$G_{\sigma(1),y_1}\otimes \cdots \otimes G_{\sigma(n),y_n}$ over $(y_1, \dots, y_n)$ of the direct summand of $G_z$ corresponding to $\sigma$ with the fiber $G_{\sigma(\tau(1)),y_{\tau(1)}}\otimes \cdots \otimes G_{\sigma(\tau(n)),y_{\tau(n)}}$
over $(y_{\tau(1)}, \dots, y_{\tau(n)})$ of the direct summand corresponding to $\sigma\circ \tau$ by permutation of the factors, for all $\sigma,\tau\in\fS_n$. Hence,

\begin{eqnarray*}
F_{z,y}&\cong & \left(\bigoplus_{\tau\in\fS_n}\bigoplus_{\sigma\in\fS_n}\left(G_{\sigma(\tau^{-1}(1)),y_1}\otimes \ \cdots \ \otimes G_{\sigma(\tau^{-1}(n)),y_n}\right)\right)^{\fS_n}
\\
& \stackrel{\gamma=\sigma\circ\tau^{-1}}{\cong} &
\left(\bigoplus_{\tau\in\fS_n}\bigoplus_{\gamma\in\fS_n}\left(G_{\gamma(1),y_1}\otimes \ \cdots \ \otimes G_{\gamma(n),y_n}\right)\right)^{\fS_n}.
\end{eqnarray*}

the fiber $F_{z,y}$ is naturally isomorphic to
\oplus_{\gamma\in\fS_n}\left(G_{\gamma(1),y_1}\otimes \ \cdots \ \otimes G_{\gamma(n),y_n}\right).

§.§.§ A relative version of the construction

Assume next that $M$ is a two dimensional projective moduli space of sheaves with a primitive Mukai vector $v$ which are
$H$-stable with respect to a $v$-generic polarization $H$ on $S$. Assume that a universal sheaf $\G$ exists over $M\times S$.
Then $\G$ is the Fourier-Mukai kernel of an equivalence $\Phi_\G:D^b(M)\rightarrow D^b(S)$.
Let $M^n\LeftArrowOf{b} \Gamma_M \RightArrowOf{q}M^{[n]}$ be the projections from the isospectral Hilbert scheme of the $K3$ surface $M$. The exterior product $\G^{\boxtimes n}$ over $M^n\times S^n$ admits a natural linearization $\rho_\boxtimes$
with respect to the diagonal action of $\fS_n$.
A relative version of the above construction produces the object

\[
\tilde{\F}:=(\StructureSheaf{\Gamma_M},\rho)\circ (\G^{\boxtimes n},\rho_\boxtimes)\circ (\StructureSheaf{\Gamma},\rho)
\]

over $M^{[n]}\times S^{[n]}$, which is the Fourier-Mukai kernel of the composition
$\tilde{\Phi}_\G^{[n]}:=BKR\circ \Phi^{\boxtimes n}_\G\circ bkr$. Using <cit.> we get the isomorphism

\begin{equation}
\label{eq-universal-sheaf-over-product-of-Hilbert-schemes}
\tilde{\F} \cong R(q\times q)_*^{\fS_{n,\Delta}}\left((b\times b)^*(\G^{\boxtimes n},\rho_\boxtimes\right),
\end{equation}

which is locally free by the flatness of $q\times q$.

The isospectral Hilbert scheme $\Gamma$ is Gorenstein, by <cit.>,  and its dualizing sheaf $\omega_\Gamma$ is the line bundle $q^*\StructureSheaf{S^{[n]}}(\delta)$, by
Hence, the right adjoint $q^!$ of $Rq_*$ is  $\omega_\Gamma\otimes Lq^*$ and so

\[
BKR^{-1}(\bullet) \cong (Rq_*^{\fS_n}\circ L_b^*)^{-1}(\bullet)\cong (Rb_*\circ q^!)(\bullet)\cong
Rb_*((\omega_\Gamma,\rho_\omega)\otimes Lq^*)(\bullet),
\]

where $\rho_{\omega}$ is the natural $\fS_n$-linearization of $\omega_\Gamma$. We have

\begin{equation}
\label{eq-linearized-dualizing-sheaf-of-isospectral-Hilbert-scheme}
(\omega_\Gamma,\rho_\omega)\cong \chi\otimes Lq^*\StructureSheaf{S^{[n]}}(\delta),
\end{equation}

since $Lq^*\StructureSheaf{S^{[n]}}(\delta)$, with its induced linearization, does not have an invariant global section, while $(\omega_\Gamma,\rho_\omega)$ does.

\begin{equation}
\label{eq-relation-betweek-BKR-inverse-and-bkr}
BKR^{-1}(\bullet) \cong \chi\otimes (Rb_*\otimes Lq^*)(\StructureSheaf{S^{[n]}}(\delta)\otimes (\bullet)) =
\chi\otimes bkr(\StructureSheaf{S^{[n]}}(\delta)\otimes (\bullet)).
\end{equation}

We conclude that the kernel of $\Phi^{[n]}:=BKR\circ \Phi^{\boxtimes n}_\G\circ BKR^{-1}:D^b(M^{[n]})\rightarrow D^b(S^{[n]})$ is the locally free

\begin{equation}
\label{eq-second-universal-sheaf-over-product-of-Hilbert-schemes}
\F \cong \pi_{M^{[n]}}^*\left(\StructureSheaf{M^{[n]}}(\delta)\right)\otimes R(q\times q)_*^{\fS_{n,\Delta}}\left((b\times b)^*(\G^{\boxtimes n},\chi\otimes\rho_\boxtimes)\right).
\end{equation}

Equation (<ref>) and Remark <ref> yield the isomorphism
$bkr(\StructureSheaf{z})\cong BKR^{-1}(\StructureSheaf{z})$,
for sky-scraper sheaves of points $z\in M^{[n]}\setminus D_{M^{[n]}}$, where $D_{M^{[n]}}$ is the divisor of non-reduced subschemes.  Consequently,

\begin{equation}
\label{eq-F-z-is-image-of-sky-scpaper-via-Phi[n]}
F_z\cong\tilde{\Phi}^{[n]}(\StructureSheaf{z})\cong \Phi^{[n]}(\StructureSheaf{z}),
\end{equation}

for $z\in M^{[n]}\setminus D_{M^{[n]}}$.

Let $X$ and $Y$ be two projective irreducible holomorphic symplectic manifolds.
Is every rational Hodge isometry $f:H^2(X,\RationalNumbers)\rightarrow H^2(Y,\RationalNumbers)$
induced by an algebraic correspondence? Shafarevich asked this question in case $X$ and $Y$ are $K3$ surfaces. Buskin, building on results of Mukai, proved that every rational Hodge isometry is algebraic [61, 11] (see [35] for another proof). Buskin's approach uses the fact the the universal bundle $\G$ as above is hyperholomorphic and it deforms over
the graph of the automorphism of the period domain of Kähler $K3$-surfaces induced by an isometry $g:H^2(M,\RationalNumbers)\rightarrow H^2(S,\RationalNumbers)$ associated to $\G$. It is natural to expect that both the vector bundle $\tilde{\F}$ given in (<ref>) and
$\F$ given in (<ref>)
have an analogous property and that the techniques  of [11] would establish the algebraicity of many rational Hodge isometries between the second cohomologies of irreducible holomorphic symplectic manifolds of $K3^{[n]}$-type. We plan to return to this question in the future.

§.§.§ The Chern character in terms of the rank and first Chern class

Let $X$ be of $K3^{[n]}$-type. Let $F$ be a vector bundle over $X$ which is obtained via a sequence of equivalences of derived categories and deformations from a sky scraper sheaf $\StructureSheaf{z}$, $z\in Y$, where $Y$ is of $K3^{[n]}$-type as well.
Then $v(F)$ belongs to $SH^*(X,\QQ)$. Set $h:=c_1(F)$. The rank of $F$ is equal $n!r_0^n$, for some rational number $r_0$, and
\Psi(v(F))=\gamma^n,

\begin{equation}
\label{eq-isotropic-gamma-for-all-n}
\gamma=r_0\alpha+\frac{h}{n!r_0^{n-1}}+\frac{(h,h)}{2(n!)^2r_0^{2n-1}}\beta
\end{equation}

is an isotropic class. In particular, $\ell(F)=\span_\QQ\{\gamma\}.$

Let $\phi:H^*(Y,\QQ)\rightarrow H^*(X,\QQ)$ be the derived parallel transport operator associated

to the sequence of derived equivalences and deformations used to obtain $F$ from the sky scraper sheaf.
Let $\widetilde{H}(\phi):\tilde{H}(Y,\QQ)\rightarrow \tilde{H}(X,\QQ)$ be the
image of $\phi$ via the functor (<ref>).

