter have experienced more population bottlenecks and thus retained less of the ancestral human diversity." Meaning that those individuals who were more connected with Africa had retained more DNA than those who had descended from migrated populations.
MailOnline reports, "The human genome has around three billion base pairs, which reside in 23 chromosomes in the heart of almost every cell in our body. The average gene in the human genome is around 765 base pairs long, meaning humans could have lost the equivalent of up to 37,000 genes since splitting from our ape cousins."
Adding to this mix is the historic breeding of modern humans with the now extinct Denisovans and Neanderthals. Segments of Denisovan and Neanderthal DNA can be identified now in modern populations. But Neanderthals and Denisovans had around 104,000 base pairs in their genomes that are not found in modern humans. Researchers found that Neanderthals and Denisovans were missing some ancient DNA as well, suggesting these extinct species had lost significant portions of genetic code.
"Their results showed that our ancestors shed about 15.8 million base pairs of DNA before leaving Africa. As populations spread across Earth's continents, they jettisoned additional chunks of DNA here and there. But certain populations have also been gaining DNA, mostly through duplication events where portions of the genetic code were accidentally copied and passed on," reports Gizmodo.
Researchers claim Neanderthals were NOT a sub-species of modern humans
Mysterious Origins – What we don't know about the emergence of humans
New theory on Hobbit species has drastic implications for Out-of-Africa theory
While modern humans may presume that the shedding of DNA has honed us to the peak of evolutionary perfection, science can show that not all of our changes have been beneficial. In one example , research suggests that our hands are actually more primitive than those of our chimp ancestors, despite our adaptations for the use of tools.
Chimpanzee hand, at left, and human hand, right.
This is the first time scientists have documented the loss (and gain) of large chunks of DNA in ancient populations. This genetic research may shed light on the enduring questions of how modern humans evolved and survived while other hominin died out.
Scientists still can only speculate as to what these results indicate, but as more research is done we can fill in the missing gaps of our understanding of the ancient story of humanity.
Featured Image: A DNA molecule that is methylated on both strands on the center cytosine. DNA methylation plays an important role for epigenetic gene regulation in development and cancer. ( C. Bock/Wikimedia Commons )
By Liz Leafloor
Liz Leafloor is Editor, Writer and Graphic Designer for Ancient Origins.
Having worked in news and online media for years, Liz covers exciting and interesting topics like ancient myth, history, technology, archaeological discoveries, life and death, and the unexplained.
Liz has been... Read More
Stephen Jones wrote on 19 August, 2015 - 00:11 Permalink
I checked out you site. Absolutely hilarious. Thanks for the laugh.
Gracie Gragg wrote on 18 August, 2015 - 23:45 Permalink
So prove evolution to me and all the others. And then show us your scientific studies on the subject. There has never been any crossing over from one species to another. Apes are one kind of species and human are another kind of species. Never been done and never will happen. BTW, check out my website at www.threeworldages.org. I do not believe in Myths. I believe in proven facts. And you have no proven facts of evolution.
The theory of gravity is "just a theory" too. I wouldn't recommend you jumping from a tall building to test it out.
Look up "scientific theory and get back" to me.
The "just a theory" crowd makes me laugh.
Sorry but you are wrong. Has not been proven. I have been researching all I can get my hands on and it is only just a THEORY. Not even a proven theory.No crossing over from one species to another one not even over long periods of time.
Sure they are. /s Evolution is a fact. It doesn't need to be proven, it already has been.
30 August, 2019 - 14:08 ashley cowie
Humans Began Mass Ecological Destruction 3,000 Years Ago
A new paper with 250 scientific authors suggests human beings had impacted Earth's fragile ecosystems much earlier and to a far greater extent than currently believed. This huge collaborative study...
Read more about Humans Began Mass Ecological Destruction 3,000 Years Ago
30 July, 2019 - 07:09 Nathan Falde
Could Humans Become Immortal by 2045?
Immortality has been a dream of human beings since the dawn of time. Mankind´s fascination with cheating death is reflected in scientific records, mythology and folklore dating back at least to...
Read more about Could Humans Become Immortal by 2045?
15 June, 2019 - 11:29 Nathan Falde
Teleportation: Will Quantum Physics Make Star Trek Tech a Success?
Most of us have seen the science fiction series Star Trek and are familiar with their concept of teleportation, in which characters are 'beamed' off the ship to another spacecraft or to the surface...
Read more about Teleportation: Will Quantum Physics Make Star Trek Tech a Success?
7 November, 2018 - 13:52 Ed Whelan
'Abundant' Occurrence of Skeletal Abnormalities Found in Early Humans
An American researcher examining ancient human remains has made a potentially ground-breaking discovery. The expert found a huge number of abnormalities and defects that seem to indicate extensive...
Read more about 'Abundant' Occurrence of Skeletal Abnormalities Found in Early Humans
22 September, 2018 - 02:17 Gisele Santos
Magnificent Tham Lod Rockshelter Sheds Light on Earliest Humans in Thailand
The Tham Lod Rockshelter (a shallow cave) in Mae Hong Son Province, in Northwest Thailand is a prehistoric area that had been the center for burial and tool–making in the late Pleistocene to the late...
Read more about Magnificent Tham Lod Rockshelter Sheds Light on Earliest Humans in Thailand
15 August, 2018 - 00:47 Joanna Gillan
Discoveries in Spanish cave suggest Neanderthals had hot water and bedrooms
Archaeologists in Spain have made a number of discoveries inside an ancient cave in Catalonia which suggest that Neanderthals had hot water and separate living quarters around 60,000 years ago. The...
Read more about Discoveries in Spanish cave suggest Neanderthals had hot water and bedrooms

