text:
\section{Introduction}

For an abelian variety $X$,
we have a notion of dual $\widehat{X}$.
It is an abelian variety
which is the moduli space $\Pic^0(X)$ of numerically trivial line bundles
on $X$, and $X$ is the moduli space of
numerically trivial line bundles on $\widehat{X}$
via the universal family ${\cal P}$:
We have an isomorphism
$X \to \Pic^0(\widehat{X})$ by sending $x \in X$ to
${\cal P}_{|\{x \} \times \widehat{X}}$.
We call this property Fourier-Mukai duality in \cite{Y:duality}, since it was a key property
of Mukai's construction of
an equivalence
${\bf D}(X) \to {\bf D}(\widehat{X})$ \cite{Mukai:1981}.
It is now known that
Fourier-Mukai duality holds for many moduli spaces $X'$ of stable sheaves
on abelian surfaces, K3 surfaces and elliptic surfaces $X$, if $\dim X=\dim X'$ \cite{BBH:1},
\cite{Mu:5}, \cite{PerverseII}.
In \cite{Y:duality}, we explained Fourier-Mukai duality on K3 surfaces by using Bridgeland stability conditions.
 In this note, by introducing a stability condition,
we shall give a similar explanation for an elliptic surface
(Theorem \ref{thm:rk=0}).
our argument is the same as in \cite{Y:duality}.
As an application, we shall recover a result in \cite{Y:7} that
a suitable twisted stability for a semi-stable 1-dimensional sheaf
is preserved under a Fourier-Mukai transform
by using Bridgeland stability condition (see Proposition \ref{prop:r=0}).
As another motivation of this note,
we discuss the relation of walls for Bridgeland stability and $\lambda$-stability
in \cite{Y:elliptic} when $X$ is an elliptic K3 surface.

Let us briefly explain the organization of this note.
In sections \ref{subsect:stability} and \ref{subsect:geometric},
we recall Bridgeland stability conditions.
In section \ref{subsect:ellipticMukai},
we introduce a symmetric pairing on $H^*(X,{\Bbb Q})$
which is preserved under Fourier-Mukai transforms.
By using our pairing, we introduce a stability condition, and show that it is equivalent to
a geometric stability condition constructed by Arcara and Bertram \cite{AB} (Proposition \ref{prop:hat}).
We next study the Fourier-Mukai transform of our stability condition in section \ref{sect:FM}, and prove that
the obtained stability condition is equivalent to
a geometric stability condition (Proposition \ref{prop:FMstability}).
Then Theorem \ref{thm:rk=0} easily follows.
All these arguments are the same as in \cite{Y:duality}.
In section \ref{sect:stability} and section \ref{sect:K3},
we give applications.

For related results, we would like to remark that Lo and Martinez \cite{Lo-M} obtained almost all results in this note
for the Fourier-Mukai transform between a Weierstrass elliptic surface and
its compactfied relative Picard scheme.

\section{Stability conditions and Fourier-Mukai transforms on an elliptic surface.}

\subsection{Stability conditions}\label{subsect:stability}
Let us briefly recall some notation on stability conditions.
For more details, see \cite{Br:stability}, \cite{Br:3} and \cite{Br:4}.
A stability condition $\sigma=(Z_\sigma,{\cal P}_\sigma)$
on ${\bf D}(X)$
consists of a group homomorphism $Z_\sigma: {\bf D}(X) \to {\Bbb C}$
and a slicing ${\cal P}_\sigma$ of ${\bf D}(X)$ such that if
$0 \ne E \in {\cal P}_\sigma(\phi)$ then
$Z_\sigma(E) =m(E)\exp(\pi \sqrt{-1} \phi)$
for some $m(E) \in {\Bbb R}_{>0}$.
The set of stability conditions has a structure of complex
manifold.
We denote this space by $\Stab(X)$.

\begin{defn}\label{defn:B-stable}
A $\sigma$-semi-stable object $E$ of phase $\phi$ is
an object of  ${\cal P}_\sigma(\phi)$.
If $E$ is a simple object of ${\cal P}_\sigma(\phi)$, then
$E$ is $\sigma$-stable.
\end{defn}

By \cite[Prop. 5.3]{Br:stability},
giving a stability condition $\sigma$ is
the same as giving a bounded $t$-structure on ${\bf D}(X)$
and a stability function $Z_\sigma$ on its heart ${\cal A}_\sigma$
with the Harder-Narasimhan property.
For $\sigma=(Z_\sigma,{\cal P}_\sigma)$,
we have the relation ${\cal A}_\sigma ={\cal P}_\sigma((0,1])$,
where ${\cal P}_\sigma((0,1])$ is the subcategory of ${\bf D}(X)$
generated by semi-stable objects $E \in {\cal P}_\sigma(\phi)$ with
$\phi \in (0,1]$.
Since the pair $(Z_\sigma,{\cal A}_\sigma)$
defines a stability condition,
we also use the symbol $\sigma=(Z_\sigma,{\cal A}_\sigma)$
to denote a stability condition.
For a $\sigma$-semi-stable object $E \in
{\cal P}_\sigma(\phi)$, we set $\phi_\sigma(E)=\phi$.
For $E \in {\cal P}_\sigma((0,1])$,
we also set $\phi_\sigma(E) \in (0,1]$
by $Z_\sigma(E)=m(E)\exp(\pi \sqrt{-1} \phi_\sigma(E))$.

For an equivalence $\Phi:{\bf D}(X) \to {\bf D}(X')$,
we have an isomorphism $\Phi:\Stab(X) \to \Stab(X')$
such that
$\Phi(\sigma)$ $(\sigma \in \Stab(X))$
is a stability condition given by
\begin{equation}\label{eq:FM-action}
\begin{split}
Z_{\Phi(\sigma)}=& Z_\sigma \circ \Phi^{-1}:{\bf D}(X') \to {\Bbb C},\\
{\cal P}_{\Phi(\sigma)}(\phi)=& \Phi({\cal P}_\sigma(\phi)).
\end{split}
\end{equation}
We also have an action of the universal covering
$\widetilde{\GL}_2^+({\Bbb R})$ of $\GL_2^+({\Bbb R})$ on
$\Stab(X)$.
Since ${\Bbb C}^{\times} \subset  \GL_2^+({\Bbb R})$,
we have an injective homomorphism
${\Bbb C} \to  \widetilde{\GL}_2^+({\Bbb R})$.
Thus
we have an action of $\lambda \in {\Bbb C}$ on
$\Stab(X)$.
For a stability condition $\sigma \in \Stab(X)$,
$\lambda(\sigma)$ is
given by
\begin{equation}\label{eq:C-action}
\begin{split}
Z_{\lambda(\sigma)}=&
\exp(-\pi  \sqrt{-1} \lambda)Z_\sigma\\
{\cal P}_{\lambda(\sigma)}(\phi)=&
{\cal P}_\sigma(\phi+\mathrm{Re}\lambda).
\end{split}
\end{equation}

\subsection{Geometric stability conditions}\label{subsect:geometric}
Bridgeland \cite{Br:3} and Arcara and Bertram \cite{AB}
constructed a stability condition
$\sigma_{(\beta,\omega)}=(Z_{(\beta,\omega)},{\cal A}_{(\beta,\omega)})$
which is characterized
by the stability of $k_x$ ($x \in X$), where
$Z_{(\beta,\omega)}(\bullet)=
\langle e^{\beta+\sqrt{-1}\omega},\ch(\bullet) \rangle:
{\bf D}(X) \to {\Bbb C}$ is the stability function and
${\cal A}_{(\beta,\omega)}$
is an abelian category which is a tilting of $\Coh(X)$
by a torsion pair $({\cal T}_{(\beta,\omega)},{\cal F}_{(\beta,\omega)})$:
\begin{enumerate}
\item
${\cal T}_{(\beta,\omega)}$ is a full subcategory of $\Coh(X)$ generated by
torsion sheaves and $\mu$-stable sheaves $E$ with
$(c_1(E)-\rk E \beta,\omega)>0$ and
\item
${\cal F}_{(\beta,\omega)}$ is a full subcategory of $\Coh(X)$ generated by
$\mu$-stable sheaves $E$ with
$(c_1(E)-\rk E \beta,\omega) \leq 0$.
\end{enumerate}
This kind of stability conditions are called geometric.
To be more precise,
for a geometric stability condition
$\sigma=(Z_\sigma,{\cal A}_\sigma)$,
we require that
$\mathrm{Re}\mho, \mathrm{Im}\mho$ span a positive definite
2-plane of $H^*(X,{\Bbb R})$,
where $Z_\sigma(\bullet)=\langle \mho,\bullet \rangle$
with $\mho \in H^*(X,{\Bbb Q})_{\alg} \otimes {\Bbb C}$.
Up to the action of $\widetilde{\GL}_2^+({\Bbb R})$,
there is $(\beta,\omega)$ such that
$\sigma=\sigma_{(\beta,\omega)}$.
We shall use this fact for a stability condition
on an elliptic surface (see subsection \ref{subsect:ellipticStab} in particular Lemma \ref{lem:geometric}).
\begin{NB}
Old:
$\sigma \in \Stab(X)$ satisfies
(1)
$Z_\sigma(\bullet)=\langle \mho,\bullet \rangle$
with $\langle \mathrm{Re}\mho^2 \rangle=
\langle \mathrm{Im}\mho^2 \rangle$,
$\langle \mathrm{Re}\mho,\mathrm{Im}\mho \rangle=0$
and
(2)
$k_x$ ($x \in X$) are $\sigma$-stable
with the phase $\phi_{\sigma}(k_x)=1$. Then
$\sigma=\sigma_{(\beta,\omega)}$ for some $(\beta,\omega)$.
\end{NB}
\begin{NB}
\begin{rem}
In order to construct a geometric stability condition $\sigma$
with $Z_\sigma=Z_{(\beta,\omega)}$,
we require that
$$
(\beta, \omega) \in
\NS(X)_{\Bbb R} \times \Amp(X)_{\Bbb R} \setminus \bigcup_{v_1 \in \Delta(X)}
\left\{(\beta,\omega) \left| \langle e^{\beta+\sqrt{-1} \omega},v_1 \rangle
\in {\Bbb R}_{\leq 0} \right. \right\},
$$
where $\Delta(X)$ is the set of  $(-2)$-vector of $H^*(X,{\Bbb Z})_{\alg}$.
\end{rem}
\end{NB}

\subsection{Mukai pairing and Fourier-Mukai transform on an elliptic surface.}\label{subsect:ellipticMukai}

Let $\pi:X \to C$ be an elliptic surface over a curve $C$
and $f$ a fiber of $\pi$.
Let
$$
H^{*}(X,{\Bbb Q})_{\alg}:=H^0(X,{\Bbb Q}) \oplus \NS(X)_{\Bbb Q} \oplus H^4(X,{\Bbb Q})
$$
be the algebraic part of the cohomology ring $H^*(X,{\Bbb Q})$.
We define $\varrho_X \in H^4(X,{\Bbb Z})$ by
$\int_X \varrho_X=1$.
Then we have
$$
H^{*}(X,{\Bbb Q})_{\alg}={\Bbb Q} \oplus \NS(X)_{\Bbb Q} \oplus {\Bbb Q}\varrho_X.
$$
For $u=x_0+x_1+x_2 \varrho_X$ ($x_0,x_2 \in {\Bbb Q}$, $x_1 \in \NS(X)_{\Bbb Q}$),
we set $u^*:=x_0-x_1+x_2 \varrho_X$.

