Fano varieties of cubic fourfolds containing a plane
Emanuele Macr\`i and Paolo Stellari
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
0909.2725.tex
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\begin{document}

\title{Fano varieties of cubic fourfolds containing a plane}

\author[E.\ Macr\`i and P.\ Stellari]{Emanuele Macr\`i and Paolo Stellari}

\address{E.M.: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA}
\email{macri@math.utah.edu}

\address{P.S.: Dipartimento di Matematica ``F. Enriques'',
Universit{\`a} degli Studi di Milano, Via Cesare Saldini 50, 20133
Milano, Italy} \email{paolo.stellari@unimi.it}


\keywords{Derived categories, cubic fourfolds}

\subjclass[2000]{18E30, 14E08}

\begin{abstract}
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:intro}

The moduli space $\kc$ of \emph{cubic fourfolds} (i.e., smooth hypersurfaces of degree $3$ in $\PP^5$) is a quasi-projective variety of dimension $20$. As pointed out in \cite{H2}, a way to produce divisors in $\kc$ is to consider \emph{special} cubic fourfolds, that is cubic fourfolds containing an algebraic surface not homologous to a complete intersection. More precisely, let $Y$ be a special cubic fourfold such that $H^4(Y,\ZZ)$ contains a primitive rank-$2$ sublattice $L$ generated by $H^2$, the self-intersection of the hyperplane section, and an algebraic surface $T$. If $d$ is a positive integer, $Y$ is contained in the divisor $\kc_d\subseteq\kc$ if there exists $L$ as above and the matrix representing the intersection form on $L$ has determinant $d$. Under some assumptions on $d$, $\kc_d$ is non-empty and irreducible (see \cite{H2}). The basic example we deal with in this paper is $\kc_8$, the divisor parametrizing cubic fourfolds containing a plane (\cite{H2,V}).

\smallskip

The \emph{Fano variety of lines} $F(Y)$ of a cubic fourfold $Y$ (i.e., the variety parametrizing the lines in $Y$) is a projective irreducible symplectic manifold of dimension $4$. A classical result of Beauville and Donagi \cite{BD} says that $F(Y)$ is deformation equivalent to the Hilbert scheme $\mathrm{Hilb^2}(S)$ of length-$2$ $0$-dimensional subschemes of a K3 surface $S$ of degree $14$. In other words, $F(Y)$ can be deformed to a fine moduli space of stable sheaves on $S$.

As the moduli space of smooth projective K3 surfaces is $19$-dimensional, $F(Y)$ cannot always be isomorphic to a moduli space of stable sheaves on a K3 surface. Nevertheless, the following result was proved in \cite{H2}:

\begin{thm}\label{thm:Hassett} {\bf (Hassett)}
	Assume that $d=2(n^2+n+2)$ where $n$ is an integer $\geq 2$ and let $Y$ be a generic cubic fourfold in $\kc_d$. Then $F(Y)$ is isomorphic to the Hilbert scheme of length-$2$ $0$-dimensional subschemes of a K3 surface and so, in particular, to a fine moduli space of stable sheaves.
\end{thm}

The case $d=8$, corresponding to cubic fourfolds $Y$ containing a plane $P$, is not covered by this result. This happens despite the fact that a very natural K3 surface is related to the geometry of $Y$. Indeed, consider the projection $Y\dashrightarrow\PP^2$ from $P$ onto a plane in $\PP^5$ disjoint from $P$. The blow up of $Y$ along $P$ yields a quadric fibration $\pi:\widetilde Y\to\PP^2$ whose fibres degenerate along a plane sextic $C\subseteq\PP^2$. Assume that $C$ is smooth (which is generically the case). The double cover $S\to\PP^2$ ramified along $C$ is a smooth projective K3 surface.

This construction provides also a natural element $\beta$ in the Brauer group of $S$ which is either trivial or of order $2$. The geometric explanation for $\beta$ goes essentially back to \cite{V}. Indeed, the K3 surface $S$ can be thought of as the `moduli space of rulings' of the quadrics in the fibration $\pi:\widetilde Y\to\PP^2$. Hence, this provides a $\PP^1$-bundle $F\to S$ parametrizing the lines $l\subseteq Y$ contained in the fibres of $\pi$. It turns out that $F$ is actually the Brauer--Severi variety corresponding to the twist $\beta$ (see Section \ref{subsec:twistedK3} for more details).

\smallskip

As many interesting examples of cubic fourfolds with a rich geometry are not considered in Hassett's result, one may wonder whether Theorem \ref{thm:Hassett} can be extended further. More precisely, following \cite{H2} and \cite{KMM}, one can raise the following natural question.

\begin{qn}\label{qn:main}
Are there other values of $d$ for which a (generic) cubic fourfold $Y$ has Fano variety $F(Y)$ isomorphic to a moduli space of stable (twisted) sheaves or complexes on a K3 surface?
\end{qn}

This paper may be considered as an attempt to answer this question in the special case of cubic fourfolds containing a plane. That, in general, $F(Y)$ is not isomorphic (and not even birational) to a fine moduli space of untwisted sheaves can be easily proved, even for $Y$ containing a plane (see, for example, Proposition \ref{prop:nospmod}).

Nevertheless, remembering the definition of the K3 surface $S$ above and passing to $\beta$-twisted sheaves and complexes (see Section \ref{subsec:twistedK3} for the precise definitions), we can state the first main result of the paper.

\begin{thm}\label{thm:main1}
If $Y$ is a generic cubic fourfold containing a plane, then $F(Y)$ is isomorphic to a moduli space of stable objects in the derived category $\Db(S,\beta)$ of bounded complexes of $\beta$-twisted coherent sheaves on $S$.
\end{thm}

In the statement, a cubic fourfold $Y$ is meant to be \emph{generic} if the plane sextic $C$ is smooth and the twist $\beta$ is not trivial. Each of these two properties is actually satisfied by the cubic fourfolds containing a plane which are in the complement of a countable union of codimension-$1$ subvarieties of $\kc_8$. Hence, the cubic fourfolds to which we refer in Theorem \ref{thm:main1} are dense in $\kc_8$ and our result can be regarded as the analogue of Theorem \ref{thm:Hassett}.

The proof of Theorem \ref{thm:main1} will be carried out in Section \ref{sec:iso}. We will show that the result could be stated in a more precise but less compact way. Indeed, the Mukai vector of the complexes parametrized by the moduli space mentioned in the statement above and the Bridgeland's stability conditions for which they are stable can be explicitly described.

\medskip

The other direction we follow in the paper consists in weakening Question \ref{qn:main}. Indeed, one can wonder if the Fano variety of lines of a cubic fourfold is birational to a fine moduli space of twisted sheaves on a K3 surface. The same dimension counting as at the beginning of the introduction shows that this cannot be true for all cubic fourfolds in $\kc$.

One geometric condition that may give a hope for a positive answer to the previous question is the presence of a K3 surface associated to the geometry of the fourfold. This guess is confirmed by the second main result of the paper.

\begin{thm}\label{thm:main2}
For all cubic fourfolds $Y$ containing a plane, the Fano variety $F(Y)$ is birationally isomorphic to a fine moduli space of twisted sheaves on the corresponding K3 surface $S$. Moreover, if the plane sextic $C$ is smooth, then such a birational isomorphism is either an isomorphism or a Mukai flop.
\end{thm}

The K3 surface $S$ and the sextic $C$ are again those presented at the beginning of the introduction. Also in this case, the Mukai vector of the stable twisted sheaves can be explicitly described. Moreover, the Hodge (and lattice) structure of the second cohomology of the moduli space will be made apparent in the course of the proof explained in Sections \ref{subsec:bir1} and \ref{subsec:bir2}.

Notice that the moduli space of complexes in Theorem \ref{thm:main1} can be regarded as a compactification of an open subset of the moduli space of twisted stable sheaves in Theorem \ref{thm:main2} via complexes. The two compactifications are related by a wall-crossing phenomenon in the derived category. In geometric terms, as observed in Remark \ref{rmk:MukaiFlopUtah}, this means that $F(Y)$ is the Mukai flop of a moduli space of twisted sheaves, for generic cubic fourfolds containing a plane.

\medskip

Some words about the plan of the paper are in order here. Section \ref{sec:prelim} studies the semi-orthogonal decomposition of the derived category of a cubic fourfold containing a plane defined in \cite{Kuz3}. In particular, we deal with some basic properties of the ideal sheaves of lines (and of complexes closely related to them) which will be fundamental for the paper. Contrary to the order in the exposition above, we first prove Theorem \ref{thm:main2} in Section \ref{sec:stab}. This is motivated by the fact that the argument descends directly from some easy properties studied in Section \ref{subsec:twistedK3} and which will be used to prove Theorem \ref{thm:main1} as well. Finally, in Section \ref{sec:iso}, we use Bridgeland's stability conditions to prove Theorem \ref{thm:main1}. For sake of simplicity, all derived functors will be denoted as if they were underived, e.g.\ for a morphism of varieties $f:X\to Z$, we will denote $f^*$ for the derived pull-back, $f_*$ for the derived push-forward, and so on.

\section{The categorical setting and the ideal sheaves of lines}\label{sec:prelim}

In this section we study the semi-orthogonal decomposition of $\Db(Y)$ given in \cite{Kuz3}, for $Y$ a cubic fourfold. When $Y$ contains a plane, some basic properties of the relevant piece of this decomposition have been analyzed in \cite[Sect.\ 5.3]{BMMS}. Here we continue this study considering some special objects in it which are used in the rest of the paper.

