\section{Introduction}

Let $\X$ be a smooth complex hypersurface of degree $3$ in projective $(d+1)$-space, $d\geqslant 3$. The geometry of $\X$ is largely determined by the Hilbert scheme $\F(\X)$ of lines on $\X$, which is is traditionally called the Fano scheme of $\X$. The scheme $\F(\X)$ is smooth of dimension $2d-4$, and its canonical sheaf is ample for $d=3$, trivial for $d=4$, and anti-ample for $d\geqslant 5$. Much is known about $\X$ and $\F(\X)$ when $d=3$ or $d=4$ (cf. \cite{Huy}), but the case $d\geqslant 5$ is largely unexplored. It is well-known \cite{KS} that every infinitesimal deformation of $\X$ is given by a family of cubic hypersurfaces, and the relative Hilbert scheme therefore defines a morphism
\begin{equation*}
\Def_{\X}\rightarrow\Def_{\F(\X)}
\end{equation*}
of the corresponding deformation functors. A remarkable result of Beauville and Donagi \cite{BD} asserts that for $d=4$ the scheme $\F(\X)$ is deformation equivalent to the Hilbert scheme of two points of a K3 surface; in particular, there are deformations of $\F(\X)$ which are not induced by $\X$, i.e. the above morphism is not surjective in this case. In this paper we address the case $d\geqslant 5$:
\begin{theorem}\label{main}
For $d\geqslant 5$ the morphism $\Def_{\X}\rightarrow\Def_{\F(\X)}$ is an isomorphism. 
\end{theorem}
One can readily reduce \Cref{main} to showing that the induced map on tangent spaces
\begin{equation*}
\H^{1}(\Theta_{\X})\rightarrow\H^{1}(\Theta_{\F(\X)})
\end{equation*}
is an isomorphism. Using Hodge theory, Huybrechts (\cite{Huy}, Corollary 2.5.10) recently proved that this map is injective for any $d\geqslant 3$.
The standard approach to computing the cohomology of sheaves on $\F(\X)$ is to use the embedding of $\F(\X)$ in the ambient Grassmannian $\G$ and the Borel-Bott-Weil theorem \cite{AK}.
Together with the explicit description of $\Theta_{\F(\X)}$ due to Clemens and Griffiths \cite{CG} and Tyurin \cite{Tyu}, Kuznetsov (\cite{Huy}, Proposition 5.2.13) used this approach
to show $h^{1}(\Theta_{\X})=h^{1}(\Theta_{\F(\X)})$ in the $d=3$ case. For $d\geqslant 5$ we have no such description of $\Theta_{\F(\X)}$, and we instead consider the exact sequence sequence in cohomology associated to
\begin{equation*}
0\rightarrow\Theta_{\F(\X)}\rightarrow\Theta_{\G}\vert_{\F(\X)}\rightarrow\cn_{\F(\X)/\G}\rightarrow 0.
\end{equation*}
It turns out that the hypercohomology spectral sequences given by the Koszul resolutions of the sheaves $\Theta_{\G}\vert_{\F(\X)}$ and $\cn_{\F(\X)/\G}$ are particularly simple to understand for $d\geqslant 5$,
which allows us to prove that the map on tangent spaces is surjective, and to show the equality  $h^{1}(\Theta_{\X})=h^{1}(\Theta_{\F(\X)})$ of the number of moduli.