\begin{defn}
\begin{enumerate}
\item[(1)]
We define the Mukai pairing $\langle \;\;,\;\; \rangle$ on
$H^*(X,{\Bbb Q})_{\alg}$
by
\begin{equation}
\langle u,v \rangle=\int_X u^* v=(x_1 \cdot y_1)-x_0 y_2-x_2 y_0,\;
u=x_0+x_1+x_2 \varrho_X, v=y_0+y_1+y_2 \varrho_X,
\end{equation}
where $(x_1 \cdot y_1)$ is the intersection pairing of $\NS(X)_{\Bbb Q}$.
\item[(2)]
For $E \in {\bf D}(X)$, we define the Mukai vector of $E$ by
\begin{equation}
v(E):=\ch E (1+\tfrac{\chi({\cal O}_X)}{2}\varrho_X) \in H^*(X,{\Bbb Q})_{\alg}.
\end{equation}
\end{enumerate}
\end{defn}

\begin{lem}
For $E, F \in {\bf D}(X)$,
\begin{equation}
-\langle v(E),v(F) \rangle=\chi(E,F(\tfrac{1}{2}K_X)).
\end{equation}
\end{lem}

\begin{proof}
\begin{equation}
\chi(E,F(\tfrac{1}{2}K_X))=\int_X (\ch E)^* \ch F e^{\frac{1}{2}K_X}\td_X=
\int_X v(E)^* v(F)=-\langle v(E),v(F) \rangle.
\end{equation}
\end{proof}

\begin{defn}\label{defn:moduli}
Let $v \in H^*(X,{\Bbb Q})_{\alg}$ be a Mukai vector (of a coherent sheaf).
\begin{enumerate}
\item[(1)]
For an ample divisor $H$ on $X$ and $\beta \in \NS(X)_{\Bbb Q}$,
let $\Mod_H^\beta(v)$ be the moduli space of
$S$-equivalence classes of
$\beta$-twisted semi-stable sheaves $E$ with
the Mukai vector $v(E)=v$
and ${\cal M}_H^\beta(v)^{ss}$ the moduli stack of
$\beta$-twisted  semi-stable sheaves $E$ with $v(E)=v$.
\item[(2)]
For a stability condition $\sigma$,
we denote the moduli stack of $\sigma$-semi-stable objects $E$ with
$v(E)=v$ by ${\cal M}_\sigma(v)$,
where we usually choose
$\phi:=\mathrm{Im}(\log Z_\sigma(v))/\pi \in (-1,1]$.
We also denote the moduli scheme by $M_\sigma(v)$ if it exists.
\end{enumerate}
\end{defn}

\begin{NB}
\begin{defn}
Let $\sigma$ be a stability condition.
For $v \in H^*(X,{\Bbb Q})_{\alg}$,
we denote the moduli stack of $\sigma$-semi-stable objects $E$ with
$v(E)=v$ by ${\cal M}_\sigma(v)$,
where we usually choose
$\phi:=\mathrm{Im}(\log Z_\sigma(v))/\pi \in (-1,1]$.
\end{defn}
\end{NB}

$M_H^\beta(v)$ is a projective scheme.
If $H$ is general in the ample cone,
$\Mod_H^\beta(v)$ and ${\cal M}_H^\beta(v)^{ss}$ are
 independent of the choice of $\beta$.

\begin{defn}
For elliptic surfaces $X \to C$, $Y \to C$
and ${\bf P} \in {\bf D}(X \times Y)$,
let $\Phi_{X \to Y}^{{\bf P}}:{\bf D}(X) \to {\bf D}(Y)$ be an integral functor
whose kernel is ${\bf P}$:
\begin{equation}
\Phi_{X \to Y}^{{\bf P}}(E)={\bf R}p_{Y*}(p_X^*(E) \otimes {\bf P}), \; \; E \in {\bf D}(X).
\end{equation}
We also set
\begin{equation}
v({\bf P}):=\ch ({\bf P})p_X^*((1+\tfrac{\chi({\cal O}_X)}{2}\varrho_X))
p_Y^*((1+\tfrac{\chi({\cal O}_Y)}{2}\varrho_Y)) \in H^*(X \times Y,{\Bbb Q}).
\end{equation}
\end{defn}

As a consequence of Grothendieck Riemann-Roch theorem, we have the following.
\begin{lem}
For $\Phi:=\Phi_{X \to Y}^{{\bf P}^{\vee}}$,
we have a commutative diagram

\begin{equation}
\begin{CD}
{\bf D}(X) @>{\Phi}>> {\bf D}(Y)\\
@V{v}VV @VV{v}V \\
H^*(X,{\Bbb Q})_{\alg} @>{\overline{\Phi}}>> H^*(Y,{\Bbb Q})_{\alg}
\end{CD}
\end{equation}
where
\begin{equation}
\overline{\Phi}(\alpha):=p_{Y*}(v({\bf P}^{\vee})p_X^*(e^{-\tfrac{1}{2}K_X})p_X^*(\alpha)).
\end{equation}
\end{lem}

\begin{NB}
Serre duality implies
\begin{equation}
\begin{split}
\chi(E,F(\tfrac{1}{2}K_X))=&\chi(F(\tfrac{1}{2}K_X),E(K_X))=\chi(F,E(\tfrac{1}{2}K_X)).
\end{split}
\end{equation}
\end{NB}

\begin{lem}\label{lem:fiber}
Assume that $\Phi$ is a relative Fourier-Mukai transform, that is,
${\bf P}_{|X \times \{ y \}}$ $(y \in Y)$ is a stable 1-dimensional sheaf on a fiber of $\pi$.
Then
for $E \in {\bf D}(X)$,
$\overline{\Phi}(v(E)e^{\lambda f})=e^{\lambda f}\overline{\Phi}(v(E))$.
\end{lem}

\begin{proof}
By
\begin{equation}
(\ch E)e^{\lambda f}=\ch E+\lambda \ch(E_{|f}),
\end{equation}
we get
\begin{equation}
v(E)e^{\lambda f}=v(E)+\lambda v(E_{|f}).
\end{equation}
Then we see that

\begin{equation}
\begin{split}
v(\Phi(E))e^{\lambda f}=& v (\Phi(E))+\lambda v(\Phi(E)_{|f})\\
=& v(\Phi(E))+\lambda v(\Phi(E_{|f}))\\
=& \overline{\Phi}(v(E))+ \overline{\Phi}(\lambda v(E_{|f}))\\
=&\overline{\Phi}( v(E)e^{\lambda f}).
\end{split}
\end{equation}
\end{proof}

\begin{prop}\label{prop:isometry}
Let $\Phi$ be an equivalence in Lemma \ref{lem:fiber}. Then
$\overline{\Phi}$ preserves
$\langle\;\;,\;\; \rangle$.
Thus
$$
\langle \overline{\Phi}(v(E_1)),\overline{\Phi}(v(E_2)) \rangle=\langle v(E_1),v(E_2) \rangle,\;
E_1,E_2 \in {\bf D}(X).
$$
\end{prop}

\begin{proof}
Let $\widehat{\Phi}:{\bf D}(Y) \to {\bf D}(X)$ be a quasi-inverse of $\Phi$.
For $E \in {\bf D}(X)$ and $F \in {\bf D}(Y)$, by using Lemma \ref{lem:fiber},
we get
\begin{equation}
\begin{split}
\chi(E,\widehat{\Phi}(F)(\tfrac{1}{2}K_X))=&\chi(E(-\tfrac{1}{2}K_X),\widehat{\Phi}(F))\\
=& \chi(\Phi(E(-\tfrac{1}{2}K_X)),F)\\
=& \chi(\Phi(E)(-\tfrac{1}{2}K_Y),F)\\
=&\chi(\Phi(E),F(\tfrac{1}{2}K_Y)).
\end{split}
\end{equation}
Hence our claim holds.
\end{proof}

\begin{NB}
\begin{lem}
$\chi(\Phi_{X \to Y}^{{\bf P}^* \otimes p_X^*(\omega_X)}(E),F)=
\chi(E, \Phi_{Y \to X}^{{\bf P}}(F))$.
\end{lem}

\begin{proof}
By the Serre duality, we get
\begin{equation}
\begin{split}
v(\Phi_{X \to Y}^{{\bf P}^* \otimes p_X^*(\omega_X)}(E))^*=& p_{Y*}(v({\bf P})^* p_X^*(v(E)e^{K_X})p_X^*(e^{-\frac{1}{2}K_X}))^*\\
=& p_{Y*}((v({\bf P}) p_X^*(v(E)^*)p_X^*(e^{-\frac{1}{2}K_X})).
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\int_Y v(\Phi_{X \to Y}^{{\bf P}^* \otimes p_X^*(\omega_X)}(E))^* v(F)e^{-\frac{1}{2}K_Y}=&
\int_Y p_{Y*}((v({\bf P}) p_X^*(v(E)^*)p_X^*(e^{-\frac{1}{2}K_X})) v(F) e^{-\frac{1}{2}K_Y}\\
=& \int_{X \times Y} v({\bf P}) p_X^*(v(E)^*)p_X^*(e^{-\frac{1}{2}K_X})
p_Y^*(v(F) e^{-\frac{1}{2}K_Y})\\
=& \int_X p_{X*}(v({\bf P}) p_X^*(v(E)^*)p_X^*(e^{-\frac{1}{2}K_X})
p_Y^*(v(F) e^{-\frac{1}{2}K_Y}))\\
=& \int_X v(E)^* e^{-\frac{1}{2}K_X}  p_{X*}(v({\bf P})
p_Y^*(v(F) e^{-\frac{1}{2}K_Y}))\\
=& \int_X v(E)^* e^{-\frac{1}{2}K_X} v(\Phi_{Y \to X}^{{\bf P}}(F))
\end{split}
\end{equation}
\end{proof}
\end{NB}

\subsection{A stability condition on an elliptic surface.}\label{subsect:ellipticStab}

Let $\pi:X \to C$ be an elliptic surface over a curve $C$.
We shall introduce a stability condition which has a better behavior than
the stability condition $\sigma_{(\beta, \omega)}$ in subsection \ref{subsect:geometric}
under the Fourier-Mukai transform $\Phi$.

\begin{defn}\label{defn:hat}
For $(\beta,\omega) \in \NS(X)_{\Bbb R} \times \Amp(X)_{\Bbb R}$ with
$(\omega^2)>\chi({\cal O}_X)$,
we set
$$
\widehat{Z}_{(\beta,\omega)}(E):=\langle e^{\beta+\sqrt{-1} \omega}, v(E) \rangle,\; E \in {\bf D}(X)
$$
and set
\begin{equation}
\widehat{\sigma}_{(\beta,\omega)}:=(\widehat{Z}_{(\beta,\omega)},{\cal A}_{(\beta,\omega)}).
\end{equation}
\end{defn}

We shall prove that $\widehat{\sigma}_{(\beta,\omega)}$ is equivalent to the stability condition
$\sigma_{(\beta,\omega')}$ by the action of $\widetilde{\GL}_2^+({\Bbb R})$.
For $H \in \Amp(X)_{\Bbb R}$, we set $\omega:=tH$ and
$s:=\sqrt{t^2-\frac{\chi({\cal O}_X)}{(H^2)}}$, where $t^2-\frac{\chi({\cal O}_X)}{(H^2)}>0$.
Let $(T,f) \in \widetilde{\GL}_2^+({\Bbb R})$ be an element such that
$T \in \GL_2^+({\Bbb R})$ acts on ${\Bbb R}^2={\Bbb R}+{\Bbb R}\sqrt{-1}$ as
\begin{equation}
T(x+y \sqrt{-1})=x+\tfrac{s}{t}y \sqrt{-1}
\end{equation}
and the increasing function $f:{\Bbb R} \to {\Bbb R}$ satisfies $f(n)=n$ for $n \in {\Bbb Z}$.
Then we get the following result.
\begin{prop}\label{prop:hat}
$(T,f) \cdot(Z_{(\beta,sH)},{\cal A}_{(\beta,sH)})=(\widehat{Z}_{(\beta,tH)},{\cal A}_{(\beta,tH)})$.
In particular $\widehat{\sigma}_{(\beta,\omega)}$ is a stability condition such that
$k_x$ $(x \in X)$ are stable objects of phase 1.
\end{prop}

\begin{proof}
We set $\sigma':=(T,f) \cdot \sigma_{(\beta,sH)}$.

\begin{equation}
\widehat{Z}_{(\beta,tH)}(E)=T^{-1} \circ Z_{(\beta,sH)}(E).
\end{equation}
We set
\begin{equation}
\ch(E)=e^\beta (r+\xi+a\varrho_X), r \in {\Bbb Z},\; \xi \in \NS(X)_{\Bbb R}, a \in {\Bbb R}.
\end{equation}
Then
\begin{equation}
\begin{split}
\widehat{Z}_{(\beta,tH)}(E)= & \langle e^\beta(1+\sqrt{-1} tH-\tfrac{t^2(H^2)}{2}\varrho_X),
e^\beta (r+\xi+a\varrho_X)(1+\tfrac{\chi({\cal O}_X)}{2}\varrho_X) \rangle\\
=& \frac{r}{2}(t^2(H^2)-\chi({\cal O}_X))-a+\sqrt{-1} t(H \cdot \xi)\\
=& \frac{r}{2}s^2 (H^2) -a+\sqrt{-1} \frac{t}{s} (sH \cdot \xi)\\
=& T^{-1} \circ Z_{(\beta,sH)}.
\end{split}
\end{equation}
By the definition of $(T,f)$, we have ${\cal P}_{\sigma'}((0,1])={\cal P}_{\sigma_{(\beta,sH)}}((0,1])=
{\cal A}_{(\beta,sH)}={\cal A}_{(\beta,tH)}$.
Hence the claim holds.
\end{proof}