\subsection{The geometric setting}\label{subsec:geometry}

Suppose that $Y$ is a cubic fourfold containing a plane $P$. As explained in the introduction and following \cite{Kuz3}, consider the diagram
\[
\xymatrix{D\ar[r]\ar[d]&\widetilde{Y}:={\rm Bl}_{P}Y\ar^{\sigma}[d]\ar[r]^{\;\;\;\;\;\;\;\;\pi}&\PP^2&\\
P\ar[r]&Y\subseteq\PP^5,
}
\]
where $\sigma:\widetilde{Y}\to Y$ is the blow-up of $Y$ along $P$, $D$ is the exceptional divisor, and $\pi:\widetilde{Y}\to\PP^2$ is the quadric fibration induced by the projection from $P$ onto a plane.

As in \cite[Sect.\ 3]{Kuz2}, to the quadric fibration $\pi$, one can associate a sheaf $\kb_0$ (resp.\ $\kb_1$)  of even (resp.\ odd) parts of the Clifford algebra. Explicitly, as sheaves of $\ko_{\PP^2}$-modules, we have
\begin{align*}
\kb_0&\cong\ko_{\PP^2}\oplus\ko_{\PP^2}(-h)^{\oplus 3}\oplus\ko_{\PP^2}(-2h)^{\oplus 3}\oplus\ko_{\PP^2}(-3h)\\
\kb_1&\cong\ko_{\PP^2}^{\oplus 3}\oplus\ko_{\PP^2}(-h)^{\oplus 2}\oplus\ko_{\PP^2}(-2h)^{\oplus 3},
\end{align*}
where $h$ denotes, by abuse of notation, both the class of a line in $\PP^2$ and its pull-back via $\pi$.
Denote by $\coh(\PP^2,\kb_0)$ the abelian category of right coherent $\kb_0$-modules and by $\Db(\PP^2,\kb_0)$ its bounded derived category.

\begin{remark}\label{rmk:spherical}
Notice that $\kb_i$ is spherical, for $i=0,1$ (i.e.,\ $\Hom(\kb_i,\kb_i)\cong\Hom(\kb_i,\kb_i[2])\cong\CC$ and $\Hom(\kb_i,\kb_i[j])=0$, when $j\neq 0,2$). Indeed, by adjunction, $\Hom(\kb_0,\kb_0[j])\cong H^j(\PP^2,\ko_{\PP^2}\oplus\ko_{\PP^2}(-h)^{\oplus 3}\oplus\ko_{\PP^2}(-2h)^{\oplus 3}\oplus\ko_{\PP^2}(-3h))$ and the functor $(-)\otimes_{\kb_0}\kb_1$ is an autoequivalence of $\coh(\PP^2,\kb_0)$ sending $\kb_0$ to $\kb_1$ (see \cite[Cor.\ 3.9]{Kuz2}).
\end{remark}

\medskip

Denote by $C$ the sextic curve in $\PP^2$ over which the fibres of $\pi$ degenerate. We will often need to restrict ourselves to the case where $Y$ satisfies the following additional requirement:
\begin{itemize}
\item[$(\ast)$] The cubic fourfold $Y$ contains a plane $P$ and the sextic $C$ is smooth.
\end{itemize}

\begin{remark}\label{rmk:stupid2}
An easy observation shows that $(\ast)$ coincides with the assumption in \cite{Kuz3}. More precisely, the cubic fourfold $Y$ satisfies $(\ast)$ if and only if the fibres of $\pi:\widetilde{Y}\to\PP^2$ have at most one singular point.	Indeed, by \cite[Prop.\ 1.2]{Bea}, the curve $C$ can have at most ordinary double points. Moreover a fibre of $\pi$ is union of two distinct planes if and only if it is projected to a singular point of $C$.
\end{remark}

If $(\ast)$ holds true, then the double cover of $\PP^2$  ramified along $C$ $$f:S\longrightarrow\PP^2$$ is a smooth K3 surface. Furthermore, by \cite[Prop.\ 3.13]{Kuz2}, there exists an Azumaya algebra $\ka_0$ on the K3 surface $S$, such that
\begin{equation}\label{eqn:Azum}
f_*\ka_0=\kb_0\qquad\mbox\qquad f_*:\coh(S,\ka_0)\isomor\coh(\PP^2,\kb_0).
\end{equation}
(See, for example, \cite[Chapter 1]{C} for the basic properties of Azumaya algebras.)

\subsection{The semi-orthogonal decomposition}\label{subsec:semiorthogonal}

Let $Y$ be a cubic fourfold and let $\Db(Y)$ be its bounded derived category of coherent sheaves.
Define $\ko_Y(iH):=\ko_{\PP^5}(iH)|_{Y}$ and $$\cat{T}_Y:=\langle\ko_Y,\ko_Y(H),\ko_Y(2H)\rangle^\perp.$$
Since the collection $\{\ko_Y,\ko_Y(H),\ko_Y(2H)\}$ is exceptional in $\Db(Y)$, we have a semi-orthogonal decomposition of $\Db(Y)$ as
\[
\Db(Y)=\langle\cat{T}_Y,\ko_Y,\ko_Y(H),\ko_Y(2H)\rangle.
\]

By \cite[Sect.\ 4]{Kuz2}, we can define a fully faithful functor $\Phi:=\Phi_{\ke'}:\Db(\PP^2,\kb_0)\to\Db(\widetilde{Y})$, $$\Phi_{\ke'}(\ka):=\pi^*\ka\otimes_{\pi^*\kb_0}\ke',$$ for all $\ka\in\Db(\PP^2,\kb_0)$, where $\ke'\in\coh(\widetilde{Y})$ is a rank-$4$ vector bundle on $\widetilde{Y}$ with a natural structure of flat left $\pi^*\kb_0$-module.
We will not need the actual definition of $\ke'$ (for which the reader is referred to \cite[Sect.\ 4]{Kuz2}) but only the presentation
\[
0\to q^*\kb_0(-2H)\to q^*\kb_1(-H)\to\alpha_*\ke'\to0,
\]
where $\alpha:\widetilde{Y}\mono\widetilde{\PP^5}$ is the natural embedding and $q:\widetilde{\PP^5}\to\PP^2$ the induced projection, $\widetilde{\PP^5}$ being the blow-up of $\PP^5$ along $P$ and $H$ denoting, again by abuse of notation, both a hyperplane in $\PP^5$ and its pull-backs to $\widetilde{Y}$ and to $\widetilde{\PP^5}$.
The calculation in \cite[Lemma 4.1]{Kuz3} yields $\ko_{\widetilde{Y}}(D)\cong\ko_{\widetilde{Y}}(H-h)$, $\widetilde{\PP}^5 \cong \PP_{\PP^2} (\ko_{\PP^2}^{\oplus 3} \oplus \ko_{\PP^2}(-h))$, the relative ample line bundle is $\ko_{\widetilde{\PP}^5}(H)$, and the relative canonical bundle is $\ko_{\widetilde{\PP}^5}(h-4H)$.

\medskip

The results of \cite{Kuz2} and \cite{Orlov} give, respectively, two semi-orthogonal decompositions:
\begin{equation}\label{eqn:semiorth}
\begin{array}{l}
	\Db(\widetilde Y)=\langle\Phi(\Db(\PP^2,\kb_0)),\ko_{\widetilde Y}(-h),\ko_{\widetilde Y},\ko_{\widetilde Y}(h),\ko_{\widetilde Y}(H),\ko_{\widetilde Y}(h+H),\ko_{\widetilde Y}(2h+H)\rangle\\
\Db(\widetilde Y)=\langle\sigma^*(\cat{T}_Y),\ko_{\widetilde Y},\ko_{\widetilde Y}(H),\ko_{\widetilde Y}(2H),i_*\ko_{D},i_*\ko_{D}(H),i_*\ko_{D}(2H)\rangle.	
\end{array}
\end{equation}
Theorem 4.3 in \cite{Kuz3} yields an equivalence $$\Db(\PP^2,\kb_0)\cong\cat{T}_Y.$$ The way this is achieved is performing a sequence of mutations which allow Kuznetsov to compare the two semi-orthogonal decompositions in \eqref{eqn:semiorth}. The details will not be needed in the rest of this paper but we just recall the two basic mutations:
\begin{itemize}
	\item[(A)] The right mutation of $\Phi(\Db(\PP^2,\kb_o))$ through $\ko_{\widetilde Y}(-h)$ leading to the semi-orthogonal decomposition
	\[
	\qquad\Db(\widetilde Y)=\langle\ko_{\widetilde Y}(-h),\Phi'(\Db(\PP^2,\kb_0)),\ko_{\widetilde Y},\ko_{\widetilde Y},\ko_{\widetilde Y}(h),\ko_{\widetilde Y}(H),\ko_{\widetilde Y}(h+H),\ko_{\widetilde Y}(2h+H)\rangle,
	\]
	where $\Phi'$ is the functor obtained composing $\Phi$ with the right mutation.
	\item[(B)] The left mutation of $\Phi'(\Db(\PP^2,\kb_0))$ through $\ko_{\widetilde Y}(h-H)$.
\end{itemize}
(For details and basic results about mutations, see, for example, \cite[Sect.\ 2]{Kuz3}.)