\section{Auxiliary results}

Let $d\geqslant 3$ be a positive integer, and  $\V$ be a complex vector space of dimension $d+2$. We write $\P=\P(\V)$ for the projective space of lines in $\V$, and
$\G=\mathrm{Gr}(2,\V)$ for the Grassmannian of $2$-planes in $\V$. We denote by
\begin{equation}\label{eq:universal}
0\rightarrow\cs\rightarrow\co\otimes\V\rightarrow\cq\rightarrow 0
\end{equation}
the universal exact sequence on $\G$. The primary tool for computing the cohomology of universal sheaves on the Grassmannian is the Borel-Bott-Weil theorem. Consider the lattice
$\bz^{d+2}$, with its natural action by the symmetric group $\mathfrak{S}_{d+2}$. For any $\nu\in\bz^{d+2}$ there exists a permutation $\sigma\in\mathfrak{S}_{d+2}$ such that $\sigma(\nu)$
is non-increasing, and if the entries of $\nu$ are pairwise distinct, then $\sigma$ is uniquely determined. We denote by $\Sigma^{\nu}$ the Schur functor corresponding to a non-increasing sequence $\nu$, and we write $\rho$ for the distinguished sequence $(d+2, d+1, \ldots, 2, 1)\in\bz^{d+2}$. 
\begin{theorem}[Borel-Bott-Weil]
Let $\nu=(\lambda,\mu)\in\bz^{d+2}$, where $\lambda\in\bz^{d}$ and $\mu\in\bz^{2}$ are non-increasing. If the entries of $\nu+\rho$ are pairwise distinct, then the only
nonvanishing cohomology group of the sheaf $\Sigma^{\lambda}\cq\otimes\Sigma^{\mu}\cs$ is
\begin{equation*}
\H^{l(\sigma)}(\Sigma^{\lambda}\cq\otimes\Sigma^{\mu}\cs)=\Sigma^{\sigma(\nu+\rho)-\rho}\V.
\end{equation*}
(Here $\sigma\in\mathfrak{S}_{d+2}$ is the unique permutation such that $\sigma(\nu+\rho)$ is non-increasing, and $l(\sigma)$ is its length.) Otherwise $\H^{\ast}(\Sigma^{\lambda}\cq\otimes\Sigma^{\mu}\cs)=0$.
\end{theorem}

\begin{example}\label{example}
The only nonvanishing cohomology groups of the sheaves $\S^{3}\cs^{\vee}$ and $\Theta_{\G}\simeq\cq\otimes\cs^{\vee}$ are $\H^{0}(\S^{3}\cs^{\vee})=\S^{3}\V^{\vee}$ and $\H^{0}(\Theta_{\G})=\End(\V)/\left\langle \mathrm{id}\right\rangle$, respectively.
\end{example}
The universal sequence (\ref{eq:universal}) induces a morphism of sheaves $\co\otimes\S^{3}\V^{\vee}\rightarrow\S^{3}\cs^{\vee}$ on $\G$, and we write
\begin{equation}\label{eq:s}
s:\S^{3}\V^{\vee}\rightarrow\H^{0}(\S^{3}\cs^{\vee})
\end{equation}
for the associated map of global sections.
\begin{lemma}\label{slemma}
The map $s$ is an isomorphism.
\end{lemma}
\begin{proof}
As $\H^{0}(\S^{3}\cs^{\vee})=\S^{3}\V^{\vee}$ by \Cref{example}, it suffices to observe that $s$ is injective.
Indeed, viewing $f\in\S^{3}\V^{\vee}$ as an element of $\H^{0}(\co_{\P}(3))$, the vanishing of $s_{f}\in\H^{0}(\S^{3}\cs^{\vee})$ means in particular that $f\vert_{\L}=0$ for every line $\L\subset\P$, and hence $f=0$.
\end{proof}
The deformation theory of a cubic hypersurface $\X\subset\P$ defined by $f\in\H^{0}(\co_{\P}(3))$ can be summarised as follows.
\begin{proposition}\label{cubic}
  (i) The restriction map
  \begin{equation*}
   \H^{0}(\co_{\P}(3))\rightarrow\H^{0}(\co_{\X}(3))
  \end{equation*}
is surjective with kernel $\left\langle f\right\rangle $; \\
(ii) the restriction map
\begin{equation*}
\H^{0}(\Theta_{\P})\rightarrow\H^{0}(\Theta_{\P}\vert_{\X})
\end{equation*}
is an isomorphism, and $\H^{1}(\Theta_{\P}\vert_{\X})=0$; \\
(iii) $\H^{q}(\Theta_{\X})=0$ for $q\neq 1$ and $h^{1}(\Theta_{\X})=\binom{d+2}{3}$.
\end{proposition}
Parts (i) and (ii) are immediate consequences of standard results about the cohomology of projective spaces, and part (iii) is a classical result of Kodaira and Spencer \cite{KS}.
We denote by $\Def_{\X}$ (resp. $\Def_{\X/\P}$) the functor of Artin rings given by the infinitesimal deformations of $\X$ (resp. of $\X$ in $\P$).
It follows from part (ii) that the forgetful morphism of deformation functors
\begin{equation*}
\Def_{\X/\P}\rightarrow\Def_{\X}
\end{equation*}
is smooth. Consider now the Hilbert scheme of lines $\F(\X)$ on $\X$, which is smooth of dimension $2d-4$.
As $\Def_{\X/\P}\rightarrow\Def_{\X}$ is smooth, every infinitesimal deformation of $\X$ is induced by an infinitesimal deformation of $\X$ in $\P$, and we can thus define a morphism of deformation functors
\begin{equation}
\Def_{\X}\rightarrow\Def_{\F(\X)}
\end{equation}
by using the relative Hilbert scheme. Viewing the Grassmannian $\G$ as the Hilbert scheme of lines $\F(\P)$, the embedding $\X\subset\P$ induces an embedding $\F(\X)\subset\G$,
and the section $f\in\H^{0}(\co_{\P}(3))$ defining $\X\subset\P$ induces by (\ref{eq:s}) a section $s_{f}\in\H^{0}(\S^{3}\cs^{\vee})$ whose zero scheme is precisely $\F(\X)\subset\G$ \cite{AK}.
It follows that the section $s_{f}$ is regular; in particular, we have a canonical isomorphism
\begin{equation*}
 \cn_{\F(\X)/\G}\xrightarrow{\sim}\S^{3}\cs^{\vee}_{\F(\X)},
\end{equation*}
where $\cs_{\F(\X)}$ denotes the restriction of $\cs$ to $\F(\X)$, and a Koszul resolution
\begin{equation*}
0\rightarrow\wedge^{4}\S^{3}\cs\rightarrow\wedge^{3}\S^{3}\cs\rightarrow\wedge^{2}\S^{3}\cs\rightarrow\S^{3}\cs\rightarrow\ci\rightarrow 0
\end{equation*}
of the ideal sheaf $\ci$ of $\F(\X)\subset\G$. For any locally free sheaf $\cf$ on $\G$, the Koszul resolution induces a hypercohomology spectral sequence
\begin{equation*}
\E^{pq}_{1}=\H^{q}(\wedge^{-p+1}\S^{3}\cs\otimes\cf)\Rightarrow\H^{p+q}(\ci\otimes\cf).
\end{equation*}
We will consider this spectral sequence when $\cf$ is $\Theta_{\G}$ or $\S^{3}\cs^{\vee}$. To compute the $\E_{1}$ terms, we need to express the sheaves $\wedge^{i}\S^{3}\cs$ in the form required by the Borel-Bott-Weil theorem. For general $n$ and $m$ the decomposition of the pletyhsm $\wedge^n \S^m$ into Schur functors is a difficult problem, but in our case $n$ and $m$ are small, and since $\cs$ has rank $2$ we get a fairly simple expression:
\begin{equation}\label{eq:plethysm}
\wedge^{2}\S^{3}\cs = \Sigma^{(5,1)}\cs\oplus\Sigma^{(3,3)}\cs, \quad \wedge^{3}\S^{3}\cs = \Sigma^{(6,3)}\cs, \quad \wedge^{4}\S^{3}\cs = \Sigma^{(6,6)}\cs.
\end{equation} 