The following claim is proved in \cite[Prop. 10.3]{Br:3}.
\begin{lem}\label{lem:geometric}
Let $H$ be a ${\Bbb Q}$-divisor with $(H^2)>0$.
Let $\sigma=(\langle e^{\beta+\sqrt{-1}tH}, v(\bullet) \rangle,{\cal A})$
be a stability condition such that $k_x$ $(x \in X)$ are stable objects of phase 1.
Then ${\cal A}={\cal A}_{(\beta,tH)}$.
\end{lem}

\begin{prop}[{\cite[Prop. 3.2.1]{MYY:2011:1}, \cite[Prop. 6.11]{Y:blowup}}]\label{prop:asymptotic}
Let $L$ be an ample ${\Bbb Q}$-divisor.
For a Mukai vector $v=e^\beta(r+\xi+a \varrho_X)$, we
decompose
$\xi$ as
$$
\xi=dL+D,\; d=\frac{(\xi \cdot L)}{(L^2)},\; D \in L^\perp.
$$
We set  $\delta:=\min\{(D \cdot L)>0 \mid D \in \NS(X) \}/(L^2)$.
Then the following claims hold.
\begin{enumerate}
\item[(1)]
Assume that $r>0$.
Then
${\cal M}_{\widehat{\sigma}_{(\beta,tL)}}(v)={\cal M}_L^{\beta-\frac{1}{2}K_X}(v)^{ss}$
if
\begin{equation}
\frac{t^2(L^2)-\chi({\cal O}_X)}{2}>
\frac{d}{\delta}
\left(d^2\frac{(L^2)}{2}-ra +r^2 \frac{\chi({\cal O}_X)}{2} \right).
\end{equation}
\item[(2)]
Assume that $r=0$. Then
${\cal M}_{\widehat{\sigma}_{(\beta,tL)}}(v)={\cal M}_L^{\beta-\frac{1}{2}K_X}(v)^{ss}$
if
\begin{equation}
\frac{t^2(L^2)-\chi({\cal O}_X)}{2}>
\frac{d}{\delta}
\left(|a|+d^2(L^2) \right).
\end{equation}
\end{enumerate}
\end{prop}

\begin{proof}
We note that
${\cal M}_{\widehat{\sigma}_{(\beta,tL)}}(v)={\cal M}_{\sigma_{(\beta,sL)}}(v)$
by Proposition \ref{prop:hat},
where $s^2 (L^2)=t^2 (L^2)-\chi({\cal O}_X)$.
By the definition of $v(E)$,
$\langle v(E)^2 \rangle=\langle \ch(E)^2 \rangle-r^2 \chi({\cal O}_X)$.
Then
\cite[Prop. 6.11]{Y:blowup} implies the claim.
\end{proof}

\begin{rem}\label{rem:delta}
If $l L \in \NS(X)$, then $\delta \geq \frac{1}{l}$.
\end{rem}

\begin{NB}
\subsection{}

If there is a section, then the wall-crossing problem is reduced to
the wall-crossing behavior of twisted stability for 1-dimensional sheaves.

It is necessary to relate the birational equivalence classes and
the wall crossing.

\begin{equation}
\begin{split}
-\langle \ch F,\ch E \rangle
=& \int_X (\ch F)^{\vee} \ch E \\
=&
\int_X \ch (F(-\tfrac{1}{2}K_X))^{\vee} \ch E (\td_X-\chi({\cal O}_X))\\
=& \chi(F(-\tfrac{1}{2}K_X),E)-\rk E \chi({\cal O}_X)
\end{split}
\end{equation}

\begin{equation}
\begin{split}
-\langle e^{\beta+\sqrt{-1}\omega},\ch E \rangle
=& \int_X e^{-(\beta+\sqrt{-1}\omega} \ch E \\
=&
\int_X e^{-(\beta-\frac{1}{2}K_X +\sqrt{-1} \omega)}\ch E (\td_X-\chi({\cal O}_X))\\
=& \chi(e^{\beta-\frac{1}{2}K_X +\sqrt{-1} \omega},E)-\rk E \chi({\cal O}_X)
\end{split}
\end{equation}

\begin{equation}
\begin{split}
-\langle e^{\beta+\sqrt{-1}\omega},\ch \Phi(E) \rangle
=& \chi(e^{\beta-\frac{1}{2}K_X +\sqrt{-1} \omega},\Phi(E))-\rk \Phi(E) \chi({\cal O}_X)\\
=& \chi(\Phi(e^{\beta-\frac{1}{2}K_X +\sqrt{-1} \omega}),E)-\rk \Phi(E) \chi({\cal O}_X)\\
=& -\langle \Phi(e^{\beta-\frac{1}{2}K_X +\sqrt{-1} \omega})(\tfrac{1}{2}K_X),\ch E \rangle
+(\rk E \rk \Phi(e^{\beta-\frac{1}{2}K_X +\sqrt{-1} \omega})-\rk \Phi(E) \chi({\cal O}_X)
\end{split}
\end{equation}

If $X=C \times D$, then
$\chi({\cal O}_X)=0$ and $\Phi(E(-\tfrac{1}{2}K_X))=\Phi(E)(-\frac{1}{2}K_X)$.
Hence $\Phi$ is an isometry.
\end{NB}

\section{Fourier-Mukai duality}\label{sect:FM}

\subsection{A Fourier-Mukai transform on an elliptic surface.}

Let $\pi:X \to C$ be an elliptic surface over a curve $C$
and $f$ a fiber of $\pi$.
Let $H$ be a relatively ample ${\Bbb Q}$-divisor such that
$(H \cdot f)=1$.
Let $v_0=\eta+b \varrho_X$ be a primitive and isotropic Mukai vector
such that $\eta=r_0 f$, that is, $\eta$ is a nef and effective divisor with
$(\eta^2)=0$.
Replacing $H$ by $H+x f$, we may assume that
$(H^2)=0$.
We shall consider $\beta$-twisted stability with respect to
$H_n:=H+nf$ $(n \gg 0)$.
We can take $e^\beta$ such that $\langle e^\beta,v_0 \rangle=0$.
Indeed we can choose $y$
satisfying
$\langle e^{\beta+yH_n},v_0 \rangle=\langle e^\beta,v_0 \rangle-y(H \cdot \eta)=0$.
Then $v_0=\eta e^\beta$.
Assume that $X':=\Mod_{H_n}^\beta(v_0)$ is a fine moduli
scheme consisting of $\beta$-twisted stable sheaves. Then
$X'$ is an elliptic surface with a fibration
$\pi':X' \to C$. We denote a fiber of $X' \to C$ by $f'$.

\begin{lem}
${\cal M}_{H+nf}^\beta(v_0)^{ss}$ depends only on
$H$ and
$\beta \mod {\Bbb Q}f+{\Bbb Q}H$.
In particular ${\cal M}_{H+nf}^{\beta-\frac{1}{2}K_X}(v_0)^{ss}={\cal M}_{H+nf}^\beta(v_0)^{ss}$.
\end{lem}

\begin{proof}
Let $E$ be a purely 1-dimensional sheaf with $v(E)=v_0$.
For an exact sequence
$$
0 \to E_1 \to E \to E_2 \to 0,
$$

$$
0=(c_1(E) \cdot f)=(c_1(E_1) \cdot f)+(c_1(E_2) \cdot f)
$$
 implies
that $(c_1(E_1) \cdot f)=(c_1(E_2) \cdot f)=0$.
Hence $(c_1(E_i) \cdot (H+yf))$ $(i=1,2)$ is independent of $y$.
Since
$$
\frac{\chi(e^{\beta+xH+yf},E_i)}{(c_1(E_i) \cdot (H+nf))}=
\frac{\chi(e^{\beta+xH+yf},E_i)}{(c_1(E_i) \cdot H)}=
\frac{\chi(e^\beta,E_i)}{(c_1(E_i) \cdot H)}-x,
$$
the twisted semi-stability is independent of the choice of $x,y$.
\end{proof}

For a universal family ${\cal E}$,
we have a Fourier-Mukai transform
$\Phi:=\Phi_{X \to X'}^{{\cal E}^{\vee}[2]}$.
We set $v_0':=v({\cal E}_{|\{ x \} \times X'}^{\vee}[1])=
-\overline{\Phi}(\varrho_X)$ and $H':=c_1(\overline{\Phi}(r_0 e^\beta))$.
Then $c_1(v_0')=r_0 f'$, $({H'}^2)=0$ and $(H' \cdot f')=1$.
\begin{NB}
$r_0 (H',f')=(H',c_1(v_0'))=\langle r_0 e^\beta,-\varrho_X \rangle=r_0$.
\end{NB}
Since $\langle \overline{\Phi}(He^\beta)^2 \rangle=0$ and
$\langle \overline{\Phi}(He^\beta),\varrho_{X'} \rangle=(H \cdot \eta)=r_0$,
there is $\beta' \in \NS(X')_{\Bbb Q}$ with
$\overline{\Phi}(He^\beta)=-r_0 e^{\beta'}$.
Then we have
\begin{equation}\label{eq:H'}
\begin{split}
\overline{\Phi}(r_0 e^\beta)&=H' e^{\beta'}\\
\overline{\Phi}(He^\beta)&=-r_0 e^{\beta'}\\
\overline{\Phi}(v_0)&=\varrho_{X'}\\
\overline{\Phi}(\varrho_X)&=-v_0'.
\end{split}
\end{equation}
By $\langle \overline{\Phi}(He^\beta),\overline{\Phi}(\varrho_X) \rangle=0$, we get
$v_0'=e^{\beta'}(r_0 f')$.
\begin{NB}
$\Phi({\cal E}_{|X \times \{ x' \}})={\Bbb C}_{x'}$.
$\Phi({\Bbb C}_x)=({\cal E}_{|\{x \} \times X'}^{\vee}[1])[-1]$.
For a torsion free sheaf $E$ on an irreducible curve $C \in |nH|$,
$\Phi(E)[-1] \in \Coh(X')$.
\end{NB}
Hence we obtain the following lemma.
\begin{lem}\label{lem:coh-FM}
For
$$
v=e^\beta(r+\xi+a \varrho_X)=e^\beta(r+pH+qf+D+a \varrho_X), D \in f^\perp \cap H^\perp,
$$
we have
$$
\overline{\Phi}(v)=e^{\beta'}(\tfrac{r}{r_0}H'-pr_0+\frac{q}{r_0}\varrho_{X'}+D'-ar_0 f'),\;
D' \in \NS(X') \cap {H'}^\perp \cap {f'}^\perp.
$$
\end{lem}

\subsection{A Fourier-Mukai transform and a stability condition.}

We take $l,l' \in {\Bbb Z}$ such that $lH \in \NS(X), l' H' \in \NS(X')$.
Since
\begin{equation}
e^{\beta+\sqrt{-1}m(H+nf)}=
e^\beta+\sqrt{-1}m( He^\beta+n f e^\beta)
-nm^2 \varrho_X,
\end{equation}
we have
\begin{equation}\label{eq:elliptic:Phi}
\begin{split}
\overline{\Phi}(e^{\beta+\sqrt{-1}m(H+nf)})
=& \frac{1}{r_0}H' e^{\beta'}+\sqrt{-1}m(-r_0 e^{\beta'}+(n/r_0) \varrho_{X'})
+nm^2 v_0'\\
=& -\sqrt{-1}r_0 m e^{\beta'+\sqrt{-1}\frac{1}{r_0^2m}(H'+r_0^2 m^2 nf')}.
\end{split}
\end{equation}
Then we have a generalization of \cite[2.5]{Y:duality}.
\begin{prop}\label{prop:FMstability}
Assume that $n>\frac{r_0^2}{2}\chi({\cal O}_X)$ and
$n>\tfrac{l r_0^3}{2n}+\frac{\chi({\cal O}_X)}{2}$. Then.
\begin{equation}
\Phi(\widehat{\sigma}_{(\beta,mH_n)}) =
\alpha( \widehat{\sigma}_{(\beta',\frac{1}{r_0^2 m}H'_{r_0^2 m^2 n})}),\;
\alpha=\frac{1}{2}+\frac{\log(r_0 m)}{\pi}\sqrt{-1},
\end{equation}
where we set $H_n':=H'+nf'$.
\begin{NB}
Old version
\begin{equation}
\Phi(\widehat{\sigma}_{(\beta,mH_n)}) =
r_0 m \exp \left(-\pi \frac{\sqrt{-1}}{2}\right)
\widehat{\sigma}_{(\beta',\frac{1}{r_0^2 m}H'_{r_0^2 m^2 n})}),
\end{equation}
\end{NB}
\end{prop}

\begin{proof}
We note that
\begin{equation}
v_0=e^\beta(d H_n+D_n), \; d_n=\frac{(\eta \cdot H_n)}{(H_n^2)}, D_n \in H_n^\perp.
\end{equation}
Since $(H_n^2)=2n$ and
$d_n=\frac{r_0}{(H_n^2)}=\frac{r_0}{2n}$, we get
\begin{equation}
\frac{m^2(H_n^2)-\chi({\cal O}_X)}{2}>
r_0 l \frac{(\eta \cdot H_n)^2}{(H_n^2)} .
\end{equation}
Proposition \ref{prop:asymptotic} implies that
${\cal E}_{|X \times \{ x' \}} \in {\cal M}_{(\beta,mH_n)}(v_0)$.
Since
$\widehat{Z}_{(\beta,mH_n)}({\cal E}_{|X \times \{ x' \}})
\in \sqrt{-1}{\Bbb R}_{>0}$,
$k_{x'}$ are $\Phi(\widehat{\sigma}_{(\beta,mH_n)})$-stable objects of
phase $\frac{1}{2}$.
By \eqref{eq:elliptic:Phi} and Lemma \ref{lem:geometric},
our claim holds.
\begin{NB}
$$
\phi_{(\beta,mH_n)}({\cal E}_{|X \times \{x'\}})
=\phi_{(\beta',\frac{1}{r_0^2 m}H'_{r_0^2 m^2 n})}(k_{x'})-\frac{1}{2}.
$$
\end{NB}
\end{proof}

\begin{rem}
If $X$ is a Weierstrass elliptic surface and $r_0=1$, then
a similar claim is proved in \cite{Lo-M} and \cite[Thm. 11.7]{Lo}.