\medskip

Recall that the left adjoint functor of $\Phi$ is
\begin{align*}
\Psi(-)&:=\pi_*((-)\otimes\ko_{\widetilde{Y}}(h)\otimes\ke[1]),
\end{align*}
where $\ke\in\coh(\widetilde{Y})$ is another rank-$4$ vector bundle on $\widetilde{Y}$ with a natural structure of right $\pi^*\kb_0$-module (see \cite[Sect.\ 4]{Kuz2}). The main property we will need is the presentation
\begin{equation}\label{eq:defofke}
0\to q^*\kb_1(-h-2H)\to q^*\kb_0(-H)\to\alpha_*\ke\to0.
\end{equation}

The following easy result will be used to control the compatibilities with the mutations (A) and (B) above.

\begin{lem}\label{lem:stupid}
	For any integer $m$ and for $a=0,-1$, we have $\Psi(\ko_{\widetilde Y}(mh+aH))=0$.
\end{lem}

\begin{proof}
	By the projection formula, \eqref{eq:defofke}, and the fact that $\pi = q \circ \alpha$,
	\begin{equation*}
	\begin{split}
	\Psi(\ko_{\widetilde{Y}}(mh+aH))&\cong\Psi(\alpha^*(\ko_{\widetilde{\PP^5}}(mh+aH)))\\
	               &\cong q_*(\ko_{\widetilde{\PP^5}}((m+1)h+aH)\otimes\alpha_*\ke[1])=0,
	\end{split}
	\end{equation*}
	for all $m$ and for $a=0,-1$. Here we are using the fact that $\widetilde{\PP^5}\to\PP^2$ is a projective bundle and $\ko_{\widetilde{\PP^5}/\PP^2}(1)\cong\ko_{\widetilde{\PP^5}}(H)$.
\end{proof}

\subsection{Ideal sheaves of lines and their relatives}\label{subsec:idealshlines}

Let $l$ be a line in $Y$ and let $I_l$ be its ideal sheaf. As remarked in \cite{KM}, while $I_l\not\in\cat{T}_Y$, the (stable) sheaf $\kf_l\in\coh(Y)$ defined by the short exact sequence
\begin{equation}\label{eqn:defFl}
0\to\kf_l\to\ko_{Y}^{\oplus 4}\to I_l(H)\to 0
\end{equation}
is in $\cat{T}_Y$. This means that, given the semi-orthogonal decomposition
\[
\Db(Y)=\langle\cat{T}'_Y,\ko_Y(-H),\ko_Y,\ko_Y(H)\rangle,
\]
the sheaf $\kf_l(-H)$ is in $\cat{T}'_Y$. Right-mutating $\cat{T}'_Y$ with respect to $\ko_Y(-H)$, one gets the semi-orthogonal decompositions
\[
\Db(Y)=\langle\ko_Y(-H),\cat{T}''_Y,\ko_Y,\ko_Y(H)\rangle=\langle\cat{T}''_Y,\ko_Y,\ko_Y(H),\ko_Y(2H)\rangle.
\]
In particular, the right mutation
\begin{equation}\label{eqn:defPl}
P_l:=\mathrm{cone}(ev\dual:\kf_l(-H)\to\mathrm{R}\Hom(\kf_l(-H),\ko_Y(-H))\dual\otimes\ko_Y(-H))[-1]
\end{equation}
is contained in $\cat{T}''_Y$ and, due to the natural identification of $\cat{T}_Y$ with $\cat{T}''_{Y}$, is an object of $\cat{T}_Y$.

\begin{remark}\label{rmk:modsp}
	Due to \cite[Prop.\ 5.5]{KM}, the Fano variety $F(Y)$ is identified with the connected component of the moduli space of stable sheaves containing the objects $\kf_l$, for any line $l$ in $Y$. Since the objects $P_l$ are obtained by tensoring $\kf_l$ and applying a right mutation (which yield an autoequivalence of $\cat{T}_Y$), $F(Y)$ can be also identified to the moduli space of the objects $P_l$.
\end{remark}

The following result which calculates the preimage
\begin{equation}\label{eq:Dan}
\ki_l:=(\sigma_*\circ\Phi')^{-1}(P_l)[1]
\end{equation}
in $\Db(\PP^2,\kb_0)$ is particularly relevant for the proof of Theorems \ref{thm:main1} and \ref{thm:main2}.

\begin{prop}\label{prop:ideallines}
	If $l$ is not contained in $P$ then $\ki_l$ is a pure torsion sheaf supported on a line in $\PP^2$. If $l\subseteq P$, then there exists an exact triangle
	\[
	\kb_0[1]\longrightarrow\ki_l\longrightarrow\kb_1,
	\]
	where the extension map $\kb_1\to\kb_0[2]$ is uniquely determined by the line $l$. Moreover, there exists a bijective correspondence between the Fano variety $P\dual$ of lines in $P$ and the isomorphism classes of extensions $\kb_1\to\kb_0[2]$.
\end{prop}

\begin{proof}
%The proof is very similar to \cite{BMMS}, Step 5, Section 3.
First observe that $(\sigma_*\circ\Phi')^{-1}(I_l)\cong\Psi(\sigma^*I_l)$.
Indeed, by Lemma \ref{lem:stupid}, the mutations (A) and (B) by $\ko_{\widetilde{Y}}(-h)$ and  $\ko_{\widetilde{Y}}(h-H)$ have no effect.
Applying $\sigma^*$ to \eqref{eqn:defFl} (tensored by $\ko_{\widetilde Y}(-H)$) and to \eqref{eqn:defPl} and observing that, again by Lemma \ref{lem:stupid},
$\Psi(\ko_{\widetilde Y}(-H))=0$, we have
\[
\ki_l=(\sigma_*\circ\Phi')^{-1}(\kf_l(-H))[1]=(\sigma_*\circ\Phi')^{-1}(I_l).
\]

The easiest case is when $l\cap P=\emptyset$. Since the rational map $Y\dashrightarrow\PP^2$ is well-defined on $l$ and map it to another line, denote by $j$ the embedding $l\mono\widetilde{Y}\mor[\pi]\PP^2$.
	Pulling back via $\sigma$ the exact sequence
	\[
	0\longrightarrow I_l\longrightarrow\ko_Y\longrightarrow\ko_l\longrightarrow 0,
	\]
	yields
	\[
	0\longrightarrow\sigma^*I_l\longrightarrow\ko_{\widetilde{Y}}\longrightarrow\sigma^*\ko_l=\ko_l\longrightarrow 0.
	\]
	By Lemma \ref{lem:stupid}, we have $\Psi(\ko_{\widetilde{Y}})=0$ and so
	\begin{equation*}
	\begin{split}
	\Psi(\sigma^*I_l)&\cong\Psi(\ko_l)[-1]\\
	    &=\pi_*(\ko_l[-1]\otimes\ko_{\widetilde{Y}}(h)\otimes\ke[1]))\\
	    &\cong j_*(j^*\ko_{\widetilde{Y}}(h)\otimes\ke |_l)\\
	    &\cong j_*(\ke |_l)\otimes\ko_{\PP^2}(h),
	\end{split}
	\end{equation*}
which is precisely the claim.

Assume now $l\cap P=\{pt\}$. A local computation shows that $\sigma^*I_l$ is a sheaf, sitting in an exact sequence
\[
0\longrightarrow\sigma^*I_l\longrightarrow\ko_{\widetilde{Y}}\longrightarrow\ko_{l\cup\gamma}\longrightarrow 0,
\]
where $\gamma:=\sigma^{-1}(pt)$ and $l$ denotes, by abuse of notation, the strict transform of $l$ inside $\widetilde{Y}$.
Now, by Lemma \ref{lem:stupid}, we have
\[
\ki_l\cong\Psi(\sigma^*I_l)\cong\Psi(\ko_{l\cup\gamma})[-1]\cong\pi_*(\ke|_{l\cup\gamma}\otimes\ko_{\widetilde{Y}}(h)).
\]
By using the exact sequence
\[
0\longrightarrow\ko_{\gamma}(-h)\longrightarrow\ko_{l\cup\gamma}\longrightarrow\ko_l\longrightarrow 0,
\]
we have an exact triangle
\begin{eqnarray}\label{eqn:coho}
\pi_*(\ke|_{\gamma})\longrightarrow\ki_l\longrightarrow\pi_*(\ke|_l),
\end{eqnarray}
where $\pi_*(\ke|_{\gamma})$ is a torsion sheaf supported on a line in $\PP^2$, since $\pi$ is a closed embedding on $\gamma$.
By \eqref{eq:defofke}, the sheaf $\ke|_{l}$ has no higher cohomologies and so $\pi_*(\ke|_{l})$ is a torsion sheaf supported on a point.
Hence $\ki_l$ is a torsion sheaf supported on a line in $\PP^2$.

Finally, let us consider the case $l\subseteq P$ and denote by $C'$ the preimage of $l$ via $\sigma$. By Lemma \ref{lem:stupid}, applied to
\[
\sigma^*I_l\longrightarrow\ko_{\widetilde{Y}}\longrightarrow\sigma^*\ko_l,
\]
we know $\ki_l\cong\Psi(\sigma^*\ko_l)[-1]$. By \cite[Prop.\ 11.12]{Huy}, we have an exact triangle
\[
\ko_{C'}(D)[1]\longrightarrow\sigma^*(\ko_{l})\longrightarrow\ko_{C'},
\]
where, as before, $D$ denotes the exceptional divisor of $\widetilde{Y}$. By the fact that $\ko_D(-C')\cong\ko_D(-H)$ and by Lemma \ref{lem:stupid} applied to the short exact sequence
\[
0\longrightarrow\ko_D(-H)\longrightarrow\ko_{\widetilde{Y}}\longrightarrow\ko_D\longrightarrow 0,
\]
we get $\Psi(\ko_D)=0$ and $\Psi(\ko_{C'})\cong\Psi(\ko_D(-H))[1]$. On the other hand, an easy calculation using again Lemma \ref{lem:stupid} shows that $\Psi(\ko_D(-H))[1]\cong\Psi(\ko_D(h-2H))[2]\cong\kb_1[1]$ (for the last isomorphism use \eqref{eq:defofke}). This gives the morphism $\ki_l\to\kb_1$.
The argument to show $\Psi(\ko_{C'}(D))\cong\kb_0$ is similar and it is left to the reader.

To prove the last claim, observe that each line $l\subseteq P$ determines uniquely the (isomorphism class of the) object $P_l$ and hence the extension $\kb_1\to\kb_0[2]$ in the above constuction. This yields an injection $P\dual\hookrightarrow\PP(\Hom(\kb_1,\kb_0[2]))$. But now, by Serre duality and adjunction, $\Hom(\kb_1,\kb_0[2])\cong \Hom(\kb_0,\kb_1)\cong\Hom(\ko_{\PP^2},\kb_1)\cong\CC^{\oplus 3}$.
\end{proof}

\section{Birationality and moduli spaces of sheaves}\label{sec:stab}

In this section we prove Theorem \ref{thm:main2}. Along the way we recall some facts about twisted sheaves and twisted K3 surfaces.