Then $v(F)=\phi(v(\StructureSheaf{z}))=\phi([pt])$, the class $[pt]$ belongs to $SH^*(Y,\QQ)$, and
$\phi$ restricts to an isomorphism from $SH^*(Y,\QQ)$ onto $SH^*(Y,\QQ)$, by <cit.>. Hence, $v(F)$
belongs to $SH^*(Y,\QQ)$.

Set $\gamma:=(\widetilde{H}(\phi))(\beta)$. Then $\Psi(v(F))=\epsilon(\widetilde{H}(\phi))\gamma^n$, by the equality $\Psi(v(\StructureSheaf{z}))=\beta^n$ of Example <ref> and the equivariance of $\Psi$ with respect to derived parallel transport operators (Theorem <ref>).
When $n$ is odd $\epsilon(\widetilde{H}(\phi))=1$, by definition, and
the fact that the rank of $F$ is positive implies that $\epsilon(\widetilde{H}(\phi))=1$ also when $n$ is even, by Remark <ref>.

Write $\gamma=r_0\alpha+\lambda+s\beta$. Then

\[
\Psi(v(F))=\gamma^n=\sum_{k=0}^n\Choose{n}{k}r_0^{n-k}\lambda^k\alpha^{n-k}+\sum_{k=1}^n\Choose{n}{k}s^k(r_0\alpha+\lambda)^{n-k}\beta^k.
\]

The coefficient of $\alpha^n$ on the left hand side is $\rank(F)/n!$ and on the right hand side it is $r_0^n$, so
The coefficient in $H^2(X,\QQ)$ of $\alpha^{n-1}$ on the left hand side is $h/(n-1)!$, since $\Psi(h)=e_h(\alpha^n/n!)=h\alpha^{n-1}/(n-1)!$, and on the right hand side it is $nr_0^{n-1}\lambda$, so $\lambda=h/(n!r_0^{n-1})$.
We have

\[
\]

We get that $s=\frac{(h,h)}{2(n!)^2r_0^{2n-1}}$ and Equation (<ref>) is verified.

Keep the notation and hypotheses of Lemma <ref> and assume that $n=2$. Then
$\gamma=r_0\alpha+\frac{h}{2r_0}+\frac{(h,h)}{8r_0^3}\beta$ is a primitive isotropic class of divisibility $1$ in the LLV lattice $\Lambda_X$ given in (<ref>). In particular, $r_0$ is an integer,
$h=r_0\gcd(2,r_0)\psi$, for a class $\psi\in H^2(X,\ZZ)$ such that $2$ divides $\div(\psi)$ if $r_0$ is odd, $\frac{2r_0}{\gcd(2,r_0)^2}$ divides $(\psi,\psi)$, and
$\frac{(\psi,\psi)\gcd(2,r_0)^2}{2r_0}\equiv r_0$ (mod $4$).

\begin{eqnarray*}
ch_2(F) & = & \frac{1}{4r_0^2}h^2-\frac{r_0^2}{12}c_2(X)
\\
ch_3(F) & = & \frac{1}{24 r_0^4}h^3-\frac{1}{24}c_2(X)h
\\
ch_4(F) & = & \left[
\frac{(h,h)^2}{2^6r_0^6}
-\frac{5(h,h)}{16 r_0^2}
\right][pt]
\\
\chi(F) & = & \left(
\frac{(h,h)+10r_0^4}{8r_0^3}
\right)^2.
\end{eqnarray*}

We have $\Psi(v(F))=\gamma^n,$ where
$\gamma=(\widetilde{H}(\phi))(\beta)=r_0\alpha+\frac{h}{2r_0}+\frac{(h,h)}{8r_0^3}\beta$, by
Lemma <ref>. We get that

\[
\gamma=B_{-\delta/2}\left(
\right).
\]

Hence, $r_0$ is an integer, $\eta=\frac{1}{2r_0}h+\frac{r_0}{2}\delta$ is a class in $H^2(X,\ZZ)$, and
$h=r_0(2\eta-r_0\delta)$. Set $\psi:=[2\eta-r_0\delta]/\gcd(2,r_0)$. Then $2$ divides $\div(\psi)$, if $r_0$ is odd.
Furthermore, the coefficient
of $\beta$ in the above displayed formula is an integer equal to

\[
\left[\frac{(h,h)}{8r_0^3}+\frac{(h,\delta)}{4r_0}-\frac{r_0}{4}\right]=
\frac{r_0^2[4(\eta,\eta)-4r_0(\eta,\delta)-2r_0^2]}{8r_0^3}+
\frac{2r_0[(\eta,\delta)+r_0]}{4r_0}-\frac{r_0}{4}=\frac{(\eta,\eta)}{2r_0}.
\]

$2r_0$ divides $(\eta,\eta)$. Now, $(\psi,\psi)=\frac{4(\eta,\eta)-2r_0^2}{\gcd(2,r_0)^2}$ and so $2r_0/\gcd(2,r_0)$ divides $(\psi,\psi)$. We have

\[
\frac{\gcd(2,r_0)^2(\psi,\psi)}{2r_0}=4\frac{(\eta,\eta)}{2r_0}-r_0,
\]

hence, $\frac{(\psi,\psi)}{2r_0}\equiv r_0$ (mod $4$).

We have $\Psi(h)=h\alpha$, $\Psi(h^2)=h^2+(h,h)\alpha\beta$, and $\Psi(h^3)=3(h,h)h\beta$. Hence,

\begin{eqnarray*}
\gamma^2 & = & r_0^2\alpha^2+h\alpha+\frac{1}{4r_0^2}h^2+\frac{(h,h)}{4r_0^2}\alpha\beta
\\
&=& \Psi\left(
\right)=\Psi(v(F)).
\end{eqnarray*}

Using $c_2(X)h^2=30(h,h)[pt]$ we get

\begin{eqnarray*}
\left(2r_0^2+h+\frac{1}{4r_0^2}h^2+\frac{1}{24r_0^4}h^3+\frac{(h,h)^2}{2^6r_0^6}[pt]\right)
\\
\frac{1}{4r_0^2}h^2-\frac{r_0^2}{12}c_2(X)\right]+
\left[\frac{1}{24r_0^4}h^3-\frac{1}{24}c_2(X)h\right]+
\left[
\frac{(h,h)^2}{2^6r_0^6}-\frac{5(h,h)}{16r_0^2}+\frac{21r_0^2}{16}
\right][pt]
\end{eqnarray*}

The formula for $\chi(F)$ follows, by Grothendieck-Riemann-Roch.

§.§ Proof of Theorem <ref>

The following will be needed for the proof of Theorem <ref>.

Let $X$ be a connected compact Kähler manifold with a trivial Picard group.
Let $V_i$, $1\leq i\leq n$, be vector bundles, possibly twisted, but if so then with respect to the same Brauer class.
Assume that $V_i$ does not have any non-zero saturated proper  subsheaf, $1\leq i \leq n$.
Assume that $V_i$ and $V_j$ are not isomorphic, if $i\neq j$.
Let $\fS_n$ act on $X^n$ by permuting the factors. Choose the linearization $\rho$ of
$W:=\oplus_{\sigma\in \fS_n}V_{\sigma(1)}\boxtimes \cdots \boxtimes V_{\sigma(n)}$ permuting the factors of each direct summand.
Then $W$ does not have any $\fS_n$-invariant non-zero saturated proper subsheaf.

The direct summands of $W$ are stable, by Proposition <ref>, are pairwise non-isomorphic, and $W$ is polystable with respect to every Kähler class on $X^n$.
Let $E$ be a saturated subsheaf of $W$. Then $\det(\SheafHom(E,W))$ is the trivial line bundle, since $\Pic(X^n)$ is trivial.
Hence, $E$ is a direct sum of a subset of the direct summands of $W$. If $E$ is $\fS_n$-invariant, then this subset must be the whole of $\fS_n$ and so $E=W$.

(of Theorem <ref>).
Being $H$-slope-stable, each $G_i$ is projectively $H$-hyperholomorphic [80] (see also [28]).
The proof is the same as that of Theorem <ref>, except that one replaces the use of
Proposition <ref> by that of Lemma <ref>
and the reference to the paragraph before Theorem <ref> by that before Theorem <ref>.