\end{rem}

\begin{rem}
For complex numbers $z,w$ in the upper half plane,
we can also prove that
\begin{equation}
\overline{\Phi}(e^{\beta+zH+w f})=
-r_0 z e^{\beta'+\tfrac{-1}{r_0^2 z}H'+wf'}
\end{equation}
and get a similar claim to Proposition \ref{prop:FMstability} if we set
$\alpha:=\frac{1}{2}+\frac{\log(r_0 z)}{\pi}\sqrt{-1}$.
\begin{NB}
It may be safe to assume that $(z,w)$ is in a neighborhood of $(m \sqrt{-1},mn \sqrt{-1})$.
\end{NB}
\begin{NB}
\begin{equation
\begin{split}
\overline{\Phi}(e^{\beta+aH+bf+\sqrt{-1}m(H+nf)})
=& \frac{1}{r_0}H' e^{\beta'}-(a+b \sqrt{-1})r e^{\beta'}+(b+mn \sqrt{-1})\tfrac{1}{r_0}\varrho_{X'}
-(a+m \sqrt{-1})(b+mn \sqrt{-1})v_0'\\
=& -r(a+m \sqrt{-1})
\exp \left(\beta'+\frac{-a}{r^2(a^2+m^2)}H'+bf'+\sqrt{-1}\frac{m}{r^2(a^2+m^2)}(H'+r^2 (a^2+m^2) nf)' \right).
\end{split}
\end{equation}
\end{NB}
\end{rem}

\begin{NB}
By \cite[Prop. 3.2.1]{MYY:2011:1} (see also \cite[Prop. 6.11]{Y:blowup}), if $H_n$ is ample and
\begin{equation}\label{eq:elliptic:star}
\frac{m^2(H_n^2)}{2}>
1+\frac{(\eta,H_n)^2}{2(H_n^2)},
\end{equation}
\begin{NB2}
$v=e^\beta \eta$.
$\eta=\frac{(\eta \cdot H_n)}{(H_n^2)}H_n+D_n$,
$D_n \in H_n^\perp$.
In particular $a_\beta=0$.

By \cite[6.2]{Y:blowup},
\begin{equation}
\frac{m^2(H_n^2)}{2}>
\frac{(\eta,H_n)}{\delta}
\left(|a_\beta|+1+\frac{(\eta,H_n)^2}{(H_n^2)} \right)
\end{equation}
Since $(f \cdot H)=1$,
\begin{equation}
\frac{m^2(H_n^2)}{2}>
\frac{(\eta,H_n)}{\delta}
\left(|a_\beta|+1+\frac{(\eta,H_n)^2}{(H_n^2)} \right)
\end{equation}

\end{NB2}
then
${\cal M}_{(\beta,mH_n)}(v_0)={\cal M}_{H_n}^\beta(v_0)^{ss}$.
In particular, for $n \gg 0$, we get the claim.

\begin{rem}
Under \eqref{eq:elliptic:star},
$H_{r^2 m^2 n}'$ is ample.
\end{rem}

\end{NB}

\subsection{Fourier-Mukai duality}
In this subsection, we shall treat Fourier-Mukai duality
for the moduli space of stable 1-dimensional sheaves.
Thus we prove the following result.
\begin{thm}[{cf. \cite[Thm. 3.2.8]{PerverseII}}]\label{thm:rk=0}
Let $\pi:X \to C$ be an elliptic surface
with a fiber $f$.
Let $e^\beta (r_0 f)$ be a primitive and isotropic Mukai vector
with $r_0 >0$ and $H+nf$ an ample divisor with $(H^2)=0$.
Assume that
$X':=M_{H+nf}^\beta(e^\beta(r_0 f))$
is a fine moduli space, and let ${\cal E}$ be a universal family.
Then for $\beta',H'$ in \eqref{eq:H'},
we have the following.
\begin{enumerate}
\item[(1)]
$H'+n' f'$ ($n' \gg 0$) is ample.
\item[(2)]
 ${\cal E}_{|\{ x \} \times X'}$ is $(-\beta')$-twisted stable with respect to
$H'+n' f'$ for all $x \in X$.
\item[(3)]
\begin{equation}
\begin{matrix}
X & \to & M_{H'+n' f'}^{-\beta'}(e^{-\beta'}(r_0 f'))\\
x & \mapsto & {\cal E}_{|\{ x \} \times X'}
\end{matrix}
\end{equation}
is an isomorphism.
\end{enumerate}
\end{thm}

\begin{proof}
We note that
\begin{equation}
v_0'=e^{\beta'}(r_0 f')=
e^{\beta'}(d' H_{r_0^2 m^2 n}'+D'),\; d'=\frac{r_0}{2 r_0^2 m^2 n}, D' \in (H_{r_0^2 m^2 n}')^\perp.
\end{equation}
If
\begin{equation}
\frac{2n}{r_0^2}>r_0 l' \frac{1}{2m^2 n}+\frac{\chi({\cal O}_X)}{2},
\end{equation}
then Proposition \ref{prop:asymptotic} implies
$$
{\cal M}_{\widehat{\sigma}_{(\beta',\frac{1}{r_0^2 m}H'_{r_0^2 m^2n})}}(v_0')
={\cal M}_{\frac{1}{r_0^2 m}H'_{r_0^2 m^2n}}^{\beta'-\frac{1}{2}K_{X'}}(v_0')^{ss}.
$$
\begin{NB}
$$
((\tfrac{1}{r_0^2 m} H'_{r_0^2 m^2 n})^2)=\frac{2n}{r_0^2},\;
\frac{(r_0 f' \cdot H_{r_0^2 m^2 n}')^2}
{({H'_{r_0^2 m^2 n}}^2)}=\frac{1}{2m^2 n}.
$$
\end{NB}

Since
\begin{equation}
\begin{split}
{\cal M}_{\widehat{\sigma}_{(\beta',\frac{1}{r_0^2 m}H'_{r_0^2 m^2n})}}(v_0')& \cong
{\cal M}_{\widehat{\sigma}_{(\beta,mH_n)}}(\varrho_X)\\
&=\{k_x \mid x \in X \}
\end{split}
\end{equation}
 for
$m^2(H_n^2)>\chi({\cal O}_X)$,
$\Phi(k_x)[-1]={\cal E}_{|\{ x\} \times X'}^{\vee}[1]$
is $\beta'$-twisted stable with respect to $H'_{r_0^2 m^2 n}$
for $n \gg 0$.
Then ${\cal E}_{|\{ x \} \times X'}$ is
$(-\beta')$-twisted stable with respect to
$H'_{r_0^2 m^2 n}$.
\end{proof}

\section{Preservation of stability}\label{sect:stability}

Let $\Phi$ be the Fourier-Mukai transform in section \ref{sect:FM}.
By Proposition \ref{prop:FMstability}, we have the following.
\begin{prop}\label{prop:FM-stable}
$\Phi$ induces an isomorphism
\begin{equation}
{\cal M}_{H_n}^{\beta-\tfrac{1}{2}K_X}(v)^{ss} \cong {\cal M}_{\widehat{\sigma}_{(\beta',\omega')}}(\overline{\Phi}(v)),
\end{equation}
where $\omega'=\frac{1}{r_0^2 m}H'_{r_0^2 m^2 n}$ and $m \gg n \gg 0$.
\end{prop}
Thus the $\lambda$-stability in \cite{Y:elliptic} is the $\widehat{\sigma}_{(\beta',\omega')}$-stability
in this proposition.

\begin{prop}[{cf. \cite[Thm. 3.15]{Y:7}, \cite[Thm. 3.13]{Y:twist2}}]\label{prop:r=0}
For $v=e^\beta(\xi+a \varrho_X)$ with $a>0$,
$\Phi[-1]=\Phi_{X \to X'}^{{\cal E}^{\vee}[1]}$ induces an isomorphism
\begin{equation}
{\cal M}_{H_n}^{\beta-\tfrac{1}{2}K_X}(v)^{ss} \cong
 {\cal M}_{H'_{n'}}^{\beta'-\frac{1}{2}K_{X'}}(-\overline{\Phi}(v))^{ss}
\end{equation}
for $n,n' \gg 0$.
\end{prop}

\begin{proof}
In the notation of Lemma \ref{lem:coh-FM},
we have
\begin{equation}
\begin{split}
\overline{\Phi}(v)=& -e^{\beta'}(pr_0+a r_0 f'-\tfrac{q}{r_0}\varrho_{X'}-D')\\
=& -
e^{\beta'}(pr_0+d(H_{r_0^2 m^2 n}-(H'-r_0^2 m^2 n f'))-D'-\tfrac{q}{r_0}\varrho_{X'}),
\end{split}
\end{equation}
where
\begin{equation}
d=\frac{ar_0}{2r_0^2 m^2 n}>0,\;
d^2 ({H_{r_0^2 m^2 n}'}^2)=\frac{a^2}{2m^2 n}.
\end{equation}
For $E \in {\cal M}_{H_n}^{\beta-\tfrac{1}{2}K_X}(v)^{ss}$,
$\widehat{Z}_{(\beta,mH_n)}(E)=-a+(\xi \cdot mH_n)\sqrt{-1}$.
Since
$\widehat{Z}_{(\beta,mH_n)}({\cal E}_{|X \times \{ x' \}}) \in {\Bbb Z}_{>0}\sqrt{-1}$,
$\phi_{\widehat{\sigma}_{(\beta,mH_n)}}({\cal E}_{|X \times \{ x' \}})=\tfrac{1}{2}<\phi_{\widehat{\sigma}_{(\beta,mH_n)}}(E)$.
Then we get $\phi_{\widehat{\sigma}_{(\beta',\omega')}}(\Phi(E))>1$, where
$\omega'=\frac{1}{r_0^2 m}H'_{r_0^2 m^2 n}$.
Thus $\Phi(E) \in {\cal A}_{(\beta',\omega')}[1]$.
Since
\begin{equation}
({\omega'}^2)=2\frac{n}{r_0^2},
\end{equation}
we can take
$m \gg n \gg 0$ such that
\begin{equation}
\frac{({\omega'}^2)-\chi({\cal O}_X)}{2}>
\frac{d}{\delta}\frac{d^2 ({H_{r_0^2 m^2 n}'}^2)-2pq}{2}.
\end{equation}
Then Proposition \ref{prop:FM-stable} and Proposition \ref{prop:asymptotic} imply the claim.
\end{proof}

\begin{rem}
\begin{enumerate}
\item[(1)]
If $a<0$, then we also see that $\Phi(E)^{\vee}$ is a twisted stable sheaf, which is treated in
\cite[Prop. 3.4.5]{PerverseII}.
\item[(2)]
If $r_0=1$, then Lo and Martinez \cite[Thm. 5.12]{Lo-M}
get the same result.
\end{enumerate}
\end{rem}

\section{Stability conditions on an elliptic K3 surface}\label{sect:K3}

\subsection{Walls for $v=1-\ell \varrho_X$ near the boundary of $\Stab(X)$
associated to the elliptic fibration. }