\subsection{Twisted K3 surfaces}\label{subsec:twistedK3}

Let $X$ be a smooth projective variety and let $\Br(X)$ be its \emph{Brauer group} (i.e., the torsion part of the
cohomology group $H^2(X,\ko_X^*)$ in the analytic topology).
Recall that any $\beta\in\Br(X)$ can be represented by a \v{C}ech cocycle on
an open analytic cover $\{U_i\}_{i\in I}$ of $X$ using the
sections $\beta_{ijk}\in\Gamma(U_i\cap U_j\cap
U_k,\mathcal{O}^*_X)$. A \emph{$\beta$-twisted coherent sheaf}
$\kf$ consists of a collection $(\{\kf_i\}_{i\in
I},\{\varphi_{ij}\}_{i,j\in I})$, where $\kf_i$ is a coherent
sheaf on $U_i$ and $\varphi_{ij}:\kf_j|_{U_i\cap
U_j}\to\kf_i|_{U_i\cap U_j}$ is an isomorphism satisfying the
following conditions:
\[
\mbox{$\varphi_{ii}=\mathrm{id}$;\;\;\;\;\;
$\varphi_{ji}=\varphi_{ij}^{-1}$;\;\;\;\;\;
$\varphi_{ij}\circ\varphi_{jk}\circ\varphi_{ki}=\beta_{ijk}\cdot\mathrm{id}$.}
\]
By $\coh(X,\beta)$ we denote the abelian category of
$\beta$-twisted coherent sheaves on $X$, while $\Db(X,\beta)$
is the bounded derived category $\Db(\coh(X,\beta))$. A \emph{twisted variety} is a pair $(X,\beta)$, where $X$ is a smooth projective variety and $\beta\in\Br(X)$. (For more details about twisted sheaves, see \cite{C}.)

If now $S$ is a K3 surface, one can lift $\beta$ to a \emph{$B$-field} $B$ (i.e., an element in $H^2(S,\QQ)$), via the exponential short exact sequence (see, for example, \cite[Sect.\ 1]{HS1}). For $\sigma$ a generator of $H^{2,0}(S)$, let $\sigma_B:=\sigma+\sigma\wedge B\in H^*(S,\CC)$ be the \emph{twisted period} of $(S,\beta)$ (see again \cite{HS1}). Let $T(S,B)\subseteq H^*(S,\ZZ)$ be the \emph{twisted transcendental lattice}, i.e.\ the minimal primitive sublattice of $H^*(S,\ZZ)$ such that $\sigma_B\in T(S,B)\otimes\CC$. (The lattice structure is given by the Mukai pairing on $H^*(S,\ZZ)$ as explained in \cite[Sect.\  5.2]{Huy}.) Moreover, we denote by $\Pic(S,B):=T(S,B)^\perp$ the \emph{twisted Picard lattice}, where the orthogonal is taken in $H^*(S,\ZZ)$ with respect to the Mukai pairing. In \cite{HS1}, it was also defined a twisted weight-$2$ Hodge structure on the total cohomology of $S$ which, together with the Mukai pairing, is then denoted by $\widetilde H(X,B,\ZZ)$.

In \cite{HS1}, using the $B$-field lift of $\beta\in\Br(S)$, the twisted Chern character ${\rm ch}^B:\coh(S,\beta)\to\widetilde H(S,B,\ZZ)$ is defined. The \emph{Mukai vector} of $\kf\in\Db(S,\beta)$ is $v^B(\kf):=\ch^B(\kf)\cdot\sqrt{\mathrm{td}(S)}\in\widetilde H(S,B,\ZZ)$.  For $\kf\in\coh(S,\beta)$ we denote by $\rk(\kf)$ and $c_1^B(\kf)$ the degree $0$ and $2$ parts of $\ch^B(\kf)$.

\begin{lem}\label{lem:NS}
	If $\beta\in\Br(S)$ has order $d$, the lattice $\Pic(S,B)$ is generated by $\Pic(S)$ and the vectors $(d,dB,0)$ and $(0,0,1)$ in $H^*(S,\ZZ)=H^0(S,\ZZ)\oplus H^2(S,\ZZ)\oplus H^4(S,\ZZ)$.
\end{lem}

\begin{proof}
Obviously the two vectors and the elements of $\Pic(S)$ are orthogonal to $\sigma_B$ and thus belong to $\Pic(S,B)$. Let $L$ be the sublattice of $\Pic(S,B)$ generated by those elements. If $c$ is the discriminant of $\Pic(S)$, an easy calculation shows that $d^2c$ is the discriminant of $L$ (see \cite[Sect.\ 1]{Ni} for the definition of the discriminant of a lattice). On the other hand, the discriminants of $\Pic(S,B)$ and $T(S,B)$ coincides up to sign (for this use \cite[Prop.\ 1.6.1]{Ni}) and $T(S,B)$ can be realized as a (non-primitive) sublattice of $T(S)$ of index $d$ (see, for example, \cite{HS1}). Therefore $\Pic(S,B)$ and $L$ have the same discriminant and so they coincide.
\end{proof}

\smallskip

Moving to the geometric situation we are interested in, assume now that $Y$ is a cubic fourfold satisfying $(\ast)$. We have observed in Section \ref{subsec:semiorthogonal} that there exists an Azumaya algebra $\ka_0$ on $S$, the K3 surface which is the double cover of $\PP^2$ ramified along $C$, such that $\Db(S,\ka_0)\cong\Db(\PP^2,\kb_0)$ and $f_*\ka_0=\kb_0$. Such an Azumaya algebra corresponds to the choice of the element $\beta$ in the $2$-torsion part of the Brauer group $\Br(S)$ of $S$ which, in turn, corresponds to the $\PP^1$-bundle $F\to S$ parametrizing the lines contained in the fibres of $\pi:\widetilde{Y}\to\PP^2$. Therefore $\Db(S,\beta)\cong\Db(\PP^2,\kb_0)$ and this equivalence is realized by the following composition of equivalences
\begin{equation}\label{eqn:equiv}
\xymatrix{
\Xi:\coh(S,\beta)\ar[rr]^{(-)\otimes_{\ko_S}\e\dual}&&\coh(S,\ka_0)\ar[rr]^{f_*}&&\coh(\PP^2,\kb_0),
}
\end{equation}
where $\e$ is a rank-$2$ locally free $\beta$-twisted sheaf such that $\ka_0\cong\ke nd(\e)$. Obviously, $\Xi(\e)=\kb_0$. For sake of simplicity we denote by $\Xi$ the induced functor on the level of derived categories. For $l\subseteq Y$ a line, set $\ke_1\in\coh(S,\beta)$ and $\kj_l\in\Db(S,\beta)$ to be such that
\[
\kb_1=\Xi(\ke_1)\qquad\mathrm{and}\qquad\ki_l=\Xi(\kj_l).
\]
For a line $l\subseteq Y$ not contained in the plane $P$, the sheaf $\kj_l$ is pure of dimension $1$ because, by Proposition \ref{prop:ideallines}, $\ki_l$ is a vector bundle supported on a line in $\PP^2$. (See \eqref{eq:Dan} for the definition of $\ki_l$.)

\begin{lem}\label{lem:calc}
	Let $Y$ be a cubic fourfold satisfying $(\ast)$. If $l$ is a line in $Y$, then $v^B(\kj_l)=(0,f^*h,s)$, for some integer $s$. Moreover, $v^B(\ke_j)=(2,K+jf^*h+2B,s_j)$, for some $K\in\Pic(S)$, $s_j\in\ZZ$ and $j=0,1$.
\end{lem}

\begin{proof}
Take a line $l\subseteq Y$ such that $l\cap P=\emptyset$. Under this assumption, by Proposition \ref{prop:ideallines}, $\kj_l$ is a vector bundle supported on a curve in the linear system of $f^*h$. Hence, assuming $l$ generic with the above property, we necessarily have that $v^B(\kj_l)$ is as required. Using this and the fact that, by Proposition \ref{prop:ideallines}, for $l\subseteq P\subseteq Y$, the complex $\kj_l$ is extension of $\e[1]$ and $\f$, it is enough to prove that $v^B(\e)=(2,K+2B,s_0)$, for $K\in\Pic(S)$. But this follows easily from the explicit description of $\Xi$ in \eqref{eqn:equiv}, Lemma \ref{lem:NS} and the fact that, as $\kb_i$ is a locally free $\ko_{\PP^2}$-module of rank $8$, the twisted sheaf $\ke_i$ is locally free of rank $2$.
\end{proof}

\subsection{Birationality I}\label{subsec:bir1}

Following for example \cite{Y}, a notion of stability for twisted sheaves can be introduced and the moduli spaces of twisted stable sheaves can be constructed. For a K3 surface $S$, given a primitive $v\in\Pic(S,B)$, we denote by
\[
M(S,v,\beta)
\]
the moduli space of $\beta$-twisted stable (with respect to a generic polarization) sheaves on $S$ with twisted Mukai vector $v$. By \cite{Y}, $M(S,v,\beta)$ is a holomorphic symplectic manifold and, as lattices,
\[
H^2(M(S,v,\beta),\ZZ)\cong v^\perp.
\]
The weight-$2$ Hodge structure on $v^\perp$, which is compatible with this isometry, is the one induced by the twisted period $\sigma_S+\sigma_S\wedge B$, for $\CC\cdot\sigma_S=H^{2,0}(S)$.