§ THE ACTION OF THE DERIVED MONODROMY GROUP OF $S$ ON THE LLV LATTICE OF $S^{[N]}$

We prove in this section Theorem <ref>
describing the composite homomorphism
$(\widetilde{H}\circ ^{[n]}):DMon(S)\rightarrow O(\widetilde{H}(S^{[n]},\QQ))$ from the derived monodromy group of a $K3$ surface to the isometry group of the rational LLV lattice of its $n$-th Hilbert scheme.

Let $\Phi^{[n]}_\chi:D^b(S^{[n]})\rightarrow D^b(S^{[n]})$ be the auto-equivalence (<ref>), which is
BKR-conjugate to tensorization by the sign character $\chi$.

$\phi^{[n]}_\chi:H^*(S^{[n]},\QQ)\rightarrow H^*(S^{[n]},\QQ)$ be the derived monodromy operator corresponding to  $\Phi^{[n]}_\chi$.
Let $\widetilde{H}(\phi^{[n]}_\chi):\widetilde{H}(S^{[n]},\QQ)\rightarrow \widetilde{H}(S^{[n]},\QQ)$ be the induced isometry via
(<ref>). Clearly, $(\widetilde{H}(\phi^{[n]}_\chi))^2=id$.
Let $\delta\in H^{1,1}(S^{[n]},\Integers)$ be half the class of the divisor of non-reduced subschemes.
Denote by $(r,\lambda,s)$ the element $r\alpha+\lambda+s\beta$ of $\widetilde{H}(S^{[n]},\QQ)$

\begin{equation}
\label{eq-notation-for-element-of-LLV-lattice}
\end{equation}

The isometry $\widetilde{H}(\phi^{[n]}_\chi)$ is $(-1)^{n+1}$ times  the reflection with respect to the hyperplane orthogonal to $u_0:=(0,\delta,n-1)=B_{-\delta/2}(0,\delta,0)$,

\[
\widetilde{H}(\phi^{[n]}_\chi)(x) = (-1)^{n+1}\left[x+\frac{(x,u_0)}{n-1}u_0\right].
\]

Let $e:\Gamma\rightarrow S^n\times S^{[n]}$ be the inclusion.
The kernel of the functor $BKR^{-1}$ is isomorphic to $(\iota_*\StructureSheaf{\Gamma})^\vee[2n]$, the derived dual $(\iota_*\StructureSheaf{\Gamma})^\vee$ is $\iota_*\omega_\Gamma[-2n]$, and $\omega_\Gamma$ is a line bundle, since $\Gamma$ is a normal Cohen-Macaulay and Gorenstein variety, by <cit.>.
So $BKR^{-1}$ has Fourier-Mukai kernel $\iota_*\omega_\Gamma$, while the functor $bkr$ given in (<ref>),
has kernel $\iota_*\StructureSheaf{\Gamma}$, each with its natural linearization.
Let $p\in S^{[n]}$ correspond to a reduced subscheme supported on the subset $\{x_1, \ \dots, \ x_n\}$ of $S$, $x_i\neq x_j$ for $i\neq j$. The restriction of $\omega_\Gamma$ to the $\fS_n$-orbit of $(x_1, \ \dots, \ x_n)$ is trivial.
Hence, $BKR^{-1}(\CC_p)\cong bkr(\CC_p)\cong
\left(\oplus_{\sigma\in\fS_n}\CC_{(x_{\sigma(1)}, \ \dots, \ x_{\sigma(n)})},\rho\right)$, for a linearization $\rho$, which is isomorphic to
$\left(\CC_{(x_{\sigma(1)}, \ \dots, \ x_{\sigma(2)})},\chi\otimes \rho\right)$ by Remark <ref>, so $\Phi^{[n]}_\chi(\CC_p)\cong\CC_p$.
Hence, $\widetilde{H}(\phi^{[n]}_\chi)$ maps the line $\ell(\CC_p)=\span_\QQ\{(0,0,1)\}$ to itself, $\Sym^n(\widetilde{H}(\phi^{[n]}_\chi))$ maps
$\Psi(v(\CC_p))$ to itself, and $\phi^{[n]}_\chi(v(\CC_p))=v(\CC_p)$. If $n$ is even, it follows that $\det(\widetilde{H}(\phi^{[n]}_\chi))=1$, by the equality
$\Sym^n(\widetilde{H}(\phi^{[n]}_\chi))\circ\Psi=\Psi\circ [\det(\widetilde{H}(\phi^{[n]}_\chi))\phi^{[n]}_\chi]$ in Theorem <ref>.

If $n$ is odd, it follows that $\ell(\CC_p)$ is an eigenline of $\widetilde{H}(\phi^{[n]}_\chi)$ with eigenvalue $1$, as the eigenvalue is $\pm 1$ as well as an $n$-th root of unity
by the equality $\Sym^n(\widetilde{H}(\phi^{[n]}_\chi))\circ\Psi=\Psi\circ \phi^{[n]}_\chi$
in Theorem <ref>.

Note that the action of $\widetilde{H}(\phi^{[n]}_\chi)$ is independent of the complex structure of the $K3$ surface $S$.
Furthermore, $\widetilde{H}(\phi^{[n]}_\chi)$ commutes with $\widetilde{H}(g^{[n]}):\widetilde{H}(S^{[n]},\QQ)\rightarrow \widetilde{H}(S^{[n]},\QQ)$, for every $g$ in $DMon(S)$, by Equation (<ref>).
Let $V$ be the subspace of $\widetilde{H}(S^{[n]},\QQ)$ spanned by the $DMon(S)$ orbit of the line
$\ell(\CC_p)=\span_\QQ\{(0,0,1)\}$. The subspace
$V$ is $\widetilde{H}(\phi^{[n]}_\chi)$-invariant and $\widetilde{H}(\phi^{[n]}_\chi)$  acts on $V$ as the identity, if $n$ is odd, and $V$ is contained in either the $1$ or $-1$ eigenspace of $\widetilde{H}(\phi^{[n]}_\chi)$, if $n$ is even.

Assume that $S$ admits an elliptic fibration
$\pi:S\rightarrow\PP^1$  and let $E_1$, …, $E_n$ be $n$ distinct smooth fibers. The abelian variety
$A:=E_1\times \cdots \times E_n$ embeds as a lagrangian subvariety of $S^{[n]}$. Let $\iota:A\rightarrow S^{[n]}$ be the embedding.
Set $\tilde{A}:=\cup_{\sigma\in\fS_n}E_{\sigma(1)}\times \cdots \times E_{\sigma(n)}$.
We get the equivariant embedding $\tilde{\iota}:\tilde{A}\rightarrow \Gamma$ into the universal subscheme $\Gamma\subset S^{[n]}\times S^n$.
Let $b:\Gamma\rightarrow S^n$ and $q:\Gamma\rightarrow S^{[n]}$ be the restriction to $\Gamma$ of the two projections.
Then $\tilde{A}$ is contained in the open subset where $b$ is an isomorphism and $q$ is étale.
Given a line bundle $\LB$ on $S^{[n]}$, then
$Lq^*(\iota_*\iota^*\LB)\cong \tilde{\iota}_*\tilde{\iota}^*(Lq^*\LB)$ in $D^b_{\fS_n}(\Gamma)$. Denote the linearization of $Lq^*\LB$ by $\lambda$. Then $\tilde{\iota}_*(\tilde{\iota}^*Lq^*\LB,\tilde{\iota}^*\lambda)$ is isomorphic to $\tilde{\iota}_*(\tilde{\iota}^*Lq^*\LB,\chi\otimes \tilde{\iota}^*\lambda)$, by Remark <ref>, as the irreducible components $E_{\sigma(1)}\times \cdots \times E_{\sigma(n)}$  are disjoint for different permutations $\sigma\in \fS_n$. Furthermore,
$Rb_*^{\fS_{2,\Delta}}\tilde{\iota}_*(\tilde{\iota}^*Lq^*\LB,\chi\otimes \tilde{\iota}^*\lambda)$ is isomorphic to
$\chi\otimes Rb_*^{\fS_{2,\Delta}}\tilde{\iota}_*(\tilde{\iota}^*Lq^*\LB,\tilde{\iota}^*\lambda)$, since $b$ is an equivariant
morphism with respect to the isomorphism $\fS_n\rightarrow \fS_n$.