redpajama_set_name: RedPajamaCommonCrawl


Take a seat and relax with this high-quality, exceptionally comfortable Hammock Chair! With the proper structure support, this swing can be hung on porches, balconies, tree limbs, or even inside your home. The chair consists of very durable poly-cotton material, featuring a foam-cushioned seat and back for extra comfort, as well as wooden arms for extra stability. This hammock chair is an easy upgrade for your home and backyard. Relaxation can be just minutes away and as close as your backyard with the purchase of this Hammock Chair!


text:
Hack Harassment adds an interdisciplinary board to tackle online hate
Author: Taylor Hatmaker / Source: TechCrunch
Hack Harassment, founded by Vox Media, Intel and Lady Gaga's youth-focused Born This Way organization, will gear up to fight online harassment with the introduction of its first advisory board. Comprised of members from anti-bullying research groups, AAA game developers, and some things in between, the board will bring the collective experience of the multi-disciplinary leadership to bear on one of the web's most intractable problems.
Members will serve as both strategists and public ambassadors for the organization, which seeks to combat online harassment with a mix of technical solutions and awareness campaigns. In February, Hack Harassment teamed up with Kesha to launch its first anti-harassment PSA, a short, punchy YouTube video that invites viewers to join the musician in the fight against online hate.
Hack Harassment's board brings together seven different voices with a range of perspectives and expertise:
Michael Beadle, Director of Public Relations at Ubisoft
Gabby Frost, Founder and CEO of Buddy Project
Iain Gray, Docker's Senior Vice President of Customer Success
Dr. Sameer Hinduja, Co-Director of the Cyberbullying Research Center
Dr. Susan M. Swearer, Co-Director, Bullying Research Network
Brittan Heller, Director of Technology and Society for the Anti-Defamation League
Stephen Balkam, Founder & CEO, Family Online Safety Institute
According to its newly-minted members, the heterogenous mix is what makes the organization uniquely suited to rise to the challenge.
"I think what's unique and different about this group is that we represent a wide cross-section of individuals who…


Both teams have been involved in high scoring games. Following the last second win over Celtics, Lakers will be highly motivated to win this game and increase their chance of making the playoffs. With Lebron back to his best and having acquired proven shooters, this game is expected to produce high scores. My first bet on this game is Over 221.5. This is a safe bet as I expect the score to be way more than 221.5. Good Luck.


From Place to Place is about my nomadic life style. As a kid, I moved around a lot, when I grew up, I always seemed to be searching for a place that felt just right; a place I could call home. I've come to realize that it's OK to have a temporary home and that home is wherever you are. Home is really where your heart is.