Our key technical result is the following application of the Borel-Bott-Weil theorem.
\begin{lemma}\label{cohomology}
For $d\geqslant 6$ the sheaves $\wedge^{n}\S^{3}\cs\otimes\Theta_{\G}$ ($1\leqslant n\leqslant 4$) and $\wedge^{m}\S^{3}\cs\otimes\S^{3}\cs^{\vee}$ ($2\leqslant m\leqslant 4$) have no cohomology. For $d=5$ the only nonvanshing cohomology groups of these sheaves are  $\H^{4}(\wedge^{2}\S^{3}\cs\otimes\Theta_{\G})=\V^{\ast}$ and $\H^{5}(\wedge^{2}\S^{3}\cs\otimes\S^{3}\cs^{\vee})=\V^{\ast}$, and for any $d\geqslant 5$ the only nonvanshing cohomology group of the sheaf $\S^{3}\cs\otimes\S^{3}\cs^{\vee}$ is $\H^{0}(\S^{3}\cs\otimes\S^{3}\cs^{\vee})=\det(\V)^{\otimes 3}$.
\end{lemma}
\begin{proof}
By applying (\ref{eq:plethysm}), $\cs^{\vee}\simeq\cs\otimes\det(\cq)$, and the Pieri rule, we obtain for all $d\geqslant 3$
\begin{align*}
\S^{3}\cs\otimes\Theta_{G} &= \Sigma^{(2,1^{d-1})}\cq\otimes(\S^{4}\cs\oplus\Sigma^{(3,1)}\cs)\\
\wedge^{2}\S^{3}\cs\otimes\Theta_{G} &= \cq\otimes(\S^{5}\cs\oplus\Sigma^{(4,1)}\cs\oplus\Sigma^{(3,2)}\cs), \\
\wedge^{3}\S^{3}\cs\otimes\Theta_{G} &= \cq\otimes(\Sigma^{(6,2)}\cs\oplus\Sigma^{(5,3)}\cs), \\
\wedge^{4}\S^{3}\cs\otimes\Theta_{G} &= \cq\otimes\Sigma^{(6,5)}\cs, \\
S^{3}\cs\otimes\S^{3}\cs^{\vee} &= \Sigma^{(3^d)}\cq\otimes(\S^{6}\cs\oplus\Sigma^{(5,1)}\cs\oplus\Sigma^{(4,2)}\cs\oplus\Sigma^{(3,3)}\cs),\\
\wedge^{2}\S^{3}\cs\otimes\S^{3}\cs^{\vee} &= \Sigma^{(3^d)}\cq\otimes (\Sigma^{(8,1)}\cs\oplus\Sigma^{(7,2)}\cs\oplus\Sigma^{(6,3)}\cs^{\oplus 2}\oplus\Sigma^{(5,4)}\cs), \\
\wedge^{3}\S^{3}\cs\otimes\S^{3}\cs^{\vee} &= \S^{6}\cs\oplus\Sigma^{(5,1)}\cs\oplus\Sigma^{(4,2)}\cs\oplus\Sigma^{(3,3)}\cs, \\
\wedge^{4}\S^{3}\cs\otimes\S^{3}\cs^{\vee} &= \Sigma^{(6,3)}\cs.
\end{align*}
The conclusion follows by applying the Borel-Bott-Weil theorem.
\end{proof}