It follows that
$BKR^{-1}(\iota_*\iota^*\LB)\cong \chi\otimes BKR^{-1}(\iota_*\iota^*\LB)$. Thus, $\Phi^{[n]}_\chi(\iota_*\iota^*\LB)\cong \iota_*\iota^*\LB$.
Now, $\ell(\StructureSheaf{A})=\span\{(0,f,0)\}$, where $f$ is the class of a fiber of $\pi$ in $H^2(S,\Integers)$ considered as the subspace of $H^2(S^{[n]},\ZZ)$ orthogonal to $\delta$, by Lemma <ref>. Thus,
We see that $\widetilde{H}(\phi^{[n]}_\chi)$ leaves invariant the lines $\span\{(0,f,s)\}$, whenever $f$ is an isotropic primitive class in $H^2(S,\ZZ)$ and there is a complex structure on $S$ such that $f$ is the class of a fiber of an elliptic fibration and there exists a line bundle $\LB$  on $S$, such that $s=(f,c_1(\LB))$.
We conclude that $\widetilde{H}(\phi^{[n]}_\chi)$ acts by multiplication by $1$ or $-1$
on $W:=\{(0,0,1),(0,\delta,0)\}^\perp$.

Assume that the elliptic fibration $\pi$ has a section and all its fibers are integral. Then $S$ is isomorphic to the moduli space $M_H(0,f,0)$
of $H$-stable  sheaves $F$  of rank $0$, $c_1(F)=f$, and $\chi(F)=0$, for a suitable choice of a polarization $H$ on $S$
[10]. Furthermore, there exists over $M_H(0,f,0)\times S$ a universal sheaf $\P$ and
$\Phi_\P:D^b(M_H(0,f,0))\rightarrow D^b(S)$ is an equivalence [10].
Choose the length $n$ subscheme $p$ of $S$ to consists of the $n$ distinct points $p_i$ of intersection of the section with the fibers $E_i$, $1\leq i\leq n$. The  Fourier Mukai transform
$\Phi_\P:D^b(S)\cong D^b(M_H(0,f,0))\rightarrow D^b(S)$
satisfies $\Phi_\P^{[n]}(\CC_{p_i})\cong \StructureSheaf{E_i}$. Hence,
$\Phi_\P^{[n]}(\CC_p)\cong \iota_*\StructureSheaf{A}$. We conclude that $W$ is contained in $V$.

Note that $V$ contains the LLV line of the vector bundle (<ref>). The latter line is not contained in $W$, by Lemma <ref>. Hence, the co-dimension of $V$ in $\tilde{H}(S^{[n]},\QQ)$ is at most $1$.
Recall that $\ell(\StructureSheaf{S^{[n]}})=\span\{(4,0,n+3)\}$, by Lemma <ref>.
Equation (<ref>) implies that $\widetilde{H}(\phi^{[n]}_\chi)$  interchanges the two lines
$\ell(\StructureSheaf{S^{[n]}})$ and $\ell(\StructureSheaf{S^{[n]}}(-\delta))=\span\{(4,-4\delta,7-3n)\}$.
Hence, $V$ is a co-dimension $1$ subspace of $\tilde{H}(S^{[n]},\QQ)$. Furthermore, $\widetilde{H}(\phi^{[n]}_\chi)$  is neither $id$, nor $-id$.
Hence, $\widetilde{H}(\phi^{[n]}_\chi)$ is either the reflection $R_V$ in $V$ or $-R_V$.
The reflection $R_V$ must map $(4,0,n+3)$ to $\pm(4,-4\delta,7-3n)$, as the two have the same  self-intersection.
Furthermore, $(4,0,n+3)+R_V((4,0,n+3))$ belongs to $V$.
If $R_V((4,0,n+3))=-(4,-4\delta,7-3n)$, then $(4,0,n+3)+R_V((4,0,n+3))=4(0,\delta,n-1)$ and so $V=(0,0,1)^\perp$.
This contradicts the fact that $V$ contains LLV lines of vector bundles of positive rank as in Lemma <ref>.
Hence, $R_V((4,0,n+3))=(4,-4\delta,7-3n)$, and $V=(0,\delta,n-1)^\perp$, since
When $n$ is even we observed above that  $\det(\widetilde{H}(\phi^{[n]}_\chi))=1$ and so $\widetilde{H}(\phi^{[n]}_\chi)=-R_V$. When $n$ is odd
we observed above that $\widetilde{H}(\phi^{[n]}_\chi)$  maps $(0,0,1)$ to itself, and so $\widetilde{H}(\phi^{[n]}_\chi)=R_V$.

Let $v=(1,0,1-n)$ be the Mukai vector of the ideal sheaf of a length $n$ subscheme of a $K3$ surface $S$.
Then the co-rank $1$ sublattice $v^\perp$ of the Mukai lattice $\widetilde{H}(S,\ZZ)$ contains $H^2(S,\ZZ)$.
Let $\theta:H^2(S,\ZZ)\rightarrow H^2(S^{[n]},\ZZ)$ be the homomorphism (<ref>).

Denote by
$\theta:H^2(S,\QQ)\rightarrow H^2(S^{[n]},\QQ)$ the induced homomorphism as well.
Let $\tilde{\theta}:\widetilde{H}(S,\QQ)\rightarrow \widetilde{H}(S^{[n]},\QQ)$ be the extension mapping $\alpha$ to $\alpha$ and
$\beta$ to $\beta$.
Let $\eta:O(\widetilde{H}(S,\QQ))\rightarrow O(\widetilde{H}(S^{[n]},\QQ))$ be the homomorphism extending an isometry by acting
as the identity on the orthogonal complement $\span_\QQ\{(0,\delta,0)\}$ of the image of $\tilde{\theta}$. Explicitly,
$\eta_g(\tilde{\theta}(x))=\tilde{\theta}(g(x))$, and $\eta_g(0,\delta,0)= (0,\delta,0)$, for all $g\in O(\widetilde{H}(S,\QQ))$.

The following was proven for $n=2$ in <cit.>.
Given an element $\phi$ of $DMon(S)$ denote by $\phi^{[n]}$ its image in $DMon(S^{[n]})$ via the functor
(<ref>). Denote the image of $\phi^{[n]}$ in $O(\tilde{H}(S^{[n]},\QQ))$ via (<ref>) by $\widetilde{H}(\phi^{[n]})$. Define $\widetilde{H}(\phi)\in O(\tilde{H}(S,\QQ))$ similarly. Set $\eta_\phi:=\eta_{\widetilde{H}(\phi)}$, for $\phi\in DMon(S)$.

B_{-\delta/2}\circ \eta_\phi\circ B_{\delta/2}
\right).$

Recall that $DMon(S)=O^+(\widetilde{H}(S,\ZZ))$, by <cit.>.
Let us first verify the theorem  for all $\phi\in O^+(\widetilde{H}(S,\QQ))$ satisfying $\phi(\alpha)=\alpha$ and $\phi(\beta)=\beta$.
Note that in this case $B_{-\delta/2}\circ \eta_\phi\circ B_{\delta/2}=\eta_\phi$.
Such $\phi$ belongs to the monodromy group $O^+(H^2(S,\ZZ))$ of $S$. Then the image $\phi^{[n]}$ of $\phi$ in the monodromy group of $S^{[n]}$ acts on $H^2(S^{[n]},\ZZ)$ via $\phi^{[n]}(\theta(x))=\theta(\phi(x))$ and $\phi^{[n]}(\delta)=\delta$. This determines the action $SH(\phi^{[n]})$
of $\phi^{[n]}$ on the subring $SH(S^{[n]},\QQ))$ generated by $H^2(S^{[n]},\QQ)$.
The action of $\widetilde{H}(\phi^{[n]})$
on the image of $\Psi$ in $\Sym^n(\widetilde{H}(S^{[n]},\QQ))$ is determined as follows.
If $n$ is odd, then $\Psi$ is equivariant, by Theorem <ref>.

If $n$ is even, then

\[
\Psi\circ \det(\widetilde{H}(\phi^{[n]}))SH(\phi^{[n]})=\Sym^n(\widetilde{H}(\phi^{[n]}))\circ\Psi,
\]

by Theorem <ref>.

Now $\Psi$ is monodromy equivariant with respect to the action of
$\eta_\phi$ on $\widetilde{H}(S^{[n]},\QQ)$.