In one of my jaunt in batangas, a sari-sari store made me realize the importance of it, for travelers not only in commodity needs of a traveler. It is a suitable place, where a traveler can ask for direction and information.
We are bound in the lakeshore town of San Nicholas, to get some fresh tilapia fish to be grilled for dinner. The sun was shinning with a clear sky, which made us stop for a halo-halo snack at a sari-sari store along the road.
While having our delicious ice-cold halo-halo, I chance upon a group who gave us some information on where to get fresh, low-priced tilapia fish. The group even gave us the direction on how to get to the place of Mang Gimo a known fresh tilapia retailer from San Nicholas town.
Well, you might get me wrong that a sari-sari store is only for information and direction. It is actually a convenience store, which is practically present in communities around Batangas and all over the Philippines.
Sari-sari, by the way means all kinds or various. The store sells all kinds of everyday essential goods in miniature packets, or as a single item, which is commonly referred as "tingi". An individual can purchase a small amount of cooking oil or a piece of egg according to his/her budget. Even a cigarette can be bought by the stick.
Examples of various essential items available are sugar, salt, coffee, canned goods, candies, soft drinks and of course ice-cold San Miguel Beer and the most like liquors like gin, brandy and rum. All vital needs that are commercially viable in the community are sold in a sari-sari store.
The store also saves members of the community in transportation costs, particularly those in far-flung areas. And if you are really dried up in cash and in good credit standing, you can take the goods on credit.
It is also a usual small-scale industry in Batangas that could serve as a primary source of income. And for starters, it could mean a secondary income source. The store requires minimal investment, while most stores are only in the side of the house, or part of the house. Family members help in the daily chores of the store, which adds additional savings to the business.
One distinctive thing about the store is that the customer is outside the store. The customer must pinpoint the goods he wants to purchase and the goods are transferred from a metal bar opening in front of the store. Goods are hung from the ceiling and small items such candies are placed in small plastic jars for the customer to have his/her choice. However, it is impossible to display all the items it sells, but most buyers have the knowledge that the most basic items are sold.
Commonly found outside the stores are wooden benches for customers to consume their purchases like snacks and their favorite drink. While some bigger stores have tables and shades for customers who would like to hang around.
In Batangas, the sari-sari store could also be a good place to hang out. Housewives while making the purchases meet and chat about latest gossip around the block. It is also a meeting place for husbands to meet after hard days work and have some cold beers or a couple of shot of their favorite gin.
So a sari-sari store is not only a place you can purchase your immediate needs, or maybe have some information and direction. It is also a place where you can have fine time with the neighborhood and get to know them.


text:
//------------------------------------------------------------------------------
// <auto-generated>
//     This code was generated by a tool.
//     Runtime Version:4.0.30319.42000
//
//     Changes to this file may cause incorrect behavior and will be lost if
//     the code is regenerated.
// </auto-generated>
//------------------------------------------------------------------------------

namespace Accelerider.Windows.I18nResources {
    using System;

    /// <summary>
    ///   A strongly-typed resource class, for looking up localized strings, etc.
    /// </summary>
    // This class was auto-generated by the StronglyTypedResourceBuilder
    // class via a tool like ResGen or Visual Studio.
    // To add or remove a member, edit your .ResX file then rerun ResGen
    // with the /str option, or rebuild your VS project.
    [global::System.CodeDom.Compiler.GeneratedCodeAttribute("System.Resources.Tools.StronglyTypedResourceBuilder", "15.0.0.0")]
    [global::System.Diagnostics.DebuggerNonUserCodeAttribute()]
    [global::System.Runtime.CompilerServices.CompilerGeneratedAttribute()]
    internal class UiResources {

        private static global::System.Resources.ResourceManager resourceMan;

        private static global::System.Globalization.CultureInfo resourceCulture;

        [global::System.Diagnostics.CodeAnalysis.SuppressMessageAttribute("Microsoft.Performance", "CA1811:AvoidUncalledPrivateCode")]
        internal UiResources() {
        }

        /// <summary>
        ///   Returns the cached ResourceManager instance used by this class.
        /// </summary>
        [global::System.ComponentModel.EditorBrowsableAttribute(global::System.ComponentModel.EditorBrowsableState.Advanced)]
        internal static global::System.Resources.ResourceManager ResourceManager {
            get {
                if (object.ReferenceEquals(resourceMan, null)) {
                    global::System.Resources.ResourceManager temp = new global::System.Resources.ResourceManager("Accelerider.Windows.I18nResources.UiResources", typeof(UiResources).Assembly);
                    resourceMan = temp;
                }
                return resourceMan;
            }
        }