We now establish an exact analogue of \Cref{cubic} for $\F(\X)\subset\G$.
\begin{proposition}\label{fano}
Let $d\geqslant 5$.
(i) The restriction map
\begin{equation*}
\H^{0}(\S^{3}\cs^{\vee})\rightarrow\H^{0}(\S^{3}\cs^{\vee}_{\F(\X)})
\end{equation*}
is surjective with kernel $\left\langle s_{f}\right\rangle $; \\
(ii) the restriction map
\begin{equation*}
  \H^{0}(\Theta_{\G})\rightarrow\H^{0}(\Theta_{\G}\vert_{\F(\X)})
\end{equation*}
is an isomorphism, and $\H^{1}(\Theta_{\G}\vert_{\F(\X)})=0$; \\
(iii) $\H^{q}(\Theta_{\F(\X)})=0$ for $q\neq 1$ and $h^{1}(\Theta_{\F(\X)})=\binom{d+2}{3}$.
\end{proposition}
\begin{proof}
(i) Let $\ci$ be the ideal sheaf of $\F(\X)$ in $\G$, and consider the spectral sequence
\begin{equation*}
\E^{pq}_{1}=\H^{q}(\wedge^{-p+1}\S^{3}\cs\otimes\S^{3}\cs^{\vee})\Rightarrow\H^{p+q}(\ci\otimes\S^{3}\cs^{\vee}).
\end{equation*}
By \Cref{cohomology} this spectral sequence degenerates at the $\E_{1}$-page, giving
\begin{equation*}
\H^{0}(\ci\otimes\S^{3}\cs^{\vee})\simeq\H^{0}(\S^{3}\cs\otimes\S^{3}\cs^{\vee}) \quad \mathrm{and} \quad \H^{1}(\ci\otimes\S^{3}\cs^{\vee})=0.
\end{equation*}
It remains to combine this with the exact sequence in cohomology associated to
\begin{equation*}
0\rightarrow\ci\otimes\S^{3}\cs^{\vee}\rightarrow\S^{3}\cs^{\vee}\rightarrow\S^{3}\cs^{\vee}_{\F(\X)}\rightarrow 0.
\end{equation*}