The following proposition proves the second statement in Theorem \ref{thm:main2}. We keep the same notation as before.

\begin{prop}\label{prop:bir1}
	If $Y$ is such that the plane sextic $C$ is smooth, then there exists a birational isomorphism $F(Y)\dasharrow M(S,v,\beta)$, where $v=(0,f^*h,s)\in\Pic(S,B)$ for some $s\in\ZZ$, which is either an isomorphism or a Mukai flop in a plane.
\end{prop}

\begin{proof}
By Remark \ref{rmk:modsp}, $F(Y)$ is identified to a connected component of the moduli space containing the stable objects $\kf_l$, for $l\subseteq Y$ a line. Take the composition of functors
\[
\Theta:=\Xi_L\circ(\sigma_*\circ\Phi)^{-1}\circ R_{\ko_{Y}(-H)}\circ((-)\otimes\ko_Y(-H)),
\]
where $\Xi_L$ is the left adjoint of $\Xi$, $(\sigma_*\circ\Phi)^{-1}$ is the functor defined in Section \ref{subsec:idealshlines}, and $R_{\ko_{Y}(-H)}$ is the right mutation in $\ko_Y(-H)$ (see the definition of $P_l$ in \eqref{eqn:defPl}). The functor $\Theta:\cat{T}_Y\to\Db(S,\beta)$ is an equivalence and, by Proposition \ref{prop:ideallines}, if $l\not\subseteq P$, then $\Theta(\kf_l)=\kj_l$. The pure torsion sheaf $\kj_l\in\coh(S,\beta)$ is stable, since its Mukai vector is primitive (see Lemma \ref{lem:calc}).

Let $P\dual\subseteq F(Y)$ be the plane dual to $P\subseteq Y$ and parametrizing the lines in $P$. The above remarks show that $\Theta$ yields an isomorphism between the open subset $U=F(Y)\setminus P\dual$ and an open subset $U'\subseteq M(S,v,\beta)$, where $v=(0,f^*h,s)$. Therefore, we have a birational isomorphism $\gamma:F(Y)\dashrightarrow M(S,v,\beta)$.

Fix $h'\in\Pic(M(S,v,\beta))$ an ample polarization. Suppose that $\gamma$ does not extend to an isomorphism. Hence, by \cite[Prop.\ 2.1 and Cor.\ 2.2]{Huy1}, for all rational curves $D\subseteq P\dual$, we have $\gamma^*h'\cdot D<0$ (see also \cite[Cor.\ 2.6]{Huy1}). Consider the Mukai flop $\gamma':M\dashrightarrow F(Y)$ of $F(Y)$ in the plane $P\dual$ and let $P'\subseteq M$ be the corresponding dual plane in $M$. Let $\gamma'':M\dashrightarrow M(S,v,\beta)$ be the composition $\gamma\circ\gamma'$. Since a Mukai flop reverses the sign of the intersection with the rational curves contained in the exceptional locus, the above remarks give that $(\gamma'')^*h'\cdot D'>0$, for any rational curve $D'$ in $P'$. Again by \cite[Prop.\ 2.1 and Cor.\ 2.2]{Huy1}, this means that $\gamma''$ extends to an isomorphism yielding the desired conclusion.
\end{proof}

The fact that we have to deal with moduli spaces of twisted sheaves with (in general) non-trivial twists is quite relevant. This is made clear by Proposition \ref{prop:nospmod} at the end of the next section. Moreover, we will see in Remark \ref{rmk:MukaiFlopUtah} that, if $\beta$ is non-trivial, then the birational isomorphism in Proposition \ref{prop:bir1} is actually a Mukai flop.

The following easy corollary is a specialization of the previous result to a dense subset of the moduli space $\kc$ of cubic fourfolds. A cubic fourfold $Y$ containing a plane $P$ is \emph{very generic} if
\[
\NS_2(Y):=H^4(Y,\ZZ)\cap H^{2,2}(Y)=\langle H^2,P\rangle,
\]
where, for simplicity, we denote by $H^2$ the self-intersection of the ample line bundle $\ko_Y(H)$. It is easy to see that a very generic $Y$ satisfies $(\ast)$ (see, for example, \cite[Sect.\ 5.3]{BMMS}).

\begin{cor}\label{cor:birgen}
	If $Y$ is very generic, then there exists a birational morphism $F(Y)\dasharrow M(S,v,\beta)$, where $v=(0,f^*h,0)$, which is either an isomorphism or a Mukai flop in a plane.
\end{cor}

\begin{proof}
By Proposition \ref{prop:bir1} and by tensoring with an appropriate power of $f^*h\in\Pic(S)$, one deduces that there exists a birational morphism $F(Y)\dasharrow M(S,v,\beta)$, where $v=(0,f^*h,\epsilon)$ and $\epsilon\in\{0,1\}$. By \cite{BD}, $\Pic(F(Y))$ contains a line bundle $L$ with self-intersection $6$ with respect to the Beauville--Bogomolov quadratic form. Moreover, by \cite{Y} and the existence of the above birational isomorphism, we have a sequence of isometries of lattices
\begin{equation}\label{eqn:NS1}
\Pic(F(Y))\cong\Pic(M(S,v,\beta))\cong\Pic(S,B)\cap v^\perp.
\end{equation}

Therefore, it is enough to show that, if $\epsilon=1$, then $\Pic(M(S,v,\beta))$ does not contain an element of self-intersection $6$. For this we use the calculations in \cite{vG1}. Indeed, denote by $e_1,e_2$ the two natural generators of the first copy of the hyperbolic lattice in the K3 lattice $\Lambda:=U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}$ (see \cite{Ni} for the definition of the lattices $U$ and $E_8$). We can identify $H^2(S,\ZZ)$ to $\Lambda$ in such a way that $f^*h$ is represented by the vector $e_1+e_2$. By \cite{vG} and using this identification, the lift $B$ of $\beta$ has the form $B=\frac{e_2+\lambda}{2}$, for some $\lambda\in U^{\oplus 2}\oplus E_8(-1)^{\oplus 2}$ and $(\lambda,\lambda)=2$. If $\epsilon=1$, then, by the last isometry in \eqref{eqn:NS1}, an easy calculation (using Lemma \ref{lem:NS}) shows that $\Pic(M(S,v,\beta))\cong\langle (0,0,1),(4,2\lambda+e_1+2e_2,1)\rangle$. Hence the intersection form on $\Pic(M(S,v,\beta))$ would not represent $6$ when $\epsilon=1$.
\end{proof}

\subsection{Birationality II: end of the proof of Theorem \ref{thm:main2}}\label{subsec:bir2}

Let us pass now to the general case of $Y$ any cubic fourfold containing a plane $P$. By \cite[Prop.\ 6]{BD}, there exists a Hodge isometry
\begin{equation}\label{eqn:fano1}
	\varphi:H^4(Y,\ZZ)_\mathrm{prim}(-1)\isomor H^2(F(Y),\ZZ)_\mathrm{prim},
\end{equation}
where $(-1)$ just reverses the signature.
Here $H^4(Y,\ZZ)_\mathrm{prim}$ is the orthogonal complement (with respect to the cup-product) of the self-intersection $H^2$ of $\ko_Y(H)$. Analogously, $H^2(F(Y),\ZZ)_\mathrm{prim}$ is the orthogonal complement (with respect to the Beauville--Bogomolov form) of the natural ample polarization coming from the embedding $F(Y)\hookrightarrow\mathrm{Gr}(2,6)$ into the Grassmannian of lines in $\PP^5$.

Let $L=\langle H^2,P\rangle^\perp\hookrightarrow H^4(Y,\ZZ)_\mathrm{prim}$. Denote by $\sigma_Y$ a generator of $H^{3,1}(Y)$ and by $\sigma_{F(Y)}$ a generator of $H^{2,0}(F(Y))$. The minimal primitive sublattices $T(Y)\hookrightarrow H^4(Y,\ZZ)$ and $T(F(Y))\hookrightarrow H^2(F(Y),\ZZ)$ with the property that $\sigma_Y\in T(Y)\otimes\CC$ and $\sigma_{F(Y)}\in T(F(Y))\otimes\CC$ are such that there is a primitive embedding $$\varphi^{-1}(T(F(Y)))(-1)=T(Y)\hookrightarrow L.$$ If $\Lambda_0\hookrightarrow\Lambda=U^{\oplus 3}\oplus E_8(-1)^{\oplus 2}$ is the orthogonal complement of a vector $k$ in the K3 lattice $\Lambda$ with $(k,k)=2$, we have, for some $B\in\Lambda\otimes\QQ$, the following diagram
\begin{equation}\label{eqn:diag}
\xymatrix{
0\ar[r]&T(F(Y))\ar@{^{(}->}[d]^{\psi_1}\ar[r]^{\;\;\;\phi_1}&T(S')\ar@{^{(}->}[d]^{\psi_2}\ar[r]^{(-,B)}&\ZZ/2\ZZ\ar@{=}[d]&\\
0\ar[r]&L(-1)\ar[r]^{\;\;\;\;\,\phi_2}&\Lambda_0\ar[r]^{(-,B)}&\ZZ/2\ZZ\ar[r]&0,
}
\end{equation}
where $T(S'):=\Pic(S')^\perp\hookrightarrow H^2(S',\ZZ)$ is the transcendental lattice of a K3 surface obtained by the surjectivity of the period map and by the Hodge structure induced by $T(F(Y))$ and $\phi_2\circ\psi_1$ on $\Lambda_0$. Here $(-,B)$ is the morphism given by the cup-product with $B$. Notice that the existence of $B$ and of the bottom line in the previous diagram is ensured by \cite[Sect.\ 1]{V} and \eqref{eqn:fano1}.

Varying the Hodge structure of $Y$ on $L$ yields a $19$-dimensional family $\kd$ of cubic fourfolds containing a plane and a corresponding $19$-dimensional family $\kd'$ of Fano varieties of lines sitting inside a connected component of the moduli space $\km$ of marked hyperk\"ahler manifolds (see \cite{Huy2} for more details about $\km$).