        /// <summary>
        ///   Overrides the current thread's CurrentUICulture property for all
        ///   resource lookups using this strongly typed resource class.
        /// </summary>
        [global::System.ComponentModel.EditorBrowsableAttribute(global::System.ComponentModel.EditorBrowsableState.Advanced)]
        internal static global::System.Globalization.CultureInfo Culture {
            get {
                return resourceCulture;
            }
            set {
                resourceCulture = value;
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Check for Update.
        /// </summary>
        internal static string AboutDialog_CheckForUpdate {
            get {
                return ResourceManager.GetString("AboutDialog_CheckForUpdate", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to A new version is available..
        /// </summary>
        internal static string AboutDialog_CheckUpdateResultAvailable {
            get {
                return ResourceManager.GetString("AboutDialog_CheckUpdateResultAvailable", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to You have the latest version. .
        /// </summary>
        internal static string AboutDialog_CheckUpdateResultLatest {
            get {
                return ResourceManager.GetString("AboutDialog_CheckUpdateResultLatest", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Email.
        /// </summary>
        internal static string AboutDialog_Email {
            get {
                return ResourceManager.GetString("AboutDialog_Email", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Github Home.
        /// </summary>
        internal static string AboutDialog_GithubHome {
            get {
                return ResourceManager.GetString("AboutDialog_GithubHome", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Github Project Home.
        /// </summary>
        internal static string AboutDialog_GithubProjectHome {
            get {
                return ResourceManager.GetString("AboutDialog_GithubProjectHome", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Mrs4s.
        /// </summary>
        internal static string AboutDialog_Mrs4s {
            get {
                return ResourceManager.GetString("AboutDialog_Mrs4s", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Server Support.
        /// </summary>
        internal static string AboutDialog_Mrs4sDescription {
            get {
                return ResourceManager.GetString("AboutDialog_Mrs4sDescription", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Release Notes.
        /// </summary>
        internal static string AboutDialog_ReleaseNotes {
            get {
                return ResourceManager.GetString("AboutDialog_ReleaseNotes", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Accelerider for Windows.
        /// </summary>
        internal static string AboutDialog_SoftwareName {
            get {
                return ResourceManager.GetString("AboutDialog_SoftwareName", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to About.
        /// </summary>
        internal static string AboutDialog_Title {
            get {
                return ResourceManager.GetString("AboutDialog_Title", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Version.
        /// </summary>
        internal static string AboutDialog_Version {
            get {
                return ResourceManager.GetString("AboutDialog_Version", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to 0.0.1-alpha1.
        /// </summary>
        internal static string AboutDialog_VersionNumber {
            get {
                return ResourceManager.GetString("AboutDialog_VersionNumber", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to ZDP.
        /// </summary>
        internal static string AboutDialog_ZDP {
            get {
                return ResourceManager.GetString("AboutDialog_ZDP", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Windows Desktop Client.
        /// </summary>
        internal static string AboutDialog_ZDPDescription {
            get {
                return ResourceManager.GetString("AboutDialog_ZDPDescription", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to LOGIN.
        /// </summary>
        internal static string AuthenticationWindow_LogIn {
            get {
                return ResourceManager.GetString("AuthenticationWindow_LogIn", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to REGISTER.
        /// </summary>
        internal static string AuthenticationWindow_SignUp {
            get {
                return ResourceManager.GetString("AuthenticationWindow_SignUp", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Confirm.
        /// </summary>
        internal static string Buttons_Confirm {
            get {
                return ResourceManager.GetString("Buttons_Confirm", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Submit.
        /// </summary>
        internal static string Buttons_Submit {
            get {
                return ResourceManager.GetString("Buttons_Submit", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to App Store.
        /// </summary>
        internal static string MainWindow_AppStore {
            get {
                return ResourceManager.GetString("MainWindow_AppStore", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Menu.
        /// </summary>
        internal static string MainWindow_Menu {
            get {
                return ResourceManager.GetString("MainWindow_Menu", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Change Password.
        /// </summary>
        internal static string ProfileDialog_ChangePassword {
            get {
                return ResourceManager.GetString("ProfileDialog_ChangePassword", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Confirm Password.
        /// </summary>
        internal static string ProfileDialog_ConfirmPassword {
            get {
                return ResourceManager.GetString("ProfileDialog_ConfirmPassword", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Email: .
        /// </summary>
        internal static string ProfileDialog_Email {
            get {
                return ResourceManager.GetString("ProfileDialog_Email", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Email Visibility: .
        /// </summary>
        internal static string ProfileDialog_EmailVisibility {
            get {
                return ResourceManager.GetString("ProfileDialog_EmailVisibility", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Multiple Online: .
        /// </summary>
        internal static string ProfileDialog_MultipleOnline {
            get {
                return ResourceManager.GetString("ProfileDialog_MultipleOnline", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to New Password.
        /// </summary>
        internal static string ProfileDialog_NewPassword {
            get {
                return ResourceManager.GetString("ProfileDialog_NewPassword", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Old Password.
        /// </summary>
        internal static string ProfileDialog_OldPassword {
            get {
                return ResourceManager.GetString("ProfileDialog_OldPassword", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Profile.
        /// </summary>
        internal static string ProfileDialog_Title {
            get {
                return ResourceManager.GetString("ProfileDialog_Title", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Username: .
        /// </summary>
        internal static string ProfileDialog_Username {
            get {
                return ResourceManager.GetString("ProfileDialog_Username", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to (.
        /// </summary>
        internal static string Punctuation_LeftParenthesis {
            get {
                return ResourceManager.GetString("Punctuation_LeftParenthesis", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to ).
        /// </summary>
        internal static string Punctuation_RightParenthesis {
            get {
                return ResourceManager.GetString("Punctuation_RightParenthesis", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to About.
        /// </summary>
        internal static string SettingsPopup_About {
            get {
                return ResourceManager.GetString("SettingsPopup_About", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Accelerider for Web.
        /// </summary>
        internal static string SettingsPopup_AcceleriderForWeb {
            get {
                return ResourceManager.GetString("SettingsPopup_AcceleriderForWeb", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Change Profile.
        /// </summary>
        internal static string SettingsPopup_ChangeProfile {
            get {
                return ResourceManager.GetString("SettingsPopup_ChangeProfile", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Help.
        /// </summary>
        internal static string SettingsPopup_Help {
            get {
                return ResourceManager.GetString("SettingsPopup_Help", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Log Out.
        /// </summary>
        internal static string SettingsPopup_LogOut {
            get {
                return ResourceManager.GetString("SettingsPopup_LogOut", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Settings.
        /// </summary>
        internal static string SettingsPopup_Settings {
            get {
                return ResourceManager.GetString("SettingsPopup_Settings", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Application.
        /// </summary>
        internal static string ShellSettingsTabItem_Application {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_Application", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Auto-shutdown the computer after the download is complete.
        /// </summary>
        internal static string ShellSettingsTabItem_AutoShutdown {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_AutoShutdown", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Auto-start application.
        /// </summary>
        internal static string ShellSettingsTabItem_AutoStart {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_AutoStart", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Basic.
        /// </summary>
        internal static string ShellSettingsTabItem_Header {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_Header", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to On close, keep the application running.
        /// </summary>
        internal static string ShellSettingsTabItem_KeepRunning {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_KeepRunning", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Language.
        /// </summary>
        internal static string ShellSettingsTabItem_Language {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_Language", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Theme.
        /// </summary>
        internal static string ShellSettingsTabItem_Theme {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_Theme", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Dark.
        /// </summary>
        internal static string ShellSettingsTabItem_ThemeOptions_Dark {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_ThemeOptions_Dark", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Light.
        /// </summary>
        internal static string ShellSettingsTabItem_ThemeOptions_Light {
            get {
                return ResourceManager.GetString("ShellSettingsTabItem_ThemeOptions_Light", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Automatic Login.
        /// </summary>
        internal static string SignInView_AutomaticLogin {
            get {
                return ResourceManager.GetString("SignInView_AutomaticLogin", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Email.
        /// </summary>
        internal static string SignInView_Email {
            get {
                return ResourceManager.GetString("SignInView_Email", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to LOGIN.
        /// </summary>
        internal static string SignInView_Login {
            get {
                return ResourceManager.GetString("SignInView_Login", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Password.
        /// </summary>
        internal static string SignInView_Password {
            get {
                return ResourceManager.GetString("SignInView_Password", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Remember Me.
        /// </summary>
        internal static string SignInView_RememberMe {
            get {
                return ResourceManager.GetString("SignInView_RememberMe", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Email Address.
        /// </summary>
        internal static string SignUpView_EmailAddress {
            get {
                return ResourceManager.GetString("SignUpView_EmailAddress", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Password.
        /// </summary>
        internal static string SignUpView_Password {
            get {
                return ResourceManager.GetString("SignUpView_Password", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Seconds.
        /// </summary>
        internal static string SignUpView_RetrySeconds {
            get {
                return ResourceManager.GetString("SignUpView_RetrySeconds", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Send Code.
        /// </summary>
        internal static string SignUpView_SendCode {
            get {
                return ResourceManager.GetString("SignUpView_SendCode", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Username (Nick).
        /// </summary>
        internal static string SignUpView_Username {
            get {
                return ResourceManager.GetString("SignUpView_Username", resourceCulture);
            }
        }