(ii) Consider the spectral sequence 
\begin{equation*}
\E^{pq}_{1}=\H^{q}(\wedge^{-p+1}\S^{3}\cs\otimes\Theta_{\G})\Rightarrow\H^{p+q}(\ci\otimes\Theta_{\G}).
\end{equation*}
By \Cref{cohomology} this spectral sequence degenerates at the $\E_{1}$-page, and we obtain
\begin{equation*}
\H^{0}(\ci\otimes\Theta_{\G}) = \H^{1}(\ci\otimes\Theta_{\G}) = \H^{2}(\ci\otimes\Theta_{\G}) = 0.
\end{equation*}
The result follows from this vanishing, \Cref{example}, and the exact sequence
\begin{equation*}
0\rightarrow\ci\otimes\Theta_{\G}\rightarrow\Theta_{\G}\rightarrow\Theta_{\G}\vert_{\F(\X)}\rightarrow 0.
\end{equation*}

(iii) We refer to Corollary 3.3.13 of \cite{Huy} for a proof of $\H^{0}(\Theta_{\F(\X)})=0$. Since the canonical sheaf $\omega_{\F(\X)}$ is anti-ample for $d\geqslant 5$, the vanishing
\begin{equation*}
 \H^{q}(\Theta_{\F(\X)})=0 \quad (q\geqslant 2)
\end{equation*}
follows from the wedge product isomorphism $\Theta_{\F(\X)}\simeq\Omega^{2d-5}_{\F(\X)}\otimes\omega_{\F(\X)}^{\vee}$ and the Kodaira–Akizuki–Nakano vanishing theorem. Using (i) and (ii), the exact sequence in cohomology of
\begin{equation*}
0\rightarrow\Theta_{\F(\X)}\rightarrow\Theta_{\G}\vert_{\F(\X)}\rightarrow\cn_{\F(\X)/\G}\rightarrow 0
\end{equation*}
can be written as
\begin{equation*}
0\rightarrow\H^{0}(\Theta_{\G}\vert_{\F(\X)})\rightarrow\H^{0}(\cn_{\F(\X)/\G})\rightarrow\H^{1}(\Theta_{\F(\X)})\rightarrow 0,
\end{equation*}
and the number of moduli of $\F(\X)$ is given by
\begin{equation*}
h^{1}(\Theta_{\F(\X)})=h^{0}(\S^{3}\cs^{\vee})-1-h^{0}(\Theta_{\G})=\binom{d+2}{3}.
\end{equation*}
\end{proof}

\begin{corollary}
The forgetful morphism $\Def_{\F(\X)/\G}\rightarrow\Def_{\F(\X)}$ is smooth.
\end{corollary}

\begin{remark}
Parts of \Cref{fano} are false for $d=3$ or $d=4$. For example
\begin{equation*}
\H^{1}(\Theta_{\G}\vert_{\F(\X)})\neq 0 \quad \mathrm{and} \quad \H^{2}(\Theta_{\F(\X)})\neq 0
\end{equation*}
(resp. $\H^{3}(\Theta_{\F(\X)})\neq 0$) for $d=3$ (resp. $d=4$).
\end{remark}
 