The bottom line of the previous diagram gives then a corresponding variation of weight-$2$ Hodge structures on $\Lambda_0$ providing a $19$-dimensional family of K3 surfaces with a (pseudo-)polarization of degree $2$. The morphism $(-,B)$ restricted to the transcendental lattice of any such K3 surface $S$ gives a (possibly non-surjective) morphism $(-,B):T(S)\to\ZZ/2\ZZ$ which can be identified to a (possibly trivial) element $\beta\in\Br(S)$ (see, for example, \cite[Lemma 5.4.1]{C}). Hence the construction produces a family $\kd''$ of twisted K3 surfaces and a bijection $\Pi:\kd\to\kd''$.

The very generic member $Y$ of $\kd$ is such that $\NS_2(Y)=\langle H^2,P\rangle$ (see Section \ref{subsec:bir1}). Let $(S,\beta)$ be the twisted K3 surface image of $Y$ via $\Pi$ and let $B$ be a lift of $\beta$. By Corollary \ref{cor:birgen}, it is easy to see that there exists a birational isomorphism
\begin{equation}\label{eqn:morbir}
\xymatrix{\gamma:F(Y)\ar@{-->}[r]& M(S,v,\beta)},
\end{equation}
where $M(S,v,\beta)$ is the moduli space of $\beta$-twisted sheaves with Mukai vector $v:=(0,f^*h,0)$. Hence, there is a Hodge isometry $\gamma^*:H^2(M(S,v,\beta),\ZZ)\isomor H^2(F(Y),\ZZ)$. As the former lattice is naturally identified to $v^\perp$ and, for any twisted K3 surface $(S',\beta')$ in $\kd'$, the twisted period $\sigma_{S'}+\sigma_{S'}\wedge B'$ is orthogonal to $v$, the construction produces a $19$-dimensional family $\kd'''$ of moduli spaces of twisted sheaves with a bijection $\Pi':\kd'\to\kd'''$. More precisely $F(Y)$ is sent to $M(S',v,\beta')$, where $(S',\beta')=\Pi(Y)$, with the additional important properties that there is a Hodge isometry $H^2(F(Y),\ZZ)\cong H^2(M(S',v,\beta'),\ZZ)$ and that, for $Y$ very generic, $\Pi'(F(Y))$ is the moduli space constructed in the previous section. Notice that $\kd'''$ is all contained in one connected component of $\km$. Moreover, since generically $F(Y)$ and $\Pi'(F(Y))$ are birational and isomorphic in codimension $2$, this connected component coincides with the one containing $\kd'$ by \cite[Thm.\ 4.7]{Huy2}.

Now the proof of Theorem \ref{thm:main2} is concluded if we show that, for any $Y$ in $\kd$, $F(Y)$ is birationally isomorphic to the moduli space $M:=M(S',v,\beta')=\Pi'(F(Y))$. Suppose not. Then, as remarked in \cite{Huy2}, there exist two open subsets $U_1$ and $U_2$ in $\km$ containing respectively $F(Y)$ and $M$, and such that $U_1\cap U_2=\emptyset$.

But this is impossible since there is a dense subset $\kl\subseteq U_1$ in $\kd'$ consisting of very generic cubic fourfolds for which \eqref{eqn:morbir} holds and such that $\Pi'(\kl)\subseteq U_2$. Again by \cite[Thm.\ 4.7]{Huy2}, if $F(Y)$ is in $\kl$, then $F(Y)$ and $\Pi'(F(Y))$ cannot be separated contradicting $U_1\cap U_2=\emptyset$.

\begin{remark}\label{rmk:Hassett}
	It might be possible that the result just proved could be deduced from \cite[Thm.\ 1.0.3]{H2}. This would probably allow to avoid completely the argument in Section \ref{subsec:bir1}. Moreover, \cite{Huy2} in the last part of the previous argument could be replaced by the recent \cite{Ve}. We stick here to our elementary approach.
\end{remark}

As an application of the remarks at the beginning of this section we have the following result that is certainly well-known to the experts.

\begin{prop}\label{prop:nospmod}
	If $Y$ is a very generic cubic fourfold containing a plane $P$, then there is no smooth projective K3 surface $S'$ such that $F(Y)$ is birationally isomorphic to a fine moduli space of stable untwisted sheaves on $S'$.
\end{prop}

\begin{proof}
Assume, by contradiction, that there exists a K3 surface $S'$ and a Mukai vector $v\in\widetilde H(S',\ZZ):=\widetilde H(S',0,\ZZ)$ such that $F(Y)$ is birational to the fine moduli space $M(S,v)$ of stable untwisted sheaves with Mukai vector $v$. Then there would be a Hodge isometry $T(F(Y))\cong T(M(S',\ZZ))\cong T(S')$ (see \cite{Y}). Hence $T(F(Y))$ would admit a primitive embedding in the K3 lattice $\Lambda$.

Now, combining \eqref{eqn:fano1} and the bottom line of \eqref{eqn:diag}, we have a short exact sequence
\begin{equation}\label{eqn:fano2}
\xymatrix{
0\ar[r]&T(F(Y))\ar[r]&T(S)\ar[r]^{(-,B)}&\ZZ/2\ZZ\ar[r]&0,
}
\end{equation}
where $S$ is again the K3 surface double cover of $\PP^2$ and $B$ is the lift of the special non-trivial $\beta\in\Br(S)$ in Section \ref{subsec:geometry}. But, due to either \cite[Cor.\ 9.4 and Sect.\ 9.7]{vG} or to the appendix of \cite{Kuz3}, the kernel of the morphism $(-,B)$ in \eqref{eqn:fano2} does not admit a primitive embedding in $\Lambda$.
\end{proof}

\section{Isomorphisms and moduli spaces of complexes}\label{sec:iso}

In the course of this section, unless clearly specified, we will always assume that all cubic fourfolds $Y$ have the restrictive property
\begin{itemize}
\item[($\ast\ast$)] $Y$ satisfies $(\ast)$ and $\beta\in\Br(S)$ is non-trivial.
\end{itemize}
As observed in the appendix to \cite{Kuz3}, the very generic cubic fourfolds containing a plane satisfy the additional condition about $\beta$. Moreover, as it follows from \cite[Prop.\ 4.7]{Kuz3}, the cubic fourfolds for which $(\ast\ast)$ does not necessarily hold true are precisely those in the codimension-$1$ subvarieties of the divisor $\kc_8$ consisting of rational cubic fourfolds with a plane studied in \cite{Ha1}.

Although relative to a different setting, most of the arguments here are inspired by \cite{AB}.

\subsection{A family of stability conditions}\label{subsec:definigstab}

Given the twisted Chern character $\ch^B$, for any torsion-free $\kf\in\coh(S,\beta)$ define its \emph{slope} in the following way:
\[
\mu^B(\kf):=\frac{c_1^B(\kf)\cdot f^*h}{\rk(\kf)}.
\]
In particular a torsion-free $\kf\in\coh(S,\beta)$ is \emph{$\mu^B$-semistable} (resp.\ \emph{$\mu^B$-stable}) if for all $\kg\hookrightarrow\kf$ in $\coh(S,\beta)$, $\mu^B(\kg)\leq\mu^B(\kf)$ (resp.\ $<$). Notice that, in the definition of slope, the ample polarization has been fixed to be equal to $f^*h$.

The following result could be proved for cubic fourfolds satisfying $(\ast)$ instead of $(\ast\ast)$. Since we will not need this generality, we just consider $(\ast\ast)$ for which the proof is straightforward.

\begin{lem}\label{prop:stab}
	The sheaves $\e$ and $\f$ are $\mu^B$-stable. Moreover, any morphism $\e\to\f$ is injective.
\end{lem}

\begin{proof}
The $\mu^B$-semistability of $\ke_j$, for $j=0,1$, follows directly form the fact that $v^B(\ke_j)=(2,K+jf^*h+2B,s_j)$ (Lemma \ref{lem:calc}) and Lemma \ref{lem:NS}. Being $\ke_j$ spherical, its Mukai vector is primitive in $\Pic(S,B)$ and so $\ke_j$ is also $\mu^B$-stable.
The fact that all morphisms $\e\to\f$ are injective follows from their $\mu^B$-stability or, directly, from \cite[Lemma 2.5]{Kuz2}.
\end{proof}

\smallskip

As we observed in \cite{HMS}, the notion of twisted Chern character allows one to generalize Bridgeland's construction in \cite{Br1} to the case of twisted K3 surfaces.

We skip all the details about the definition of stability conditions which are not relevant in this paper and for which we refer to \cite{Br,HMS}. So we just recall that, by \cite[Prop.\ 5.3]{Br} giving a stability condition on a triangulated category $\cat{T}$ is equivalent to giving a bounded $t$-structure on $\cat{T}$ with heart $\cat{A}$ and a group homomorphism $Z:K(\cat{A})\to\CC$ such that $Z(\kg)\in\HH$, for all $0\neq \kg\in\cat{A}$, and with Harder--Narasimhan filtrations (see \cite[Sect.\ 5.2]{Br}). Here $\HH:=\{ z\in\CC^*:z=|z|\exp(i\pi\phi), \, 0< \phi \leq 1 \}$. More precisely, any $0\neq\kg\in\cat{A}$ has a well-defined \emph{phase} $\phi(\kg):=\mathrm{arg}(Z(\kg))\in(0,1]$.
For $\phi\in(0,1]$, an object $0\neq\kg\in\cat{A}$ is then in $\kp(\phi)$ if and only if, for all $\kg\epi\kg'\neq 0$ in $\cat{A}$, $\phi=\phi(\kg)\leq\phi(\kg')$.
A stability condition is called \emph{locally-finite} if there
exists some $\epsilon > 0$ such that, for all $\phi\in\RR$, each
(quasi-abelian) subcategory  $\mc{P} ((\phi - \epsilon , \phi +
\epsilon))$ is of finite length. In this case $\mc{P} (\phi)$ has
finite length so that every object in $\mc{P} (\phi)$ has a finite
\emph{Jordan--H\"older filtration} into stable factors of the same phase.