        /// <summary>
        ///   Looks up a localized string similar to Verification Code.
        /// </summary>
        internal static string SignUpView_VerificationCode {
            get {
                return ResourceManager.GetString("SignUpView_VerificationCode", resourceCulture);
            }
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Mortgage lenders worried about commonhold insolvency
Mortgage lenders are reluctant to lend money to homebuyers in a commonhold set-up over insolvency fears, says the Law Commission, which has been tasked by the government to find ways to make the leasehold alternative work.
Commonhold allows a person to own a freehold flat and be a member of the company which owns and manages the shared areas and structure of the building. However, it has struggled to gain traction since it was introduced in 2002 – there are only 150 commonhold units in England and Wales.
The commission told a stakeholder engagement event last week that mortgage lenders are reluctant to lend to commonhold buyers because they are concerned about the association potentially becoming insolvent. Nick Roberts, one of the commission's lawyers, said: 'Our priority must be to prevent commonhold associations becoming insolvent in the first place. The first obvious thing to do is require commonhold associations to take out public liability insurance, which is not a requirement under the 2002 act.'
In 2002, Roberts said, it was suggested that commonhold associations might struggle to get public liability insurance. 'Bearing in mind risk management companies and other leasehold-controlled companies, we do not think that's likely. If people are having problems we would like to know about them.'
Commonhold associations should also be required to have 'reserve' or 'sinking' funds, Roberts said.
The commission has been looking at commonhold as part of a wider residential leasehold and commonhold project, which includes leasehold enfranchisement and the right to manage. Its commonhold consultation closes on 10 March.
Kindly shared by Law Society Gazette
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Landlords warning: 3 weeks until tough buy-to-let licensing rules kick in
New Homes England statistics show overall housing starts up in 2019-20