If $n$ is odd, the theorem follows for $\phi$.
If $n$ is even we get that $\det(\widetilde{H}(\phi^{[n]}))=1$ and so
\eta_\phi & \mbox{if} & \det(\widetilde{H}(\phi))=1,
\\
-\eta_\phi & \mbox{if} & \det(\widetilde{H}(\phi))=-1.
\end{array}\right.$

It remains to verify the theorem for the index $2$ subgroup $SO^+(\widetilde{H}(S,\ZZ))$ of $DMon(S)$.
The statement of the theorem for this subroup is equivalent to the statement that the composition

\[
B_{-\frac{\delta}{2}}\circ \tilde{\theta}:\widetilde{H}(S,\QQ)\rightarrow \widetilde{H}(S^{[n]},\QQ)
\]

is $SO^+(\widetilde{H}(S,\ZZ))$-equivariant, since $SO^+(\widetilde{H}(S,\ZZ))$ does not have any non-trivial characters, and so must act trivially on the one-dimensional orthogonal complement $\span\{u_0\}$, $u_0:=(0,\delta,n-1)=B_{-\frac{\delta}{2}}(0,\delta,0)$, of its image.
Note first the equality

\begin{equation}
\label{eq-image-of-tensorization-by-lb-via-[n]}
\end{equation}

for all $\lambda\in H^2(S,\QQ)$.
It suffices to prove (<ref>) for integral $\lambda$, and by varying the complex structure, we may assume that $\lambda$ is of type $(1,1)$.
Given a line bundle $L$ on $S$, the equivariant line bundle
$(L\boxtimes \ \cdots \ \boxtimes L,\rho_\boxtimes)$ over $S^n$ is a pullback of a line bundle $\LB$  on the symmetric product
$S^{(n)}$.  We have the commutative diagram

\[
\xymatrix{
& \Gamma \ar[dl]_{b} \ar[dr]^{q}
\\
S^n \ar[dr]_{\bar{q}} & & S^{[n]} \ar[dl]^{\bar{b}}
\\
& S^{(n)}.
\]

Hence, $BKR((L\boxtimes \ \cdots \ \boxtimes L,\rho_\boxtimes))\cong
Rq_*^{\fS_n}(Lq^*L\bar{b}^*\LB)\cong \bar{b}^*\LB$.
We get that

\begin{eqnarray*}
BKR((F\boxtimes \ \cdots \ \boxtimes F,\lambda)\otimes (L\boxtimes \ \cdots \ \boxtimes L,\rho_\boxtimes))  \cong
Rq_*^{\fS_n}(Lb^*(F\boxtimes \ \cdots \ \boxtimes F,\lambda)\otimes Lq^*L\bar{b}^*\LB) \cong
\\
Rq_*^{\fS_n}(Lb^*(F\boxtimes \ \cdots \ \boxtimes F,\lambda))\otimes L\bar{b}^*\LB
\cong  BKR(F\boxtimes \ \cdots \ \boxtimes F,\lambda)\otimes BKR((L\boxtimes \ \cdots \ \boxtimes L,\rho_\boxtimes)),
\end{eqnarray*}

where the second isomorphism is via the projection formula.
The equality (<ref>) thus follows for $\lambda=c_1(L)$ from the well known equality

Passing to Lie algebras we get the homomorphism

\begin{equation}
\label{eq-Lie-algebra-homomorphism[n]}
{}^{[n]}:\LieAlg{so}(\widetilde{H}(S,\QQ))\rightarrow \LieAlg{so}(\widetilde{H}(S^{[n]},\QQ))
\end{equation}

and the
equality $e_\lambda^{[n]}=e_{\theta(\lambda)}$.
As $B_\lambda^{[n]}$ commutes with $\widetilde{H}(\phi^{[n]}_\chi)$, by Equation (<ref>),
so does $e_\lambda^{[n]}$ and hence so does $e_{\theta(\lambda)}$. This agrees with the fact  that the eigenspace $V:=(0.\delta,n-1)^\perp$ of $\widetilde{H}(\phi^{[n]}_\chi)$
is $e_{\theta(\lambda)}$-invariant, for all $\lambda\in H^2(S,\QQ)$.

Every non-trivial $24$-dimensional representation of $\LieAlg{so}(\widetilde{H}(S,\QQ))$ is isomorphic to
$\widetilde{H}(S,\QQ)$. Hence there exists an isomorphism

\[
\gamma : \widetilde{H}(S,\QQ)\rightarrow V
\]

of $\LieAlg{so}(\widetilde{H}(S,\QQ))$-modules, which must also be an isomorphism of $SO^+(\widetilde{H}(S,\QQ))$-representations.
The element $\beta$ of $\widetilde{H}(S,\QQ)$ spans the common kernel of
$e_\lambda$,  $\lambda\in H^2(S,\QQ)$. Hence,
the element $\gamma(\beta)$ must belong to the common kernel $\span\{\beta\}$
of $e_{\theta(\lambda)}$, $\lambda\in H^2(S,\QQ)$. We may normalize $\gamma$ to satisfy $\gamma(\beta)=\beta$.

The Lie algebras appearing in (<ref>) are subalgebras of $\LieAlg{gl}(H^*(S,\QQ))$ and
of $\LieAlg{gl}(H^*(S^{[n]},\QQ))$ and Hodge sub-modules. We have the commutative diagram

\[
\xymatrix{
\LieAlg{gl}(H^*(S,\QQ)) \ar[r] &\LieAlg{gl}(H^*(S^n,\QQ)) \ar[r]& \LieAlg{gl}(H^*(S^{[n]},\QQ))
\\
\LieAlg{so}(\widetilde{H}(S,\QQ)) \ar[rr]_{[n]} \ar[u]^{\cup} & & \LieAlg{so}(\widetilde{H}(S^{[n]},\QQ)), \ar[u]_{\cup}
\]

where the top left horizontal arrow is the map from the endomorphism algebra of a vector space to that of its $n$-th tensor power.
The top right horizontal arrow sends  a correspondence in $H^*(S^n\times S^n,\QQ)$ to its convolution with the correspondence
$\sqrt{td_{S^{[n]}}} ch(\StructureSheaf{\Gamma})\sqrt{td_{S^n}}$ in $H^*(S^{[n]}\times S^n,\QQ)$
and its adjoint in $H^*(S^n\times S^{[n]},\QQ)$ associated to the dual object $\StructureSheaf{\Gamma}^\vee$ (??? need to account for the difference between $Rq_*$ and $Rq_*^{\fS_n}$ ???).
Assign $H^*(S^{[n]},\QQ)$ a coarser Hodge structure of weight $0$,
where $H^{p,q}(S^{[n]})$ has weight $(p-\frac{p+q}{2},q-\frac{p+q}{2})$. Endow $\LieAlg{gl}(H^*(S^{[n]},\QQ))$ with the coarser Hodge structure as well. All Hodge classes has weight $(0,0)$ with respect to the coarser Hodge structure.
The homomorphism ${}^{[n]}$ is compatible with the coarser Hodge structures, as it is induced by rational Hodge classes.

Now, the $\LieAlg{so}(\widetilde{H}(S,\QQ))$-action determines the

Hodge structure of weight $0$ introduced in Section <ref>. Similarly,
the $\LieAlg{so}(\widetilde{H}(S^{[n]},\QQ))$-action determines the coarse Hodge structure of
$\widetilde{H}(S^{[n]},\QQ)$. We conclude that $\gamma$ is an isomorphism of the coarse Hodge structures.

Let $\tilde{\gamma}:\widetilde{H}(S,\QQ)\rightarrow \widetilde{H}(S,\QQ)$ be the composition
$\tilde{\gamma}=\tilde{\theta}^{-1}\circ B_{\delta/2}\circ \gamma$, where $\tilde{\theta}^{-1}:(0,\delta,0)^\perp\rightarrow \widetilde{H}(S,\QQ)$ is the left inverse of $\tilde{\theta}$. Then $\tilde{\gamma}$ is $SO^+(H^2(S,\ZZ))$-equivariant (the two actions, via ${}^{[n]}$ and via $\eta$ coincide for this subgroup, as demonstrated in the first paragraph of the proof).
both $\tilde{\theta}^{-1}$ and $B_{\delta/2}$ are $SO^+(H^2(S,\ZZ))$-equivariant and  $\gamma$ is  $SO^+(H^2(S,\ZZ))$-equivariant as it is $SO^+(\widetilde{H}(S,\QQ))$-equivariant by assumption.
Hence, $\tilde{\gamma}$ must map $H^2(S,\QQ)$ to itself and act on it by $\pm 1$.
Being an isometry,  $\tilde{\gamma}$  maps the orthogonal complement
$U:=\span\{\alpha,\beta\}$ of  $H^2(S,\QQ)$ to itself.
$\tilde{\gamma}(\beta)=\beta$, and $\tilde{\gamma}$ commutes with $e_\lambda$, for all $\lambda\in H^2(S,\QQ)$.