\smallskip

Assume from now on that $Y$ is a cubic fourfold containing a plane $P$ and satisfying $(\ast\ast)$. We want to produce a family of stability conditions on the derived category $\Db(S,\beta)$, where $(S,\beta)$ is the twisted K3 surface associated to $Y$. Observe that, by the definition of a lift of a Brauer class, we can assume further that the $B$-field lift of $\beta$ is such that $B\cdot f^*h=\frac{1}{2}$. By Lemma \ref{lem:calc}, we can write $v^B(\e)=(2,K+2B,s_0)\in\widetilde H(S,B,\ZZ)$. Define
\[
Z_{m}:\Pic(S,B)\longrightarrow\CC\qquad v\longmapsto\left\langle\exp\left(\frac{K}{2}+\left(\frac{1}{4}+im\right)f^*h+B\right),v\right\rangle,
\]
where $\langle-,-\rangle$ denotes the Mukai pairing and $m\in\RR_{>0}$.

\begin{remark}\label{rmk:realimstab}
An easy calculation using that $\e$ and $\f$ are spherical (see Remark \ref{rmk:spherical}) shows that the real and imaginary parts of $Z_m^B(v^B(\kj_l))$ satisfy
\[
\mathrm{Re}(Z_m(v^B(\kj_l)))=\mathrm{Re}(Z_m((0,f^*h,s)))=0\qquad \mathrm{Im}(Z_m(v^B(\kj_l)))=2m,
\]
for any line $l\subseteq Y$. Moreover, $\mathrm{Re}(Z_{m}(v^B(\f)))=\frac{1}{4}\left(-2+8m^2-\frac{1}{2}\right)$ and so, for $m_0:=\frac{\sqrt{5}}{4}$,
$\mathrm{Re}(Z_{m_0}(v^B(\f)))=0$.
\end{remark}

To define an abelian category which is the heart of a bounded $t$-structure on $\Db(S,\beta)$, fix the rational number
\[
\mu:=\left(B+\frac{K}{2}+\frac{1}{4}f^*h\right)\cdot f^*h.
\]
Applying the previous remark, it is not difficult to see that $\mu=s$, where $s$ is the degree-$4$ part of $v^B(\kj_l)$.

Let
$\cat{T},\cat{F}\subseteq\coh(S,\beta)$ be the following two full additive
subcategories: The non-trivial objects in $\cat{T}$ are the twisted
sheaves $\ka$ such that their torsion-free part have $\mu^B$-semistable Harder--Narasimhan factors of slope $\mu^B>\mu$. A
non-trivial twisted sheaf $\ka$ is an object in $\cat{F}$ if $\ka$ is
torsion-free and every $\mu^B$-semistable Harder--Narasimhan factor of $\ka$  has slope $\mu^B\leq\mu$. It is easy to see that $(\cat{T},\cat{F})$ is a torsion theory.

Following \cite{Br1}, we define the heart of the induced $t$-structure as the abelian category
\[
\cat{A}:=\left\{\ka\in\Db(S,\beta):\begin{array}{l}
\bullet\;\;\kh^i(\ka)=0\mbox{ for }i\not\in\{-1,0\},\\\bullet\;\;
\kh^{-1}(\ka)\in\cat{F},\\\bullet\;\;\kh^0(\ka)\in\cat{T}\end{array}\right\}.
\]

\begin{remark}\label{rmk:objstab}
Again a direct computation shows that
\[
\mu^B(\e)=\frac{(K+2B)\cdot f^*h}{2}<\mu<\frac{(K+f^*h+2B)\cdot f^*h}{2}=\mu^B(\f).
\]
Being $\e$ and $\f$ $\mu^B$-stable (Lemma \ref{prop:stab}), we have $\e[1],\f\in\cat{A}$. Moreover, due to Proposition \ref{prop:ideallines}, we also have $\kj_l\in\cat{A}$, for all lines $l\subseteq Y$.
\end{remark}

We conclude this section with the following result.

\begin{lem}\label{lem:exstab}
The pair $\sigma_m:=(Z_m,\cat{A})$ defines a locally-finite stability condition for all $m>\epsilon$, where $0\ll\epsilon<m_0$.
\end{lem}

\begin{proof}
Let $\ka\in\coh(S,\beta)$ be a torsion-free $\mu^B$-stable sheaf with $\mu^B(\ka)=\mu$ so that $$\mathrm{Im}(Z_m(v^B(\ka)))=m\left(c-r\left(\frac{K}{2}+\frac{1}{4}f^*h+B\right)\right)\cdot f^*h=0,$$ where $v^B(\ka)=(r,c,d)$. As $r>0$, by Lemma \ref{lem:NS}, we have $F:=\left(c-r\left(\frac{1}{2}K+\frac{1}{2}f^*h+B\right)\right)\in\Pic(S)$. Thus, by the Hodge Index theorem, $F\cdot F\leq 0$. Since $\ka$ is $\mu^B$-stable, we have $\chi(\ka,\ka)\leq 2$. By definition,
\[
\mathrm{Re}(Z_m(v^B(\ka)))=\frac{1}{2r}(-\chi(\ka,\ka)+2r^2m^2-F\cdot F).
\]
If $\ka$ is not spherical (i.e., $\chi(\ka,\ka)<2$), then $\mathrm{Re}(Z_m(v^B(\ka)))>0$ and so $Z_m(v^B(\ka[1]))\in\HH$. If $\ka$ is spherical, then $-2+2r^2m_0^2-F\cdot F>0$, because, by Lemma \ref{lem:NS}, $r\geq 2$. This is enough to prove that, for $m>\epsilon$ and $0\ll\epsilon<m_0$, we have $\mathrm{Re}(Z_m(v^B(\ka)))>0$.

As in \cite[Lemma 6.2]{Br1}, this suffices to show that $Z_m(\ka)\in\HH$, for any $\ka\in\cat{A}$ and $m$ as in the statement. The existence of the Harder--Narasimhan filtrations (which is what we need to conclude that the pair $(Z_m,\cat{A})$ is a locally-finite stability condition) follows now from the same argument as in \cite[Prop.\ 7.1 and Sect.\ 11]{Br1} adapted to the twisted setting according to \cite[Sect.\ 3.1]{HMS}.
\end{proof}

We leave it to the reader to verify that, omitting $(\ast\ast)$, the previous result does not hold anymore.

\subsection{Proof of Theorem \ref{thm:main1}}\label{subsec:proofthm2}

Denote by $I:=(\epsilon,\infty)$ the interval in Lemma \ref{lem:exstab}. Under the above assumption, we then have the following.

\begin{lem}\label{lem:stabB01}
	The objects $\e[1]$ and $\f$ are $\sigma_m$-stable for any $m\in I$.
\end{lem}

\begin{proof}
As the proofs for $\e[1]$ and $\f$ are the same, we just deal with $\f$ and leave the other to the reader. Suppose $\f$ is not $\sigma_m$-stable, for some $m\in I$. Then there exists a short exact sequence in $\cat{A}$
\begin{equation}\label{eqn:sta1}
0\longrightarrow\ka\longrightarrow\f\longrightarrow\kb\longrightarrow 0,
\end{equation}
where $\ka\neq 0$ destabilizes $\f$ and can be assumed to be $\sigma_m$-stable. In particular, $\ka$ is a sheaf.

An easy calculation shows that $\mathrm{Im}(Z_m(v^B(\f)))=m$. Since $\frac{1}{m}\mathrm{Im}(Z_m(v^B(-)))$ is an integral function which is additive on triangles, $\mathrm{Im}(Z_m(v^B(\ka)))$ is either $0$ or $m$. The former is impossible because $\f$ is locally free and $\ka$ would then be supported on points. If the latter holds true, then $\mathrm{Im}(Z_m(v^B(\kb)))=0$ and so, as soon as $\kb\neq 0$, \eqref{eqn:sta1} does not destabilizes $\f$.
\end{proof}

For $l$ a line in $Y$, consider the following sheaves in $\coh(S,\beta)$
\[
\kk_l:=\left\{\begin{array}{ll}
\kj_l&\mbox{if } l\not\subseteq P\\
\mathrm{Coker}(s_l:\e\to\f) &\mbox{otherwise,}
\end{array}\right.
\]
where $s_l$ is the morphism corresponding (non canonically) via Serre duality to the morphism $\ke_1\to\ke_0[2]$ associated to the line $l\subseteq P$ according to Proposition \ref{prop:ideallines}. Notice that, by Lemma \ref{prop:stab}, all morphisms $\e\to\f$ are injective.

\begin{lem}\label{lem:st0}
For any line $l\subseteq Y$, the sheaf $\kk_l$ is pure of dimension $1$.
\end{lem}

\begin{proof}
If $l\not\subseteq P$, this was already observed in Section \ref{subsec:twistedK3}. Indeed, by Proposition \ref{prop:ideallines}, $\kk_l$ is actually a vector bundle supported on a curve. If $l\subseteq P$, then $\kk_l$ has a locally free resolution of length $1$ and so it cannot have torsion supported on points.
\end{proof}

The following is the first step in the proof of Theorem \ref{thm:main1}.

\begin{lem}\label{lem:st1}
{\rm (i)} If $m>m_0=\frac{\sqrt{5}}{4}$, then $\kk_l$ is $\sigma_m$-stable, for any line $l\subseteq Y$.

{\rm (ii)} Assume $m=m_0$. If $l\not\subseteq P$ is a line in $Y$, then $\kk_l$ is $\sigma_m$-stable. If $l\subseteq P$, then $\kk_l$ is $\sigma_{m_0}$-semistable and
\begin{equation}\label{eqn:sta4}
\f\longrightarrow\kk_l\longrightarrow\e[1]
\end{equation}
is the Jordan--H\"older filtration of $\kk_l$.
\end{lem}

\begin{proof}
Suppose that there is a line $l\subseteq Y$ and $m\geq m_0$ such that $\kk_l$ is not $\sigma_m$-stable. This means that there is a destabilizing short exact sequence in $\cat{A}$
\begin{equation}\label{eqn:sta2}
0\longrightarrow\ka\longrightarrow\kk_l\longrightarrow\kb\longrightarrow 0,
\end{equation}
where $\ka\neq 0$ is $\sigma_m$-stable and so $\chi(\ka,\ka)\leq 2$. Obviously, $\ka\in\coh(S,\beta)$. Set $v^B(\ka)=(r,c,d)$.