How Dogs by Debin's Dog Rehoming Service Works….
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These are just a FEW of the reasons to utilize our service! Although we know every dog has a different story, we pride ourselves in marketing them to families that will be a good pairing. Please read below to see how our service works.
Before the dog goes into it's new home, it must be heartworm tested, spayed/neutered and microchipped. Dogs by Debin can provide these at the same time we are setting the dog up to market him/her for adoption.
Marketed the dog on Craigslist.org, contacted the shelter near where dog was found and proof you have adequately tried to find the owner, including having it scanned for a microchip. After at least 5 days, please feel free to contact us to get the ball rolling to place the dog up for adoption.
After you have read this in it's entirety and completed the form at the end, please email DogsbyDebin@gmail.com with 2-3 photos of the dog and as much information as you you can about the dog, including what you have done to locate the owner (if a stray). If the dog needs the vetting services listed above, we can help make that happen as well! Otherwise, we encourage you to visit your local Veterinarian. Like mentioned before, every dog is different so we will work with you to accommodate each person/dog's individual needs.
Are you still contemplating using our service? GREAT! Please keep reading so we can get started.
We can not guarantee that a home will be found but we can assure you we will do our best, including a professional photography session within 10-days of signing up for our services at no cost to you. Still interested? Why wouldn't you be? Visit FB.com/DogsbyDebin to like our page and read our reviews. Although we never profess to be perfect, we offer top notch customer service and are readily available to contact so you never feel alone.