Let $\pi:X \to C$ be an elliptic K3 surface.
Then Bayer and Macri \cite{BM:2} classified Bridgeland walls.
In this subsection, we shall study walls for the Mukai vector $1-\ell \varrho_X$ near a boundary of $\Stab(X)$.
For a primitive Mukai vector $v$,
let $P$ be the positive cone in $v^\perp$ and $\overline{P}$ its closure.
Let ${\cal W} \subset \Stab(X)$ be a wall for $v$ defined by a
Mukai vector $u \in H^{*}(X,{\Bbb Z})_{\alg}$.
Then ${\cal W}$ defines a wall $W:=u^\perp$ in $\overline{P}$.
For $(\beta, \omega) \in \NS(X)_{\Bbb R} \times \Amp(X)_{\Bbb R}$ with
$(\omega^2)>\chi({\cal O}_X)=2$,
we set
\begin{equation}
\xi(\beta,\omega):=[
\mathrm{Im} (\overline{\langle e^{\beta+\sqrt{-1} \omega},v) \rangle}
e^{\beta+\sqrt{-1} \omega})] \in P/{\Bbb R}_{>0},
\end{equation}
where $[u]:={\Bbb R}_{>0}u$ is the equivalence class of $u \in P$ in
$P/{\Bbb R}_{>0}$.
We have an identification
\begin{equation}
\theta_v:v^\perp \cong H^2(M_H(v),{\Bbb Z})_{\alg}
\end{equation}
by using the universal family of stable sheaves.
Under this identification, the boundaries of the movable cone and the nef cones of
minimal models of $M_H(v)$ are $W=u^\perp$ defining
walls for $\Stab(X)$.
For $v=1-\ell \varrho_X$,
we have
\begin{equation}
v^\perp={\Bbb Z}\nu+\NS(X),\;\nu:=1+\ell \varrho_X.
\end{equation}
In this case, $[f] \in \overline{P}$ and
defines a fibration $M_H(v) \to S^{\ell+1} {\Bbb P}^1$.
Let us classify walls near $[f]$.

\begin{lem}\label{lem:f-wall}
Let $W$ be a wall for $v=1-\ell \varrho_X$ in $\overline{P}$ such that $f \in W$.
Then there is a Mukai vector $u=\xi+a \varrho_X$ such that $W=u^\perp$ and one of the following hold:
\begin{enumerate}
\item[(1)]
$\xi \in {\Bbb Z}f$ and $-\ell-1 \leq a \leq 0$ or
\item[(2)]
$\xi$ or $-\xi$ is an effective $(-2)$-divisor in a fiber of $\pi$ and
$-\ell \leq a \leq 0$,
\end{enumerate}
\end{lem}

\begin{proof}
Let $u=r+\xi+a \varrho_X$ $(r,a \in {\Bbb Z}, \xi \in \NS(X))$ be a Mukai
vector such that $u^\perp =W$. We note that $(v-u)^\perp=u^\perp$.
By \cite{BM:2}, we may assume that
\begin{enumerate}
\item
$\langle u^2 \rangle=-2$ and
$0 \leq \langle v,u \rangle \leq \ell$, or
\item
 $\langle u^2 \rangle=0$ and
$0 < \langle v,u \rangle \leq \ell$, or
\item
$0<\langle u^2 \rangle$ and $2\langle u^2 \rangle+1 \leq \langle v,u \rangle \leq \ell$.
\end{enumerate}
By $f \in u^\perp$, we have
$(\xi,f)=0$.
Assume that $ra<0$.
Since $\langle v,u \rangle=r\ell-a$, we see that
$\langle v,u \rangle<0$ or $\langle v,u \rangle>\ell$.
Therefore $ra \geq 0$.
Then we get $\langle u^2 \rangle=(\xi^2)-2ra \leq 0$. Thus we may assume that $\langle u^2 \rangle=0$
or $\langle u^2 \rangle=-2$.

We fitst treat the case where $\langle u^2 \rangle=0$.
In this case $\xi=kf$ and $ra=0$.
If $a=0$, then $\langle v,u \rangle=r\ell$. Hence $r=1$.
In this case, $w:=v-u=-kf-\ell \varrho_X$
satisfies $\langle w^2 \rangle=0$ and $\langle v,w \rangle=\ell$.
If $r=0$, then $\langle v,u \rangle=-a$ implies $0 \geq a \geq -\ell$.
Hence $u=kf+a \varrho_X$ with $0 \geq a \geq -\ell$.

We next treat the case where $\langle u^2 \rangle=-2$.
In this case $(\xi^2)=-2$ and $ra=0$ or $\xi=kf$ and $ra=1$.
For the first case,
if $a=0$, then we also see that $u=1+\xi$ and $w:=v-w=-\xi-\ell \varrho_X$
satisfies $\langle w^2 \rangle=-2$ and
$\langle v,w \rangle=\ell$.
If $r=0$, then $u=\pm D+a \varrho_X$ with $0 \geq a \geq -\ell$, where
$D$ is an effective $(-2)$-curve in a fiber.
For the second case,
if $r=a=1$, then $\langle v,u \rangle=\ell-1$.
Then $w:=v-u=-kf-(\ell+1)\varrho_X$, $\langle w^2 \rangle=0$ and $\langle v,w \rangle=\ell+1$.
Therefore our claim holds.
\end{proof}

\begin{lem}\label{lem:xi}
Assume that $v=1-\ell \varrho_X$. Then
for $(\beta,\omega) \in \NS(X)_{\Bbb R} \times \Amp(X)_{\Bbb R}$,
\begin{equation}
\xi(\beta,\omega)= [(\beta \cdot \omega)\nu+
(\beta \cdot \omega)\beta+(\ell+\tfrac{(\omega^2)}{2}-\tfrac{(\beta^2)}{2})\omega].
\end{equation}
\end{lem}

\begin{proof}
Since
\begin{equation}
\begin{split}
& \overline{\langle e^{\beta+\sqrt{-1} \omega},v \rangle} e^{\beta+\sqrt{-1}\omega}\\
= & \left(\ell-\frac{(\beta^2)-(\omega^2)}{2}+\sqrt{-1}(\beta \cdot \omega) \right)
(1+(\beta+\sqrt{-1}\omega)+
(\tfrac{(\beta^2)-(\omega^2)}{2}+\sqrt{-1}(\beta \cdot \omega))\varrho_X),\\
\end{split}
\end{equation}
we get
\begin{equation}
\mathrm{Im} (\overline{\langle e^{\beta+\sqrt{-1} \omega}, v \rangle}
e^{\beta+\sqrt{-1} \omega})
= (\beta \cdot \omega)\nu+
(\beta \cdot \omega)\beta+(\ell+\tfrac{(\omega^2)}{2}-\tfrac{(\beta^2)}{2})\omega.
\end{equation}
\end{proof}

\begin{NB}
We set $v:=(r,\xi,a)=\ch(E)(1,0,\tfrac{\chi({\cal O})}{2})$.
Since
\begin{equation}
\begin{split}
& \overline{\langle e^{\beta+\sqrt{-1} \omega},(r,\xi,a) \rangle} e^{\beta+\sqrt{-1}\omega}\\
= & \left((\beta \cdot \xi)-a-r\frac{(\beta^2)-(\omega^2)}{2}+\sqrt{-1}((r\beta- \xi)\cdot \omega) \right)(1,\beta+\sqrt{-1}\omega,
\tfrac{(\beta^2)-(\omega^2)}{2}+\sqrt{-1}(\beta \cdot \omega))\\
\end{split}
\end{equation}
we get
\begin{equation}
\begin{split}
& \mathrm{Im} (\overline{\langle e^{\beta+\sqrt{-1} \omega},(r,\xi,a) \rangle}
e^{\beta+\sqrt{-1} \omega})\\
=& ((\beta-\tfrac{\xi}{r}) \cdot \omega)((r,0,-a)+(0,r\beta,(\beta \cdot \xi)))
+(\tfrac{(\omega^2)}{2}-\tfrac{(\beta^2)}{2}+(\beta \cdot \tfrac{\xi}{r})-\tfrac{a}{r})(0,r\omega,(\omega \cdot \xi))\\
=& ((\beta-\tfrac{\xi}{r}) \cdot \omega)((r,0,-a)+(0,r\beta,(\beta \cdot \xi)))
+(\tfrac{(\omega^2)}{2}-\tfrac{((\beta-\tfrac{\xi}{r})^2)}{2}+\tfrac{\langle v^2 \rangle}{2r^2})(0,r\omega,(\omega \cdot \xi)).
\end{split}
\end{equation}
\end{NB}

\subsection{Relation with walls in Bridgeland stability for an elliptic K3 surface.}

Let $\Phi:{\bf D}(X) \to {\bf D}(X')$ be the Fourier-Mukai transform in section \ref{sect:FM} and assume that
$\Phi(v)=1-\ell \varrho_{X'}$.
In this subsection, let us study chamber structure for $1-\ell \varrho_{X'}$ near $[f']$.
For
\begin{equation}
\omega':=\frac{1}{r_0^2 m}(H'+r_0^2 m^2 n f')=\frac{1}{r_0^2 m}H'+mn f',
\end{equation}
we have
\begin{equation}
(\beta' \cdot \omega')=\frac{1}{r_0^2 m}(\beta' \cdot H')+mn(\beta' \cdot f')
\end{equation}
and

\begin{equation}
\frac{(\beta' \cdot \omega')}{mn(\ell+\frac{({\omega'}^2)}{2}-\frac{({\beta'}^2)}{2})}=
\frac{\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')}{\ell+\frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}.
\end{equation}
Hence
\begin{equation}\label{eq:behavior}
\xi(\beta',\omega')=
\left[
\frac{\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')}{\ell+\frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}(\nu+\beta')
+\frac{1}{r_0^2 m^2 n}H'+f' \right] \to [f'] \quad (n, m/n \to \infty).
\end{equation}
Thus $\xi(\beta',\omega')$ is an ample divisor of
$M_{\widehat{\sigma}(\beta',\omega')}(1-\ell \varrho_{X'})$
which is very close to $[f']$.

Let $\Delta$ be a finite polyhedral cone in $\overline{P}$ which contains
$[f'],[H'],-[\nu+\beta']$.
In a small neighborhood $U$ of $[f']$ in $\Delta$,
all walls contain $[f']$ (see \cite[Prop. 2.2]{MY} and the paragraph after \cite[Conj. 1.1]{MY}), and hence
walls are defined by Mukai vectors $u$ in Lemma \ref{lem:f-wall}.
We assume that $(\beta' \cdot f')<0$.
Let us consider a family of stability conditions
$\widehat{\sigma}_{(\beta',t \omega')}$ $(1 \leq t)$ and study the wall crossing.
We note that
$$
\xi(\beta',t\omega')=
\left[
\frac{\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')}{\ell+t^2 \frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}(\nu+\beta')
+\frac{1}{r_0^2 m^2 n}H'+f' \right].
$$
Since $m \gg n \gg 0$, we may assume that
$\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')<0$ and
$\xi(\beta',t\omega')$ belongs to the neighborhood $U$.
For a Mukai vector $u=(0,\xi,a)$ in Lemma \ref{lem:f-wall},
$\xi(\beta',t\omega') \in u^\perp$ if and only if
$$
\frac{\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')}{\ell+t^2 \frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}=
\frac{(H' \cdot \xi)}{r_0^2 m^2 n (a-(\beta' \cdot \xi))}.
$$
Hence $u$ satisfies
\begin{equation}\label{eq:f-u}
\frac{\frac{(\beta' \cdot H')}{r_0^2 m^2 n}+(\beta' \cdot f')}{\ell+\frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}<
\frac{(H' \cdot \xi)}{r_0^2 m^2 n (a-(\beta' \cdot \xi))}<0.
\end{equation}
Then the wall crossing along $t \geq 1$ is the same as the wall crossing for $\lambda$-stability
in \cite[sect. 5]{Y:elliptic}.
\begin{NB}
\begin{lem}
There are finitely many $u$ satisfying \eqref{eq:f-u}.
\end{lem}

\begin{proof}
Let $D_1,D_2,...,D_N$ be representatives of $(-2)$-classes in $f^\perp/{\Bbb Z}f$.
Then we have $\xi=D_i+kf$, where $1 \leq i \leq N$ and $k \in {\Bbb Z}$.
If $(H' \cdot \xi')>0$, then $a-(\beta' \cdot \xi)<0$ implies $k \geq 0$ and
$-\ell-1 \leq a<(\beta' \cdot D_i)+k(\beta' \cdot f)$ Hence the choice of $k$ is finite.
If $(H' \cdot \xi') < 0$, then $a-(\beta' \cdot \xi)>0$ implies $k \leq 0$ and
$a>(\beta' \cdot D_i)+k(\beta' \cdot f)$. Hence the choice of $k$ is finite.
\end{proof}
\end{NB}