\section{Proof of the Theorem}

\begin{proof}
Consider the commutative diagram of functorial morphisms
\begin{equation}\label{eq:compatibility}
\begin{tikzcd}
\Def_{\X/\P}  \arrow[r] \arrow[d] & \Def_{\X}  \arrow[d]  \\
\Def_{\F(\X)/\G} \arrow[r] & \Def_{\F(\X)},
\end{tikzcd}
\end{equation}
where the horizontal morphisms are the forgetful ones, and the vertical morphisms are given by taking the relative Hilbert scheme. By \Cref{cubic} (iii) and \Cref{fano} (iii), the functors $\Def_{\X}$ and $\Def_{\F(\X)}$ are prorepresentable. To show that $\Def_{\X}\rightarrow\Def_{\F(\X)}$ is an isomorphism, it thus suffices to show that the map of first-order deformations
\begin{equation}\label{eq:tangent}
\H^{1}(\Theta_{\X})\rightarrow\H^{1}(\Theta_{\F(\X)})
\end{equation}
is an isomorphism. By \Cref{cubic} (iii) and \Cref{fano} (iii) $\X$ and $\F(\X)$ have the same number of moduli, 
\begin{equation*}
h^{1}(\Theta_{\X})=h^{1}(\Theta_{\F(\X)}).
\end{equation*}
It remains to show that (\ref{eq:tangent}) is surjective. Consider the diagram
\begin{equation*}
\begin{tikzcd}
\H^{0}(\cn_{\X/\P}) \arrow[r] \arrow[d] & \H^{1}(\Theta_{\X})  \arrow[d]  \\
\H^{0}(\cn_{\F(\X)/\G}) \arrow[r] & \H^{1}(\Theta_{\F(\X)}),
\end{tikzcd}
\end{equation*}
induced by (\ref{eq:compatibility}). By Proposition 3.2.9 of \cite{Ser} the horizontal maps can be identifed with the connecting morphisms of the long exact sequences in cohomology associated to
\begin{gather*}
0\rightarrow\Theta_{\X}\rightarrow\Theta_{\P}|_{\X}\rightarrow\cn_{\X/\P}\rightarrow 0, \\
0\rightarrow\Theta_{\F(\X)}\rightarrow\Theta_{\G}|_{\F(\X)}\rightarrow\cn_{\F(\X)/\G}\rightarrow 0,
\end{gather*}
which are surjective by \Cref{cubic} (ii) and \Cref{fano} (ii). By \Cref{cubic} (i) every first order deformation of $\X$ in $\P$ can be represented in the form $f+\varepsilon g=0$ for some $g\in\H^{0}(\co_{\P}(3))$,
and the map $\H^{0}(\cn_{\X/\P}) \rightarrow \H^{0}(\cn_{\F(\X)/\G})$
takes it to the first order deformation $s_{f}+\varepsilon s_{g}=0$ of $\F(\X)$ in $\G$. It follows that we have a commutative diagram
\begin{equation*}
\begin{tikzcd}
\H^{0}(\co_{\P}(3)) \arrow[r] \arrow[d, swap, "s"] & \H^{0}(\cn_{\X/\P})  \arrow[d]  \\
\H^{0}(\S^{3}\cs^{\vee}) \arrow[r] & \H^{0}(\cn_{\F(\X)/\G}),
\end{tikzcd}
\end{equation*}
where the horizontal maps are given by restriction, and $s$ is an isomorphism by \Cref{slemma}. Using \Cref{cubic} (i) and \Cref{fano} (i),
it follows that $\H^{0}(\cn_{\X/\P}) \rightarrow \H^{0}(\cn_{\F(\X)/\G})$ is exactly the isomorphism $\H^{0}(\co_{\P}(3))/\langle f\rangle\xrightarrow{\sim}\H^{0}(\S^{3}\cs^{\vee})/\langle s_{f}\rangle$ induced by $s$.
\end{proof}

\begin{remark}
It would be worthwhile to consider the case where $\X$ has isolated singularities. In this case the section $s_{f}$ is still regular (\cite{AK}, Theorem 1.3), and our proof gives an exact sequence
\begin{equation*}
0\rightarrow\Hom(\Omega^{1}_{\F(\X)},\co)\rightarrow\H^{0}(\Omega^{1}_{\G}\vert_{\F(\X)})\rightarrow\H^{0}(\cn_{\F(\X)/\G})\rightarrow\Ext^{1}(\Omega^{1}_{\F(\X)},\co)\rightarrow 0.
\end{equation*}
Here one should study the exact sequence
\begin{equation*}
0\rightarrow\H^{1}(\Theta_{\F(\X)})\rightarrow\Ext^{1}(\Omega^{1}_{\F(\X)},\co)\rightarrow \H^{0}(\SExt^{1}(\Omega^{1}_{\F(\X)},\co))\rightarrow\H^{2}(\Theta_{\F(\X)})
\end{equation*}
coming from the local-to-global spectral sequence. (If $\X$ has exactly one ordinary double point, the singular locus of $\F(\X)$ is a smooth complete intersection of type $(2,3)$ in projective $d$-space \cite{CG},
and so has $\binom{d+2}{3}$ moduli.) One could also consider the analogous problem for other types of complete intersections, and the Hilbert schemes of planes of higher rank (cf. \cite{DM} for some basic results about these schemes).This seems especially interesting for the intersection of two even-dimensional quadrics with corresponding hyperelliptic curve $\C$. Here the deformation problem is in a sense \cite{DR} parallel to the analogous problem for the moduli space of stable vector bundles on $\C$, which was studied by Narasimhan and Ramanan \cite{NR}.
\end{remark}

\noindent 
\textbf{Acknowledgments.}
We are indebted to D. Huybrechts for several helpful exchanges on cubic hypersurfaces and especially his book \cite{Huy}, which proved to be an invaluable reference. Equally, we thank R. Thomas for a number of discussions on deformation theory. The author is supported by the grant SNF-200020-182181.

{\small