Thus $\tilde{\gamma}(\alpha)=\alpha$, since $\alpha$ is the unique isotropic class in $U$ which pairs to $-1$ with $\beta$.
Finally, $\tilde{\gamma}(\lambda)=\tilde{\gamma}(e_\lambda(\alpha))=e_\lambda(\tilde{\gamma}(\alpha))=e_\lambda(\alpha)=\lambda$. Hence $\tilde{\gamma}$ is the identity.

Hence $B_{-\frac{\delta}{2}}\circ \tilde{\theta}$ is $SO^+(\widetilde{H}(S,\QQ))$-equivariant.
The theorem is thus verified also for
the subgroup $SO^+(\widetilde{H}(S,\ZZ))$ of $DMon(S)$.

The vector bundle $F_z$  in Lemma <ref> satisfies

Note that $n!r_0^{n-1}(\theta(\lambda)-r_0\delta/2)$ is $n!r_0^{n-1}$ times the direct summand of $B_{-\frac{\delta}{2}}(\tilde{\theta}(v))$ in $H^2(S^{[n]},\QQ)$, where $v=(r_0,\lambda,(\lambda,\lambda)/2)$
is the Mukai vector of $G_i$, $1\leq i\leq n$.
Theorem <ref>(<ref>) reduces the proof of the Corollary to that of the equality
Let $\phi$ be the element of $DMon(S)$ corresponding to the equivalence $\Phi$.
Denote by $\beta_1=(0,0,1)\in\widetilde{H}(S,\QQ)$ the Mukai vector of a point. Then $v=\widetilde{H}(\phi)(\beta_1)$,
since $G_i$ is isomorphic to $\Phi(\StructureSheaf{p_i})$.

The equality $\ell(F_z)=\widetilde{H}(\phi^{[n]})(\span\{\beta\})$ follows from
Equation (<ref>).
Theorem <ref> implies that
the embedding $B_{-\frac{\delta}{2}}\circ\tilde{\theta}:\widetilde{H}(S,\QQ)\rightarrow \widetilde{H}(S^{[n]},\QQ)$ is  equivariant up to sign with respect to (it spans a character of) $DMon(S):=\Aut_{K3^{[1]}}(S)$ acting via the functor $\widetilde{H}$ on the domain and via the functor $\widetilde{H}\circ ^{[n]}$ on the co-domain.  This equivariance explains the third equality below.

\begin{eqnarray*}
\ell(F_z)&=&
\widetilde{H}(\phi^{[n]})(\span\{\beta\})=
\widetilde{H}(\phi^{[n]})(\span\{B_{-\frac{\delta}{2}}\circ\tilde{\theta}(\beta_1)\})=
\span\{B_{-\frac{\delta}{2}}(\tilde{\theta}(\phi(\beta_1))\}
\\
&=& \span\{B_{-\frac{\delta}{2}}(\tilde{\theta}(v)\}.
\end{eqnarray*}

§ A REFLEXIVE TORSION FREE FOURIER-MUKAI IMAGE OF A LAGRANGIAN LINE BUNDLE WITH A POSITIVE LLV LINE

Let $S$ be a $K3$ surface of degree $6$ in $\PP^4$. The Fano variety of lines on a smooth cubic $3$-folds $C$ containing $S$
embeds as a smooth Lagrangian surface $Z$ in the Hilbert scheme $S^{[2]}$.
In Section <ref> we state
the main result of this section that the images $E_k$ of the natural line bundles $\StructureSheaf{Z}(k)$ on $Z$ via a particular Fourier-Mukai auto-equivalence $\Phi^{[2]}_{\P,\chi}$
of $D^b(S^{[2]})$ are reflexive torsion free sheaves, for $k$ sufficiently large,  and we compute the LLV line of $E_k$
(Proposition <ref>).
The auto-equivalence $\Phi^{[2]}_{\P,\chi}$ is the BKR-conjugate of the auto-equivalence of $D^b(S\times S)$ induced by tensoring with the sign character $\chi$ of $\fS_2$ the cartesian-square of the spherical twist $\Phi_\P$ of $D^b(S)$ with respect to $\StructureSheaf{S}$.
In Section <ref>
we describe in more detail the equivalences $BKR:D^b_{\fS_2}(S\times S)\rightarrow D^b(S^{[2]})$ and
$\Phi_{\P\boxtimes\P}:D^b(S\times S)\rightarrow D^b(S\times S)$.
In section <ref> we provide a local coordinate description of
strict transforms of certain diagonals in the cartesian square $\widehat{S\times S}\times \widehat{S\times S}$
of the blow-up $\widehat{S\times S}$ of the diagonal in $S\times S$.
In Section <ref> we provide an explicit complex over $S^{[2]}\times S^{[2]}$ representing  the Fourier-Mukai kernel of $\Phi^{[2]}_{\P,\chi}$ (Lemma <ref>).
In Section <ref> we provide a general criterion
for the Fourier-Mukai image via $\Phi^{[2]}_{\P,\chi}$ of a line-bundle over a lagrangian surface in $S^{[2]}$ to be a reflexive torsion free sheaf (Lemma <ref>).
We then verify that the hypotheses of this criterion are satisfied for sufficiently large powers $\StructureSheaf{Z}(k)$ of the natural ample line bundle over the Fano variety of lines $Z$ over a cubic $3$-fold. In Section <ref>
we complete the proof of Proposition <ref>.

§.§ Fourier-Mukai images of line bundles on the Fano variety of lines on a cubic 3-fold

Let $S$ be a $K3$ surface. Let $\Delta:S\rightarrow S\times S$ the diagonal embedding, and let $\Phi_\P:D^b(S)\rightarrow D^b(S)$ be the auto-equivalence with Fourier-Mukai kernel the complex $\P$ given by
$\StructureSheaf{S\times S}\rightarrow \Delta_*\StructureSheaf{S}$ supported in degrees $0$ and $1$. $\P$ is quasi-isomorphic to the ideal sheaf of the diagonal. We will denote $\Delta_*\StructureSheaf{S}$ by $\StructureSheaf{\Delta_S}$ and use a similar abuse of notation for subschemes of $S^4$ and $S^{[2]}\times S^{[2]}$.
$\Phi_\P$ is the spherical twist with respect to the spherical object $\StructureSheaf{S}$. We get the
auto-equivalence $\Phi_\P\boxtimes \Phi_\P:D^b_{\fS_2}(S\times S)\rightarrow D^b_{\fS_2}(S\times S)$ of $\fS_2$-equivariant
Denote by $\Phi_\P^{[2]}:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$ its conjugate (<ref>) via the BKR equivalence.
Let $\Phi_\chi$ be the auto-equivalence of $D^b_{\fS_2}(S\times S)$ corresponding to tensorization with the sign character $\chi$ of $\fS_2$ and let $\Phi^{[2]}_\chi:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$ be its conjugate (<ref>) via the BKR equivalence.

\[
\Phi^{[2]}_{\P,\chi}:=\Phi^{[2]}_\chi\circ \Phi^{[2]}_\P.
\]

Let $Q$ be a quadric in $\PP^4$ and let $C$ be a cubic in $\PP^4$ satisfying the following.

* $Q$ and $C$ are smooth and their intersection $S:=Q\cap C$ is smooth.

* $S$ does not contain any line.

* $C$ does not have any Eckhardt points, i.e., through every point of $C$ pass only finitely many lines on $C$.

Note that $S$ is a $K3$ surface.
The Fano variety $F(C)$ of lines on $C$ is a smooth surface, by [17].

\begin{equation}
\label{eq-g}
g:F(C)\rightarrow S^{[2]}
\end{equation}

be the morphism sending a line $\ell$ on $C$ to the length $2$ subscheme $\ell\cap Q$ of $S$.

Let $\StructureSheaf{F(C)}(1)$ be the pull back of the line bundle $\StructureSheaf{Gr(2,5)}(1)$ via the inclusion $F(C)\subset Gr(2,5)$. Set $h:=c_1(\StructureSheaf{S}(1))$ and let $\tilde{h}:=\theta(h)\in H^2(S^{[2]},\Integers)$ be its image via (<ref>).

Set $Z:=g(F(C))$ and $\StructureSheaf{Z}(1):=g_*\StructureSheaf{F(C)}(1)$.
Keep the notation for elements of the rational LLV lattice as triples introduced in Equation (<ref>).
Following is the main result of this section.