By Remark \ref{rmk:realimstab} we have $\mathrm{Im}(Z_m(v^B(\kj_l)))=2m$ and so $J:=\mathrm{Im}(Z_m(v^B(\ka)))\in\{0,m,2m\}$. If $J=0$, then $\ka$ is a torsion sheaf supported on points. But this contradicts Lemma \ref{lem:st0}. If $J=2m$, then $\kb$ has phase $1$ and \eqref{eqn:sta2} would not destabilize $\kk_l$ if $\kb\neq 0$. Hence $J=m$ and, due to $(\ast\ast)$ and Lemma \ref{lem:NS}, $r\geq 2$.

Reasoning as in the proof of Lemma \ref{lem:exstab}, this also implies that, as $\ka$ destabilizes $\kk_l$,
\begin{equation}\label{eqn:re}
\mathrm{Re}(Z_m(v^B(\ka)))=\frac{1}{2r}(-\chi(\ka,\ka)+2r^2m^2-F\cdot F)\leq 0,
\end{equation}
where $F:=\left(c-r\left(\frac{1}{2}K+\frac{1}{2}f^*h+B\right)\right)\in\Pic(S)$.
Since $K=m$, the Hodge Index theorem yields $F\cdot F\leq\frac{1}{2}$. In particular, by Remark \ref{rmk:realimstab},
\begin{equation}\label{eqn:re1}
\mathrm{Re}(Z_m(v^B(\ka)))\geq\frac{1}{4}\left(-2+8m^2-\frac{1}{2}\right)=\mathrm{Re}(Z_m(v^B(\f)))\geq 0,
\end{equation}
for $m\geq m_0$. Therefore, \eqref{eqn:re} is not verified unless $m=m_0$. This proves (i).

If $m=m_0$, the only possibility for \eqref{eqn:sta2} to destabilize $\kk_l$ is that $r=2$, $\chi(\ka,\ka)=2$ and $\mathrm{Re}(Z_{m_0}(v^B(\ka)))=\mathrm{Re}(Z_{m_0}(v^B(\f)))=0$. Under these assumptions, $E:=c_1^B(\ka)-c_1^B(\f)=c-K-f^*h-2B$ is in $\Pic(S)$ (use again Lemmas \ref{lem:NS} and \ref{lem:calc}) and $F=E+\frac{1}{2}f^*h$. As $\mathrm{Im}(Z_{m_0}(v^B(\ka)))=\mathrm{Im}(Z_{m_0}(v^B(\f)))=m_0$, we have $E\cdot f^*h=0$. Then either $E$ is trivial or $E\cdot E<0$. The latter cannot be true because
$\frac{1}{2}=F\cdot F=E\cdot E+\frac{1}{2}<\frac{1}{2}$.

If $E$ is trivial, then the fact that $\chi(\ka,\ka)=\chi(\f,\f)=2$ implies that $v^B(\ka)=v^B(\f)$ and $v^B(\kb)=v^B(\e[1])$. Thus, by Lemma \ref{lem:stabB01},  $\ka\cong\f$ and $\kb\cong\e[1]$. Therefore, by Serre duality and Proposition \ref{prop:ideallines}, \eqref{eqn:sta2} destabilizes $\kk_l$ if and only if $l\subseteq P$. In this situation, $\mathrm{Re}(Z_{m_0}(v^B(\e[1])))=\mathrm{Re}(Z_{m_0}(v^B(\f)))=0$ and $\mathrm{Im}(Z_{m_0}(v^B(\e[1])))=\mathrm{Im}(Z_{m_0}(v^B(\f)))=m_0$. So, due to Lemma \ref{lem:stabB01}, \eqref{eqn:sta4} is the Jordan--H\"older filtration of $\kk_l$.
\end{proof}

By the discussion in \cite[Sect.\ 9]{Br1} and by Lemma \ref{lem:st1} (ii), for $\epsilon$ sufficiently close to $m_0$ and for a line $l\not\subseteq P$, we have that $\kj_l(=\kk_l)$ is $\sigma_m$-stable, for all $m\in I=(\epsilon,\infty)$ (use that $v^B(\kj_l)$ is primitive).

Let now $l\subseteq Y$ be a line contained in $P$. By Lemma \ref{lem:stabB01}, the triangle
\[
\e[1]\longrightarrow\kj_l\longrightarrow\f
\]
is the Harder--Narasimhan filtration of $\kj_l$ for $m>m_0$. This means that $\kj_l$ is $\sigma_{m_0}$-semistable with Jordan--H\"older factors $\e[1]$ and $\f$. As a consequence, there exists $\epsilon< m_0$ but sufficiently close to $m_0$ such that $\kj_l$ is $\sigma_m$-stable, for all  $m\in(\epsilon,m_0)$. Indeed, if not, by \cite[Prop.\ 9.3]{Br1}, the Harder--Narasimhan factors of $\kj_l$ in the stability condition $\sigma_m$, for $m\in(\epsilon,m_0)$, would survive in the stability condition $\sigma_{m_0}$. This would contradict the $\sigma_{m_0}$-semistability of $\kj_l$.

\smallskip

To finish the proof of Theorem \ref{thm:main1}, first of all we observe that all the arguments in \cite{Toda} generalize to the twisted setting.
In particular, for all $m>\epsilon$, it makes sense to speak about the moduli space $M^v(S,\sigma_m)$ of $\sigma_m$-stable objects in $\cat{A}$ with Mukai vector $v:=v^B(\kj_l)$ as algebraic space over $\CC$. Moreover, by the results in \cite[Sect.\ 7.2]{I}, $M^v(S,\sigma_m)$ is a smooth hyperk\"ahler manifold of dimension $4$. Hence, the only thing we need to prove is that, for $m\in(\epsilon,m_0)$, the objects $\kj_l$ are the only $\sigma_m$-semistable objects in $\cat{A}$ with Mukai vector $v$.

Let $\kg\in\cat{A}$ be a $\sigma_m$-semistable object, for some $m\in(\epsilon,m_0)$, with $v(\kg)=v$.
By \cite[Prop.\ 9.3]{Br1}, up to replacing $\epsilon$ uniformly, we can assume that $\kg$ is $\sigma_{m_0}$-semistable.
If $\kg$ is $\sigma_{m_0}$-stable, then, by \cite[Prop.\ 9.4]{Br1}, $\kg$ is $\sigma_m$-stable, for some $m>m_0$.
Hence $\kg\in M(S,v,B)$ which is irreducible. As remarked in Section \ref{sec:stab}, $\kg$ is then isomorphic to $\kk_l$, for some $l\subseteq Y$, and so, as observed above, to $\kj_l$.

Assume then that $\kg$ is $\sigma_{m_0}$-semistable (and not stable).
Since $\Im(Z_{m_0}(\kg))=2m_0$, $\kg$ must have two $\sigma_{m_0}$-stable factors $\ka_0$ and $\ka_1$ with
\[
0\longrightarrow\ka_0\longrightarrow\kg\longrightarrow\ka_1\longrightarrow 0
\]
an exact sequence in $\cat{A}$ and $\Im(Z_{m_0}(\ka_0))=\Im(Z_{m_0}(\ka_1))=m_0$.
By the same argument as above, we have an extension
\[
0\longrightarrow\ka_1\longrightarrow\widetilde{\kg}\longrightarrow\ka_0\longrightarrow 0
\]
which is $\sigma_m$-stable, for some $m>m_0$.
Hence, $\widetilde{\kg}$ is isomorphic to $\kk_l$, for some $l\subseteq Y$ and is strictly $\sigma_{m_0}$-semistable.
By Lemma \ref{lem:st1}, $\widetilde{\kg}$ is then isomorphic to the cokernel of a map $\e\to\f$.
In particular, $\ka_0\cong\e[1]$, $\ka_1\cong\f$ and so $\kg$ is isomorphic to $\kj_l$, for some $l\subseteq Y$, as wanted.
This completes the proof of Theorem \ref{thm:main1}.

\begin{remark}\label{rmk:MukaiFlopUtah}
As in \cite{AB}, we can be more precise about the relation between $M^v(S,\sigma_m)\cong F(Y)$, for $m\in(\epsilon,m_0)$, and $M(S,v,B)$.
Indeed, we can show that $F(Y)$ \emph{is} the Mukai flop of $M(S,v,B)$ in the plane $\PP(\Hom(\e,\f))$.
More precisely, if $\widetilde{M}$ denotes the Mukai flop of $M(S,v,B)$ in the plane $\PP(\Hom(\e,\f))$, then, by mimicking \cite[Sect.\ 5]{AB}, we can construct a universal family $\widetilde{U}$ on $\widetilde{M}$ so that the pair $(\widetilde{M},\widetilde{U})$ represents the functor
parametrizing $\sigma_m$-stable objects in $\cat{A}$ with Mukai vector $v$.
Hence, by Theorem \ref{thm:main1}, $\widetilde{M}$ is isomorphic to $M^v(S,\sigma_m)$ and so to $F(Y)$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

{\small\noindent {\bf Acknowledgements.} It is a great pleasure to thank Aaron Bertram, Bert van Geemen, and Daniel Huybrechts for very interesting and useful discussions and suggestions. This paper was concluded during the authors' stay at the Mathematische Institut of the University of Bonn whose warm hospitality is gratefully acknowledged. During their visit to Bonn they were supported by the grant SFB/TR 45.}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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