\begin{rem}
\begin{enumerate}
\item[(1)]
By \eqref{eq:behavior},
$$
\xi(\beta',\omega') \sim
\left[f'+\frac{(\beta' \cdot f')}{\ell+\frac{n}{r_0^2}-\frac{({\beta'}^2)}{2}}(\nu+\beta') \right],\;  (m \gg 0).
$$
Hence
$\delta+\beta'$ determines the chamber where $\xi(\beta',\omega')$ belongs.
\item[(2)]
If all fibers of $\pi$ are irreducible, then
$u=(0,kf',a)$ with
$-\ell-1 \leq a<0$ and $a<k(\beta' \cdot f')$.
\end{enumerate}
\end{rem}

\begin{rem}
We also have a similar description of walls for an elliptic abelian surface by results in \cite{movable}.
\end{rem}

\begin{NB}

\section{}

\begin{equation}
\langle e^{\beta+\sqrt{-1}(t^{-1}H+nt f)},e^\beta(r+\xi^\beta+a^\beta) \rangle=
nr-a^\beta+\sqrt{-1}((t^{-1}H+ntf) \cdot \xi).
\end{equation}

\begin{equation}
\begin{split}
& \frac{a^\beta-nr}{nt(\xi^\beta \cdot f)+t^{-1}(\xi^\beta \cdot H)}-
\frac{a_1^\beta-nr_1}{nt(\xi_1^\beta \cdot f)+t^{-1}(\xi_1^\beta \cdot H)}\\
=& \frac{n^2 t(r_1(\xi^\beta \cdot f)-r(\xi_1^\beta \cdot f))+
nt((-a_1^\beta (\xi^\beta \cdot f)-(-a)(\xi_1^\beta \cdot f))}
{(nt(\xi^\beta \cdot f)+t^{-1}(\xi^\beta \cdot H))(nt(\xi_1^\beta \cdot f)+t^{-1}(\xi_1^\beta \cdot H))}\\
& +\frac{n t^{-1}(r_1(\xi^\beta \cdot H)-r(\xi_1^\beta \cdot H))+
t^{-1}((-a_1^\beta (\xi^\beta \cdot H)-(-a)(\xi_1^\beta \cdot H))}
{(nt(\xi^\beta \cdot f)+t^{-1}(\xi^\beta \cdot H))(nt(\xi_1^\beta \cdot f)+t^{-1}(\xi_1^\beta \cdot H))}
\end{split}
\end{equation}

For $t \gg n \gg 0$,

We set $\omega:=t^{-1}H+tn f$.

Let $E$ be a $\widehat{\sigma}_{(\beta,\omega)}$-semi-stable object with $v(E)=e^\beta(r+\xi+a\varrho_X)$
with $(\xi \cdot f)>0$.
Assume that $H^{-1}(E) \ne 0$.
We set $v(H^{-1}(E)[1]):=e^\beta(r_1+\xi_1 +a_1 \varrho_X)$.
Then $r_1<0$ and $(\xi_1 \cdot (H+t^2 n f)) \geq 0$ for all $t \gg n \gg 0$.
In particular $(\xi_1 \cdot f) \geq 0$.
Then $r_1(\xi \cdot f)-r(\xi_1 \cdot f)<0$, which implies $E$ is not semi-stable.
Therefore $E$ is a sheaf.

For an exact sequence
\begin{equation}
0 \to E_1 \to E \to E_2 \to 0
\end{equation}
in $\Coh(X)$, we set
$v(E_i):=e^\beta(r_i+\xi_i +a_i \varrho_X)$.
Since $E \in {\cal A}_{(\beta,\omega)}$ for $t \gg n \gg 0$,
$(\xi_2 \cdot(H+nt^2 f))>0$ for $t \gg n \gg 0$.
Thus $(\xi_2 \cdot f) \geq 0$.
If $E$ is not $f$-semi-stable, then by taking the Harder-Narashimhan filtration
of $E$,
we may assume that $E_1 \in {\cal A}_{(\beta,\omega)}$,
$r_2>0$ and $r_2 (\xi \cdot f)-r(\xi_2 \cdot f) \geq 0$.
By $\widehat{\sigma}_{(\beta,\omega)}$-semi-stability of $E$,
$r_2 (\xi \cdot f)-r(\xi_2 \cdot f) \leq 0$.
Therefore $r_2 (\xi \cdot f)-r(\xi_2 \cdot f)=0$.
Thus $E$ is $f$-semi-stable.

Assume that $\pi:X \to C$ is an elliptic abelian surface.
Then $(0,f,0)$ defines a fibration $M_H(v) \to S^{\ell} C$.

Mukai vectors $u$ such that
$\langle u^2 \rangle=0$ and
$0 < \langle v,u \rangle \leq \ell$,
$0<\langle u^2 \rangle$ and $2\langle u^2 \rangle+1 \leq \langle v,u \rangle \leq \ell$.

Let $X=C_1 \times C_2$ be a product of elliptic curves.

\begin{equation}
\beta:=xf+yg,\;\omega:=zf+wg.
\end{equation}

\begin{equation}
e^{\beta+\sqrt{-1}\omega} \Longleftrightarrow
\begin{pmatrix}
1 & x+\sqrt{-1} z\\
y+\sqrt{-1} w & (x+\sqrt{-1}z)(x+\sqrt{-1} w)
\end{pmatrix}
=
\begin{pmatrix}
1\\
y+\sqrt{-1} w
\end{pmatrix}
\begin{pmatrix}
1 & x+\sqrt{-1} z
\end{pmatrix}
\end{equation}

\begin{equation}
\begin{split}
& \mathrm{Im} (\overline{\langle e^{\beta+\sqrt{-1} \omega},(1,0,-\ell) \rangle}
e^{\beta+\sqrt{-1} \omega})\\
=& (xw+yz)(1,0,\ell)+(0,(\ell z+z^2 w+x^2 w)f+(\ell w+zw^2+y^2 z)g,0)
\end{split}
\end{equation}

We consider a Fourier-Mukai transform
on $C_1$ associated to
\begin{equation}
A:=
\begin{pmatrix}
r & r' \\
d & d'
\end{pmatrix} \in \SL(2,{\Bbb Z}).
\end{equation}
Then $\Phi$ induces an autoequivalence of $X$:
\begin{equation}
A \begin{pmatrix}
1 & q\\
p & pq
\end{pmatrix}
=A
\begin{pmatrix}
1\\
p
\end{pmatrix}
\begin{pmatrix}
1 & q
\end{pmatrix}
=(a+bp)
\begin{pmatrix}
1\\
\frac{c+dp}{a+bp}
\end{pmatrix}
\begin{pmatrix}
1 & q
\end{pmatrix}.
\end{equation}

\begin{equation}
\Phi(e^{pg+qf})=(a+bp) e^{\frac{c+dp}{a+bp}g+qf}.
\end{equation}

\begin{equation}
\beta+\sqrt{-1} \omega=(x+\sqrt{-1}z) f+(y+\sqrt{-1} w)g=
qf+pg.
\end{equation}

Assume that $w \gg 0$ and $z/w \gg 0$.

\begin{equation}
\begin{split}
\frac{c+dp}{a+bp}g+qf= & \left(\frac{(a+by)(c+dy)+bd w^2}{b^2 w^2+(a+by)^2}g+xf \right)+
\sqrt{-1} \left(\frac{w}{b^2 w^2+(a+by)^2}g+zf \right)\\
\sim & \frac{d}{b}g+xf+\sqrt{-1} \left(\frac{1}{b^2 w}g+zf \right).
\end{split}
\end{equation}

We set $v_0:=(0,r' f,-r)=e^\beta (r' f)$, where $\beta:=\frac{-r}{r'}\sigma$.
Then $\Phi(v_0)=-(0,0,1)$.
We set $\beta':=\frac{d'}{r'}\sigma$.
Then $v_0'=e^{\beta'} (r' f)$ and
$\Phi(\sigma)=r'e^{\beta'}$.

We see that $\delta+\beta' \in (0,r' f,d')^\perp$.

For $E \in {\bf D}(X)$,
$r(E)$ is the rank of $E$ and $d(E):=(c_1(E) \cdot f)$ is the relative degree of $E$.

Let $\Phi:{\bf D}(X) \to {\bf D}(Y)$ be a relative Fourier-Mukai transform
such that
\begin{equation}
\begin{pmatrix}
r(\Phi(E))\\
d(\Phi(E))
\end{pmatrix}
=
\begin{pmatrix}
c & a\\
d & b
\end{pmatrix}
\begin{pmatrix}
r(E)\\
d(E)
\end{pmatrix}
\end{equation}
Then \cite[4.3]{Br:1} implies
$\Phi((0,pf,q))=(0,(cp+aq)f,dp+bq)$.

\begin{NB2}
Let $\pi_X':X \times_C Y \to X$ and $\pi_Y':X \times_C Y \to Y$ be projections.

\begin{lem}
$\ch \Phi(E(\lambda f))=e^{\lambda f}\ch \Phi(E)$.
\end{lem}

\begin{proof}
Let ${\bf P}$ be the kernel.
Then we see that
\begin{equation}
\begin{split}
\ch ({\bf P}) \otimes {\pi_Y'}^*( e^{\lambda \pi_X^{-1}(c)})=&
\ch ({\bf P})+\lambda \ch ({\bf P}) (\pi_X^{-1}(c) \times \pi_Y^{-1}(c))\\
 \ch ({\bf P}) \otimes {\pi_X'}^*( e^{\lambda \pi_Y^{-1}(c)})=&
\ch ({\bf P})+\lambda \ch ({\bf P}) (\pi_X^{-1}(c) \times \pi_Y^{-1}(c)),
\end{split}
\end{equation}
which implies
\begin{equation}
\ch ({\bf P}) \otimes {\pi_Y}^*( e^{\lambda \pi_X^{-1}(c)})=
\ch ({\bf P}) \otimes {\pi_X}^*( e^{\lambda \pi_Y^{-1}(c)}).
\end{equation}
Therefore the claim holds.
\end{proof}

\end{NB2}

\section{Wall crossing for moduli of 1-dimensional sheaves}

\begin{equation}
\frac{\chi-(D \cdot \eta)}{(D \cdot H)}=\frac{\chi'-(D' \cdot \eta)}{(D' \cdot H)}
\iff \chi (D' \cdot H)-\chi' (D \cdot H)=(D' \cdot H)(D \cdot \eta)-(D \cdot H)(D' \cdot \eta)
\end{equation}

If $H=H_0+nf$ and $\eta=xH_0$, then
\begin{equation}
x=\frac{\chi(D' \cdot f)-\chi' (D \cdot f)}{(D \cdot H_0)(D' \cdot f)-(D' \cdot H_0)(D \cdot f)}+
\frac{1}{n}\frac{\chi(D' \cdot H_0)-\chi' (D \cdot H_0)}{(D \cdot H_0)(D' \cdot f)-(D' \cdot H_0)(D \cdot f)}.
\end{equation}

\subsection{An elliptic ruled surface}

Let $\varpi:X \to C$ be an elliptic ruled surface with a minimal section $\sigma$
with $(\sigma^2)=1$. Let $g$ be a fiber of $\varpi$.

\begin{lem}
Let $D$ be an effective divisor.
Then
\begin{enumerate}
\item
$D$ is nef.
\item
$(D^2)=0$ if and only if $D \in {\Bbb Z}_{>0} (2\sigma-g)$ or $D \in {\Bbb Z}_{>0} g$.
\item
$(D^2)=1$ if and only if $D=\sigma$.
\end{enumerate}
\end{lem}

\begin{lem}
Let $D_1,D_2$ be effective divisors.
\begin{enumerate}
\item
$(D_1,D_2)=0$ if and only if
$D_1, D_2 \in {\Bbb Z}_{>0} (2\sigma-g)$ or $D_1,D_2 \in {\Bbb Z}_{>0} g$.
\item
$(D_1,D_2)=1$ if and only if one of the following holds.
\begin{enumerate}
\item
$D_1=D_2=\sigma$.
\item
$\{ D_1,D_2 \}=\{2\sigma-g,\sigma+n(2\sigma-g)\}$, $n \in {\Bbb Z}_{\geq 0}$.
\item
$\{D_1,D_2 \}=\{g,\sigma+ng \}$,  $n \in {\Bbb Z}_{\geq 0}$.
\end{enumerate}
\end{enumerate}
\end{lem}

\begin{proof}
Assume that $(D_1 \cdot D_2)=1$ and $(D_1^2)=0$.
Then $D_1=2\sigma-g,g$.
If $D_1=2\sigma-g$, then
$(\sigma \cdot (2\sigma-g))=1$ implies
$D_2-\sigma=n(2\sigma-g)$ $(n \in {\Bbb Z})$.
If $\sigma+\xi$ is effective, then $n \geq 0$.
Therefore $D_2=\sigma+n(2\sigma-g)$ $(n \in {\Bbb Z}_{\geq 0})$.