The image $g(F(C))$ is a smooth lagrangian surface in $S^{[2]}$ and
$g_*\StructureSheaf{F(C)}$ has a rank $1$ cohomological obstruction map

(Definition <ref>(<ref>)). For all integers $k$ sufficiently large the object $\Phi_{\P,\chi}^{[2]}(g_*\StructureSheaf{F(C)}(k))$ is isomorphic to a reflexive torsion free sheaf $E_k$ over $S^{[2]}$ of rank $45k^2$ and  $c_1(E_k)=-15k\tilde{h}-\frac{45k(k+1)}{2}\delta$. The
sheaf $E_k$  has a rank $1$ cohomological obstruction map

and $\ell(E_k)$ is spanned by $\left(45k^2,-15k\tilde{h}-\frac{45k(k-1)}{2}\delta,-\frac{45k(k-2)}{4}\right)$. The sheaf $E_k$ fits in a short exact sequence

\begin{equation}
\label{eq-short-exact-sequence-of-E-k}
0\rightarrow E_k \rightarrow
\left[
H^0(Z,\StructureSheaf{Z}(k+1))\otimes_\CC \StructureSheaf{S^{[2]}}
\right]
\oplus
\left[
H^0(Z,\StructureSheaf{Z}(k))\otimes_\CC \StructureSheaf{S^{[2]}}(-\delta)
\right]
\RightArrowOf{a_k} F_k\rightarrow 0,
\end{equation}

where $F_k$ is a torsion free sheaf of rank $12$ and $c_1(F_k)=15k\tilde{h}-6\delta$. The restriction of $a_k$ to the first direct summand is surjective onto $F_k$, while the image of its restriction to the second summand is isomorphic to $F_{k-1}\otimes \StructureSheaf{S^{[2]}}(-\delta)$.

The proposition is proved at the end of this section. Note that $E_k$ is the image of $g_*\StructureSheaf{F(C)}$ via an auto-equivalence, hence $\dim\Ext^i(E_k,E_k)=\dim\Ext^i(g_*\StructureSheaf{F(C)},g_*\StructureSheaf{F(C)})=\dim H^i(F(C),\ComplexNumbers)$, where the latter equality follows from <cit.>.
In particular,
$E_k$ is simple, i.e., $\Hom(E_k,E_k)\cong\CC$,  $\dim\Ext^1(E_k,E_k)=10$, and $\dim\Ext^2(E_k,E_k)=45$, by Example
We do not know[We check in Remark <ref> that
the intersections of $E_k$ with each of the two direct summands in (<ref>) do not slope-destabilize $E_k$ for polarizations in an open subcone of the ample cone.
if $E_k$ is slope-stable for line bundles in some open subcone of the ample cone. If it is, then Theorem <ref> would apply and the Azumaya algebra $\SheafEnd(E_k)$ would deform with $S^{[2]}$ to every irreducible holomorphic symplectic manifold of $K3^{[2]}$-type.

Assumption <ref> does not rule out the case where $S$ contains a smooth conic and hence admits an elliptic fibration onto the pencil of hyperplane sections containing the conic. In this case $S^{[2]}$ admits a lagrangian fibration the generic fiber of which is the product of two elliptic curves. Techniques for proving stability of vector bundles over IHSMs admitting lagrangian fibrations were developed in <cit.>.

§.§ Explicit description of the isospectral Hilbert scheme for $n=2$

Let $S$ be a $K3$ surface. Let $b:\widehat{S\times S}\rightarrow S\times S$ be the blow-up of the diagonal,  and $q:\widehat{S\times S}\rightarrow S^{[2]}$ the quotient map by the natural $\fS_2$-action.
The BKR equivalence, given in (<ref>),

\[
:D^b_{\fS_2}(S\times S)\rightarrow D^b(S^{[2]})
\]

is the composition $Rq_*^{\fS_2}\circ Lb^*$,

$Rq_*^{\fS_2}:D^b_{\fS_2}(\widehat{S\times S})\rightarrow D^b(S^{[2]})$ is the equivariant pushforward associated to the homomorphism $\fS_2\rightarrow \{1\}$, and
$Lb^*:D^b_{\fS_2}(S\times S)\rightarrow D^b_{\fS_2}(\widehat{S\times S})$  is the equivariant pullback.

The functor $Rq_*^{\fS_2}$ takes an $\fS_2$-equivariant object $F$ to the $\fS_2$-invariant direct summand $(Rq_*(F))^{\fS_2}$ of the object

The morphism $(b,q):\widehat{S\times S}\rightarrow (S\times S)\times S^{[2]}$ embeds $\widehat{S\times S}$ isomorphically onto
the isospectral Hilbert scheme $\Gamma$.
The sheaf $\StructureSheaf{\Gamma}$ admits a natural $\fS_2\times 1$ linearization $\rho$ and $(\StructureSheaf{\Gamma},\rho)$
is the Fourier-Mukai kernel of the functor $BKR$ given in (<ref>).

Its inverse has Fourier-Mukai kernel $(\StructureSheaf{\Gamma}^\vee[2],\rho^\vee)\cong(\omega_{\Gamma},\rho_\omega)$ in $D^b_{1\times \fS_2}(S^{[2]}\times (S\times S))$, where $\rho_\omega$ is the natural linearization (see <cit.>).

The functor
$\Phi_{(\omega_{\Gamma},\rho_\omega)}:D^b(S^{[2]})\rightarrow D^b_{\fS_2}(S\times S)$ is the composition of tensorization by $\StructureSheaf{S^{[2]}}(\delta)$, followed by
$R^{\fS_2}b_*\circ  \chi\otimes Lq^*$, where $\chi$ is the non-trivial character of $\fS_2$, by Equation (<ref>).

\begin{eqnarray}
\label{eq-tensorization-by-chi-interchanges-to-lines}
BKR(\StructureSheaf{S\times S},1)&\cong &\StructureSheaf{S^{[2]}}, \  \mbox{and}
\\
\nonumber
BKR(\StructureSheaf{S\times S},\chi)&\cong &\StructureSheaf{S^{[2]}}(-\delta),
\end{eqnarray}

as observed already in (<ref>),
and so
$BKR^{-1}(\StructureSheaf{S^{[2]}})\cong (\StructureSheaf{S\times S},1)$ and
$BKR^{-1}(\StructureSheaf{S^{[2]}}(-\delta))\cong (\StructureSheaf{S\times S},\chi)$, which agrees with the above description of $BKR^{-1}$.

The following lemma describes the involutive auto-equivalence $\Phi^{[2]}_\chi:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$, given in (<ref>), which is the BKR conjugate of tensorization with the sign character $\chi$.

The following isomorphisms holds for every object $E$ in $D^b(S^{[2]})$ and every object $(F,\rho)$ in $D^b_{\fS_2}(S\times S)$.

\begin{eqnarray*}
\Phi^{[2]}_\chi(E):=BKR(BKR^{-1}(E)\otimes\chi)&\cong& (BKR\left[\left(BKR^{-1}(E))^\vee\right]\right)^\vee\otimes\StructureSheaf{S^{[2]}}(-\delta),
\\
BKR(F,\rho\otimes\chi)&\cong& (BKR(F^\vee,\rho^\vee))^\vee\otimes\StructureSheaf{S^{[2]}}(-\delta).
\end{eqnarray*}

The existence of the second isomorphism is equivalent to that of  the first, by setting $E=BKR(F,\rho)$. We prove the second.

\begin{eqnarray*}
[BKR(F^\vee,\rho^\vee)]^\vee &\cong &\left[Rq_*^{\fS_2}(Lb^*(F^\vee,\rho^\vee))\right]^\vee
\cong
Rq_*^{\fS_2}((\omega_{\widehat{S\times S}},\rho_\omega)\otimes Lb^*(F,\rho))
\\
Rq_*^{\fS_2}\left[\chi\otimes Lq^*\StructureSheaf{S^{[2]}}(\delta)\otimes Lb^*(F,\rho)\right]
\cong
\StructureSheaf{S^{[2]}}(\delta)\otimes Rq_*^{\fS_2}(Lb^*(F,\rho\otimes\chi))
\\
\StructureSheaf{S^{[2]}}(\delta)\otimes BKR(F,\rho\otimes\chi),
\end{eqnarray*}

where the first isomorphism is a definition, the second is by equivariant Grothendieck-Verdier duality, the third by the isomorphism $(\omega_{\widehat{S\times S}},\rho_\omega\otimes\chi)\cong Lq^*\StructureSheaf{S^{[2]}}(\delta)$, the fourth by the projection formula,

and the fifth by definition of $BKR$.