If $D_1=g$, then we also see that $D_2=\sigma+ng$ $(n \in {\Bbb Z}_{\geq 0})$.
\end{proof}

\end{NB}

redpajama_set_name: RedPajamaArXiv


Carol Duncan is a friend I met from my days lurking around Australian site, sourdough.com. A couple of days ago, Carol tweeted to me, sharing that she would be visiting New York within days. So we met offline. Lovely to transform the world of virtual friendships to granularity of sharing actual real world space. And so we did. She turned up with multiple folks, including husband, sons, and friends, leading to a great evening.


text:
Katsuhiro Otomo at the Sendai Airport – Riding the Waves
Last updated: / Ceramics / Art by Asian artists, Art in Asia, Art in Japan, Art in public, Artworks by Japanese artists, Comic art, CREARE Atami-Yugawara Studio, Cute art, Earthquake-inspired art, Fukushima disaster art, Katsuhiro Otomo, manga art
Katsuhiro Otomo – Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2015, ceramic relief, 8.7 x 2.8 meter, installation view, International Arrival Lobby on 1st Floor, Passenger Terminal Building, Sendai Airport, Japan
1 Katsuhiro Otomo's relief
2 What the relief shows
3 A disaster inspired its creation
4 Why Otomo was the perfect artist for this project
Katsuhiro Otomo's relief
Katsuhiro Otomo, the creator of Akira, the 1988 Japanese animated cyberpunk action film, is one of the most well-known manga artists, animators, screenwriters, and film directors in the world.
Though he is best known for his manga series and film adaptations, Otomo crossed over into the sculpture world when he was asked to design a relief for the 1st floor of the terminal building at Sendai Airport in Japan.
The completed mural was based on an illustration created by Otomo. Even though the mural features numerous components of traditional Japanese art, the relief still managed to retain the artist's dark, playful style, which can be seen in this relief in the form of the boy wearing thick-rimmed glasses.
Katsuhiro Otomo – Kinka Doji Riding the Waves Accompanied by Fujin and Raijin (detail), 2015, ceramic relief, 8.7 x 2.8 meter, installation view, International Arrival Lobby on 1st Floor, Passenger Terminal Building, Sendai Airport, Japan
What the relief shows
The mural at the Sendai Airport depicts a boy riding a robotic goldfish in the face of the storm while he's being bordered by Fujin and Raijin, the formidable Japanese gods of weather. Specifically, Raijin is the god of storms, lightning, and thunder and is usually depicted in many artworks holding hammers that are surrounded by drums.
Original painting of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin
In these depictions, Raijin is typically seen with three fingers on each hand to symbolize the past, present, and future. As you can expect, Raijin is one of the most dreaded deities in Japan.
Fujin, on the other hand, is the god of wind, which is why he is habitually portrayed holding a bag full of wind. That's why Fuijin has wild hair and a messy appearance, due to all the wind surrounding him. Throughout history, Japan has always dealt with massive typhoons that cause extensive damage and, worse, deaths in the hundreds.
Katsuhiro Otomo with a sketch of Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2014
A disaster inspired its creation
Based on the two gods, Fuijin and Raijin, the mural at the Sendai Airport was therefore created to commemorate the Tōhoku earthquake and tsunami, a severe natural disaster that took place in March 2011.
Model of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2014
The earthquake and tsunami killed 20,000 individuals and is still recorded as the strongest earthquake in Japan's history. The quake struck 81 miles east of Sendai, the largest and most populated city in Tohoku, which is situated in the northern part of Honshu island.
Production of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2014
The natural disaster ruthlessly crippled the country's infrastructure while also destroying homes, roads, businesses, railways, and more in their thousands. The tsunami also led to the meltdown of three nuclear reactors situated at the Fukushima Daiichi Nuclear Power Plant.
Cutting, shaping, backside cutting, drying, unglazed firing of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2014
This breakdown consequently released radioactive materials into the environment, thus causing the residents to evacuate everything that they had built behind.
Color determination, glazing of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2014
Why Otomo was the perfect artist for this project
Otomo was the perfect person to design the mural because he grew up in Tome, Miyagi Prefecture, which is part of the Tohoku region. The relief was designed and created by a team of 10 artists at the popular CREARE Atami-Yugawara Studio, thanks to the financial support given by the Japan Lottery Association.
Final firing of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2015, ceramic relief, 8.7 x 2.8 meter
The planning process was undertaken by the Japan Traffic Culture Association, which has always served its mission of installing public works throughout Japan's stations and airports. The 24-square-meter mural was completed in March 2015 just in time to commemorate that fateful day of the earthquake and tsunami.
Preliminary assembly, Installation, completion of Katsuhiro Otomo's Kinka Doji Riding the Waves Accompanied by Fujin and Raijin, 2015
All images: Courtesy of Japan Traffic Culture Association unless otherwise noted
At age 95, Fumiko Hori created this mural for the Fukushima Airport
Marina Abramović walks China's Great Wall only to break up
McDonald's did not pay for Brad Downey's mural
Francis Alÿs - Can faith move a whole mountain?
Fischli & Weiss: The groundbreaking way things go
Pieter Hugo interview - How can art change the world?
Nabana no Sato's winter illumination - Everything you need to know
This was Ugo Rondinone's iconic Hell, Yes!

redpajama_set_name: RedPajamaCommonCrawl


Prague-based online gambling games developer Endorphina launched its latest slot Thursday, a 4-row, 5-reel, 40-line game titled Lucky Streak 1 and the first in an intended series of three fruit-themed slots.
The developer says the key to the series is easy, uncomplicated "old school" slot action with a traditional theme.
Endorphina has found that this sort of game is among the most favoured by its slots fans and the developer expects that this latest example will prove equally popular, enhanced as it is with colourful graphics and intricate animation.
Endorphina, the Prague-based online gambling software provider, has announced a content deal that will see it supplying games to Asian slot game supplier XIN Gaming….


text:
import { ImproveRelevance24 } from "../../";

export = ImproveRelevance24;

redpajama_set_name: RedPajamaGithub


text:
The super bowl since and broncos core
Posted by admin on November 21, 2018 in Bartolo Colon Jersey 0 Comment
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But you ask him, 'What did this do?', he'll give you the stats and tell you how much he played.Rio, athletes be able to use an NFC-enabled payment ring to make purchases by simply tapping their ring at any NFC-enabled payment terminal.Tocchet: I 't believe you can read a book on analytics and go behind the bench and coach that way.I think that we made some strides and improvements http://www.texasrangersprostore.com/WOMENS-BARTOLO-COLON-JERSEY.html last year as far as the Authentic Bartolo Colon Jersey defense.True to the attitude of the commercial, however, her hits the car, setting off alarms.
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First, Their Breast Implants Made Them Sick. Then They Were Hit with the Bill.
September 9, 2020 We Are QuotedBII, breast implant illness, breast implant removal, Cosmopolitan, explant surgery, explants, insurance, plastic surgery, removalBIadmin
Catherine Guthrie, Cosmopolitan: August 18, 2020.
Annie Reynoso wanted a tummy tuck. But her doctor said no. Uterine fibroids meant she wasn't a good candidate for abdominal surgery. He had an idea though, the doctor. If what Annie was looking for was a physical boost, there were other options out there. Like breast implants.
Then it was a few years later, and odd things started happening: Her breasts swelled to a G-cup. She had fatigue that would knock her out for days. Sudden dizzy spells made it scary for her to drive. She'd get short of breath after walking just a few steps. She had night sweats that soaked her mattress. And there were even stranger symptoms: nausea if she ate before noon, and pain in her chest, neck, ears, and jaw that felt like the worst sunburn of her life. She kept tubes of Aspercreme in her purse, coat pockets, and desk drawer. At least a dozen times a day, she had to slather herself in ointment.
That's when something finally changed: A friend of Annie's started feeling better after her implants were removed, and a rheumatologist told Annie some of her symptoms aligned with Sjögren's syndrome, an autoimmune disease potentially triggered by her implants. After that, all she could think about was getting them out.
Once again, she waited in a surgeon's office. A new one this time. She'd gotten her implants in the Dominican Republic, where her family lives and they could help her through the recovery. Now she sat alone in New Jersey, consulting with a doctor about having her implants removed in a procedure called an explant. She was hopeful. Until she was told it would cost $7,500—and that insurance wasn't likely to cover a dime.
And yet, there is evidence to suggest that breast implant illness might actually be an autoimmune disorder caused by implants. In 2018, a study in the International Journal of Epidemiology found that women with silicone breast implants had more than a 20 percent increased risk of being diagnosed with an autoimmune or rheumatic disorder.
There is also overwhelming anecdotal evidence online, in Facebook groups like Breast Implant Illness and Healing by Nicole, where hundreds of thousands of women share their stories. And of course, there are the endless comments on posts like Chrissy Teigen's recent Instagram update about having her implants removed. Women saying they wish they could too—women saying, "Having them is making me sick."
For sick women who are financially able to get explant surgery, it can change everything. In a July 2020 study in Annals of Plastic Surgery, researchers found that those experiencing symptoms of breast implant illness saw an improvement in their health within a month of having their implants removed. Like Lauren Dearman, who had explant surgery last November.
She was just 20 when her parents offered to pay $8,000 for her C-cup breast implants. Six years later, she began to have severe abdominal issues and chest pain, bad enough that she went to the ER. She was tired all the time and had such trouble focusing that she couldn't even write a to-do list. She couldn't climb the stairs to her third-floor apartment without gripping the railing. Her boyfriend told her she was out of shape. Her boyfriend, she knew, was wrong.
When she came across a Facebook group where women were talking about breast implant illness, the stories read like her own medical file. Within weeks, she made an appointment with a surgeon in Chicago who agreed that her implants could be at the root of her health issues. Lauren cried when she learned how much the surgery would cost. Her parents couldn't pitch in financially this time, so Lauren withdrew $2,500 from savings and took out a personal loan for $6,500 so she could have the procedure. At $190 a month, she'll be paying it off for the next three years. But she feels so much better now.
In a health care system that has no problem paying for Viagra (and Viagra overdoses), the fact that women like Lauren have to finance their medical care is infuriating, says Cari M. Schwartz, a lawyer at the firm Kantor & Kantor in California.
Schwartz is working on bringing a class-action lawsuit against insurance companies that deny explant coverage. She's interviewed hundreds of women across the country desperate to get their implants out. In every case, insurers denied coverage, even when physicians deemed removals medically necessary. Schwartz has seen women rack up credit-card debt, borrow money, lose relationships and jobs, and go bankrupt in an effort to save their health. "Women are essentially told that their health issues are their fault," she says, "because they chose to get implants."
In 2019, the FDA said it was putting more effort into educating doctors and patients on the "systemic symptoms" many women with implants experience. Although they also still say they don't have "definitive evidence demonstrating breast implants cause these symptoms."
Doctors, too, remain reluctant to get onboard. "To recognize breast implant illness is to kill the goose that laid the golden egg," says H. Jae Chun, MD, a Newport Beach, California, surgeon who specializes in explant surgery. "And many doctors just aren't going to mess with the goose."
In the meantime, one of the only options for desperate patients is the National Center for Health Research's program to help women navigate the path to explant. They don't give out money, but if a woman has health insurance, they'll do what they can to coach her and her plastic surgeon through the insurance maze. Often, that means helping fill out paperwork explaining why the procedure is medically necessary. It's a process that can take months and still results in a denial the majority of the time, says Diana Zuckerman, PhD, the group's president. But so far, they've helped more than 1,500 women get the explants they need.
← The Number Of Women Having Their Breast Implants Removed Is on the Rise	FDA Updates Breast Implant Labeling Recommendations to Help Inform Patients About Dangerous Potential Side Effects →


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GREAT BARRINGTON SAVINGS BANK & another vs. RICHARD H. GENS & others.
The defendants challenge on appeal the correctness of an order of a District Court judge, sitting by designation in Superior Court, granting the plaintiffs' motion for summary judgment. We conclude that summary judgment was properly entered.
The judge correctly ruled that all the matters presented by the defendants in opposition to the plaintiffs' motion for summary judgment "involve legal issues that may be resolved on the pleadings and affidavits or are not supported by any facts set forth in any of the affidavits or exhibits presented [in accordance with Mass.R.Civ.P. 56(e), 365 Mass. 825 (1974)] to the Court."
Based on the stipulation offered by the parties, relevant pleadings and the application of settled principles (see Noyes v. Quincy Mut. Fire Ins. Co., 7 Mass. App. Ct. 723 , 725-726 [1979]), it is apparent that there is only one issue raised by the defendants which deserves detailed treatment -- whether the defendants signed the note solely as accommodation parties for the plaintiffs' benefit. See G. L. c. 106, Section 3-415.
We think this case is controlled in all material respects by Community Natl. Bank v. Dawes, 369 Mass. 550 , 553-556 (1976). The defense is bottomed on the "principle that one who draws or indorses
commercial paper for the accommodation of another is not liable on that paper to the party accommodated, regardless of what their relationship on the instrument may indicate." Id. at 560. The defendants accordingly attempt to defeat the plaintiffs' motion for summary judgment by suggesting that certain oral understandings reached by all the respective parties establish the existence of material issues of fact as to whether the defendants signed the note as an accommodation to the plaintiffs. The defendants' efforts, however, are totally inadequate, as their counter affidavits are bereft of "specific facts" showing that there is a genuine, triable issue. See O'Brion, Russell & Co. v. LeMay, 370 Mass. 243 , 245 (1976); John B. Deary, Inc. v. Crane, 4 Mass. App. Ct. 719 , 722-723 (1976); Doff v. Brunswick Corp., 372 F.2d 801, 805 (9th Cir. 1966), cert. denied, 389 U.S. 820 (1967). See also 10 Wright & Miller, Federal Practice and Procedure Section 2738, at 709-710 (1973). For instance, the defendants fail to disclose, with any degree of specificity, which persons, acting in what capacity on behalf of the plaintiffs, said what, to whom, when and where. Commonwealth Bank & Trust Co. v. Plotkin, 371 Mass. 218 , 221-222 (1976). D & P Equipment Corp. v. Harvey Constr. Co., 5 Mass. App. Ct. 851 (1977). Compare Wagner v. Lectrox Corp., 4 Mass. App. Ct. 815 (1976). Moreover, save for the Silver affidavit, all the counter affidavits are defective on their faces, as they are not based on the respective affiants' personal knowledge (e.g., "evidence that has been made available to me"). See rule 56(e). See also Automatic Radio Mfg. Co. v. Hazeltine Research, Inc., 339 U.S. 827, 831 (1950); Antonio v. Barnes, 464 F.2d 584, 585 (4th Cir. 1972).


Nextoilrush.com: Broker predicts 440% rise for Cashed up Junior: Morocco, Gabon and now Madagascar. Is it a brave prediction?
This is a pretty big price prediction, and most likely based on recent news that PVD has just secured US $15 million in cash from a sign on bonus, bagged some highly prospective new acreage in Madagascar, raised $7 million in an oversubscribed capital raising and had their Moroccan acreage valued at US $1.8 billion.
PVD is up nearly 40% since the team at The Next Oil Rush called it in August in our exclusive report Why has this stock got analysts predicting four thousand per cent gains? This report goes into great detail on why we like PVD and why we have added PVD to our portfolio as a LONG TERM PLAY… we are certainly hoping that these broker predictions come true.
There are a number of signs pointing to potential share price growth in the coming months, on the back of the major happenings in the PVD camp.
Today we would like to update our readers on the PVD announcements that have recently come out of Morocco, Gabon and… news just in, Madagascar!
Correct, PVD have strategically expanded their African acreage to Madagascar. More on that below.
Regular readers would remember the deal struck by PVD that would see a US $15 million bonus paid upon government approval of the joint venture deal with US exploration giant Freeport-McMoRan in Morocco.
Well the joint venture deal was signed months ago, and since then PVD and the entire investment community have been awaiting government approval for the cheque to come through. After some hiccups dealing with an unfortunately timed change of Moroccan ministers and typical African bureaucracy, the bonus has been finalized!
The Next Oil Rush can confirm that the government approval has been obtained and US $15 million has hit the PVD bank account!
This significant capital injection into PVD proves they can come through and get the right deals done. Current investors are watching as the crowd who were waiting on the sidelines will now likely pile into the stock with this news. Needless to say, these movements could translate into potential gains in share price.
In addition to the US $ 15 million bonus going to PVD, there has been a frenzy of recent activity offshore Morocco, with super majors scrambling to claim their stakes in the area. PVD is perfectly placed to take advantage of the deals being struck all around their acreage. More on this below.
Turning to PVD's interests in Gabon, the US $155 billion dollar French oil super major, Total S.A., recently made a pre salt gas/condensate discovery nearby PVD's acreage in Gabon. This is no insignificant feat as confirms the existence of a working petroleum system in the same region as PVD.
But it doesn't end there, as PVD can add to this the AUD $7 million from their capital raising announced in September 2013. The AUD $7 million placement on the ASX was oversubscribed multiple times over. Apparently PVD were in the enviable position of picking and choosing strategic investors to contribute. This backing from major investors is a sign that PVD continues to represent real potential.
Finally, The Next Oil Rush can confirm that PVD has completed a strategic farm-in deal in MADAGASCAR, in a proven oil producing basin. A first in the country for PVD, it comes hot on the heels of PVD's capital injections described above, and is a wise play in diversifying their acreage in Africa.
A recent interview with Damon Neaves via BRR Media summarises all the latest deals at PVD.
All signs point to a potential upward trend in the PVD share price. In summary, PVD is fully funded and the next stage is to sit back and watch the drilling!
The Next Oil Rush has covered PVD extensively over the past few months, and our original report Why has this stock got analysts predicting four thousand per cent gains? provides a comprehensive review of the background and potential price catalysts for PVD.
Certainly a great insider's look at these prospects when viewed via the computer models!
US listed Plains E&P (NYSE:PXP) agreed to fund the entire costs for drilling two offshore wells in PVD's Mazagan field up to a maximum of US$215 million, in return for 52% of the block.
In addition to funding the drilling Plains E&P has paid US$15 million to PVD upfront in cash.
As part of the deal, PVD retain 23% of the block, and can now sit back and watch the well results come through - Great news for PVD and their investors who won't pay a cent in exploration costs!
Now government approval of the deal has been secured and the US $15 million has been banked by PVD. This leaves PVD in a very enviable capital position to pursue further opportunities in the region.
The Australian Financial Review also put in their 2 cents on the PVD deal, with the article 'Pura Vida one signature away from big Moroccan pay day' released in mid-October. All excellent news for our junior explorer, PVD.
Kosmos, owner of 3 blocks surrounding PVD (see the locations above), has just announced a rather large farm-in partner, none other than BP (NYSE:BP). In a similar move to Plains E&P's two well carry for PVD, BP will be carrying Kosmos for 3 exploratory wells, one well in each of their 3 blocks.
For Kosmos, the three drills will be subject to a maximum spend of US $120 million per well. In the event a second well is drilled on any block, BP will pay 150% of its share of costs up to a cap of US $ 120 million per well. Great news for Kosmos, and great news for PVD, who are already invested in a region BP is only just stepping into now.
Just south of PVD and the Kosmos / BP blocks, Cairn Energy are ramping up their own drilling program, recently spudding the FD-1 exploration well in Foum Draa (south of PVD's Mazagan permit). This will be the first test of a Cretaceous turbidite play with a potential of around 140 mmbo. The well is aiming for a different play to PVD's Toubkal prospect so whilst PVD investors will watch the well with interest, a negative outcome on this well is not going to take any of the momentum from Toubkal. Cairn aren't stopping there, with an additional 1 to 3 exploration wells planned for 2014.
Cairn are staying very busy in the region, with yet another deal struck in Morocco. The recent announcement relates to their farm-in agreement in the Cap Boujdour block, south of the Canary Islands. Under this deal, Cairn is funding various operations including a new 3D, an exploration well plus 2 optional appraisal wells.
All this activity coupled with the announcement of BP as a farm in partner in the blocks next door to PVD translates to potential price catalysts and interesting times for PVD investors.
PVD are expanding rapidly and have recently moved into Madagascar. The recent ASX announcement 'Pura Vida to Acquire 50% Interest In Ambilobe Block, Offshore Madagascar' has all the details of this deal. The Australian Financial Review also covered the Madagascar deal extensively.
In essence, PVD has entered into an agreement with Sterling Energy (AIM:SEY) to farm-in with 50% of the Ambilobe PSC block, off the coast of Madagascar. The block is huge, with an acreage footprint of 17,650 km2. PVD's moved in next door to major ExxonMobil, who are involved in a block to the south of the Ambilobe block.
PVD sees the Ambilobe block as having a variety of plays relating to salt with potential for large scale discoveries. PVD's depth of knowledge relating to salt basins and how best to explore them makes the Ambilobe block a natural addition to their growing portfolio.
PVD have chosen this region strategically as it is located in a proven oil producing basin. One glance at the figure below shows the East African discoveries along with the location of PVD's Ambilobe basin.
Exploration onshore Madagascar has already resulted in the discovery of numerous heavy oil fields. One look at all those red and green dots and anyone would think that Christmas has come early in Madagascar! The Next Oil Rush believes it might have for PVD, as these onshore discoveries prove the "existence of an abundant oil source".
Over in Gabon some interesting developments are going on too in the PVD camp.
In addition to the US $15 million PVD will receive related to Morocco, September 2013 saw a capital raising of AUD $7 million via placement on the ASX.
Investor response was strong. We hear that the offer was oversubscribed multiple times, and PVD had the luxury of picking and choosing a suite of long term, strategic investors who together put AUD $7 million on the table and into PVD. Apparently this included funds out of the UK, Asia as well as Australia. This demonstrates PVD's evolution from a small-cap to a serious player in the sector, backed by institutional money.
The AUD $7 million investment will be used chiefly to invest in the potentially highly lucrative Nkembe block off Gabon. Combined with the US $15 million, the capital raising puts PVD in an enviable financial position ahead of coming price catalysts.
It's difficult to have a more positive signal of investor sentiment when highly sophisticated investors are so convinced of the potential success of PVD. As you can see below there is a lot of activity offshore in Gabon, both above and below PVD's exploration acreage.
The detail in red in the above slide shows where French oil super-major Total S.A. recently made a very impressive pre-salt discovery on their block. This is big news as it is the first time that a discovery in the deep pre-salt reservoir has been made offshore in Gabon.
The Diaman-1B pre-salt discovery successfully confirms the existence of a working petroleum system in the region.
The discovery proves the new pre-salt play in Gabon and proves one of the main plays which extends into PVD's acreage. Luckily, Total's discovery in the pre-salt, deep-water portion is the same target type that PVD is looking to drill. This is a timely development given PVD is currently in farmout discussions.
Gabon oil experts already know that the onshore pre-salt oil reserves are proven, with an estimated 800 to 900 million barrels. The offshore pre-salt potential in Gabon looks very encouraging, especially looking at the players already investing in this hot market.
There are no less than 10 wells to be drilled over the next 12 months in the region, so investors will be watching closely the results. The team at The Next Oil Rush expects some clear movement in the PVD price based on these Gabon wells.
PVD have staked their claim in Gabon and the team at The Next Oil Rush is watching developments closely.
WHAT ARE THE BROKERS SAYING?
Followers of PVD may remember back in mid-2013, the stock analysts' positive target price predictions for PVD. If the mid-year brokers target prices had slipped your mind, here are a few repeated again. For those desperate to know the latest prediction from Morgans, just scroll down.
Things are continuing to look up for PVD.
With two capital injections of US $15 million and AUD $7 million, PVD is fully funded and will be in a strong position for the coming months of potential price catalysts.
PVD's entry into Madagascar confirms their presence in three potentially lucrative upcoming oil regions.
Total's discovery offshore in Gabon is another positive sign for PVD, and confirms the potential for sizeable pre-salt discoveries to be made in Gabon.
The next stop is continued DRILLING in both Morocco and Gabon, with further seismic data acquisition slated for Madagascar. The Next Oil Rush will keep you posted on all the developments as they come through.
For further investor information and an in-depth analysis of PVD, remember to refer back to our original article Why has this stock got analysts predicting four thousand per cent gains?
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I'm New to Intimacy. How Do I Approach This Situation?
I'm New to Intimacy. How Do I Approach This Situation?
I am a 25-year-old male dating a 29-year-old woman. I've never had a relationship in my life and am very new to intimacy. I really like this woman, but I fear that when it comes time for intimacy, especially sexual relations, she will be disappointed and this will end things.
Do you have any advice on how I approach the situation?
To quote a famous president, you have nothing to fear but fear itself.
In other words, it is your anxiety, not her judgements, that is the root of your problem. And if you move too fast, these anxieties can disrupt the physical part of you.
So, you've got to move slowly. Build trust. Build communication. Don't jump into bed, but spend a few good nights with kissing and fondling. Let your body relax and take over.
And tell her she means something to you and that's why you'd like to move slowly. She'll respect that.
If you take the time to build a friendship and some good sexual communication skills, you'll reduce the fear that she'll judge you.