Let $\phi^{[2]}_\chi:H^*(S^{[2]]},\QQ)\rightarrow H^*(S^{[2]]},\QQ)$ be the automorphism induced by $\Phi^{[2]}_\chi$.
Let $\widetilde{H}(\phi^{[2]}_\chi):\widetilde{H}(S^{[2]},\QQ)\rightarrow \widetilde{H}(S^{[2]},\QQ)$ be the induced isometry via
(<ref>), which appeared in Lemma <ref>.

The isometry $\widetilde{H}(\phi^{[2]}_\chi)$ is minus the reflection with respect to the hyperplane orthogonal to $u_0:=(0,\delta,1)=B_{-\delta/2}(0,\delta,0)$,

\[
\widetilde{H}(\phi^{[2]}_\chi)(x) = -[x+(x,u_0)u_0].
\]

Let $e:\Gamma\rightarrow S^n\times S^{[n]}$ be the inclusion.
The kernel of the functor $BKR^{-1}$ is isomorphic to $(\iota_*\StructureSheaf{\Gamma})^\vee[2n]$, the derived dual $(\iota_*\StructureSheaf{\Gamma})^\vee$ is $\iota_*\omega_\Gamma[-2n]$, and $\omega_\Gamma$ is a line bundle, since $\Gamma$ is a normal Cohen-Macaulay and Gorenstein, by <cit.>.
Let $p\in S^{[2]}$ correspond to a reduced subscheme supported on the subset $\{x_1,x_2\}$ of $S$, $x_1\neq x_2$.
We have $BKR^{-1}(\CC_p)\cong bkr(\CC_p)\cong \left(\CC_{(x_1,x_2)}\oplus \CC_{(x_2,x_1)},\rho\right)$, for a linearization $\rho$, which is isomorphic to
$\left(\CC_{(x_1,x_2)}\oplus \CC_{(x_2,x_1)},\chi\otimes \rho\right)$ by Remark <ref>, so $\Phi^{[2]}_\chi(\CC_p)\cong\CC_p$.
Hence, $\widetilde{H}(\phi^{[2]}_\chi)$ maps the line $\ell(\CC_p)=\span\{(0,0,1)\}$ to itself, $\Sym^2(\widetilde{H}(\phi^{[2]}_\chi))$ restricts as the identity to
$\Psi(v(\CC_p))$, and $\phi(v(\CC_p))=v(\CC_p)$, where $\phi:H^*(S^{[2]},\QQ)\rightarrow H^*(S^{[2]},\QQ)$ is the derived monodromy operator corresponding to  $\Phi^{[2]}_\chi$. It follows that $\det(\widetilde{H}(\phi^{[2]}_\chi))=1$, since we must have
$\Sym^2(\widetilde{H}(\phi^{[2]}_\chi))\circ\Psi=\Psi\circ [\det(\widetilde{H}(\phi^{[2]}_\chi))\phi]$, by <cit.>.

Note that the action of $\widetilde{H}(\phi^{[2]}_\chi)$ is independent of the complex structure on the $K3$ surface $S$.
If $\pi:S\rightarrow\PP^1$  is an elliptic fibration and $E_1$ and $E_2$ are two distinct smooth fibers, then
$E_1\times E_2$ embeds as a lagrangian surface in $S^{[2]}$. Let $\iota:E_1\times E_2\rightarrow S^{[2]}$ be the embedding.
Set $Z:=E_1\times E_2\cup E_2\times E_1$. We get the equivariant embedding $\hat{e}:Z\rightarrow \widehat{S\times S}$.
Given a line bundle $\LB$ on $S^{[2]}$, then
$Lq^*(\iota_*\iota^*\LB)\cong \hat{e}_*\hat{e}^*(Lq^*\LB)$ in $D^b_{\fS_2}(\widehat{S\times S})$. Denote the linearization of $Lq^*\LB$ by $\lambda$. Then $\hat{e}_*(\hat{e}^*Lq^*\LB,\hat{e}^*\lambda)$ is isomorphic to $\hat{e}_*(\hat{e}^*Lq^*\LB,\chi\otimes \hat{e}^*\lambda)$, by Remark <ref>, as $E_1\times E_2$ and $E_2\times E_1$ are disjoint. Furthermore,
$Rb_*^{\fS_{2,\Delta}}\hat{e}_*(\hat{e}^*Lq^*\LB,\chi\otimes \hat{e}^*\lambda)$ is isomorphic to
$\chi\otimes Rb_*^{\fS_{2,\Delta}}\hat{e}_*(\hat{e}^*Lq^*\LB,\hat{e}^*\lambda)$, since $b$ is an equivariant
morphism with respect to the isomorphism $\fS_2\rightarrow \fS_2$.

It follows that
$BKR^{-1}(\iota_*\iota^*\LB)\cong \chi\otimes BKR^{-1}(\iota_*\iota^*\LB)$. Thus, $\Phi^{[2]}_\chi(\iota_*\iota^*\LB)\cong \iota_*\iota^*\LB$.
Now, $\ell(\StructureSheaf{E_1\times E_2})=\span\{(0,f,0)\}$, where $f$ is the class of a fiber of $\pi$ in $H^2(S,\Integers)$ considered as the subspace of $H^2(S^{[2]},\ZZ)$ orthogonal to $\delta$, by Lemma <ref>. Thus,
We see that $\widetilde{H}(\phi^{[2]}_\chi)$ leaves invariant the lines $\span\{(0,f,s)\}$, whenever $f$ is an isotropic primitive class in $H^2(S,\ZZ)$ and there is a complex structure on $S$ such that $f$ is the class of a fiber of an elliptic fibration and there exists a line bundle $\LB$  on $S$, such that $s=(f,c_1(\LB))$.
We conclude that $\widetilde{H}(\phi^{[2]}_\chi)$ acts by multiplication by $1$ or $-1$
on $M:=\{(0,0,1),(0,\delta,0)\}^\perp$.

Recall that $\ell(\StructureSheaf{S^{[2]}})=\span\{(4,0,5)\}$, by Example <ref>.
Equation (<ref>) implies that $\widetilde{H}(\phi^{[2]}_\chi)$  interchanges the two lines
$\ell(\StructureSheaf{S^{[2]}})\}$ and $\ell(\StructureSheaf{S^{[2]}}(-\delta))=\span\{(4,-4\delta,1)\}$.
Set $N:=\span_\QQ\{(4,0,5),(0,\delta,1)\}$.
The LLV lattice $\Lambda=B_{-\delta/2}(\widetilde{H}(S^{[2]},\ZZ))$  of $S^{[2]}$ given in (<ref>) is invariant under the derived monodromy group, hence it is $\widetilde{H}(\phi^{[2]}_\chi)$-invariant.
Thus, so is $N_\ZZ:=N\cap\Lambda$.
Now $B_{\delta/2}(0,\delta,1)=(0,\delta,0)$, $B_{\delta/2}(4,0,5)=2(2,\delta,2)$, and

\[
B_{\delta/2}(N_\ZZ)=\span_\QQ\{(0,\delta,0),(2,\delta,2)\}\cap \widetilde{H}(S^{[2]},\ZZ)=\span_\ZZ\{(0,\delta,0),(1,0,1)\}
\]

is $B_{\delta/2}\circ\widetilde{H}(\phi^{[2]}_\chi)\circ B_{-\delta/2}$-invariant.
There are precisely $4$ elements of self intersection $-2$ in $B_{\delta/2}(N_\ZZ)$, namely $\{\pm(0,\delta,0),\pm(1,0,1)\}$.
An isometry of $B_{\delta/2}(N_\ZZ)$, which extends to $\widetilde{H}(S^{[2]},\ZZ)$, must take $(0,\delta,0)$ to $\pm(0,\delta,0)$
and $(1,0,1)$ to $\pm(1,0,1)$, as the divisibility of $(1,0,1)$ in $\widetilde{H}(S^{[2]},\ZZ)$ is $1$ and that of $(0,\delta,0)$ is $2$.
Hence, such an isometry must be one of

\[
\{\pm 1, \pm R_{(0,\delta,0)},\pm R_{(1,0,1)},\pm R_{(0,\delta,0)}R_{(1,0,1)}\},
\]

where $R_u$ is the reflection with respect to $u^\perp$. Of those, only
$\pm R_{(0,\delta,0)}$ interchange the lines spanned by
$B_{\delta/2}(4,0,5)=(4,2\delta,4)$ and $B_{\delta/2}(4,-4\delta,1)=(4,-2\delta,4)$.
Note that $R_{(0,\delta,0)}=R_{B_{\delta/2}(u_0)}=B_{\delta/2}\circ R_{u_0}\circ B_{-\delta/2}$.
We conclude that $\widetilde{H}(\phi^{[2]}_\chi)$
restricts to $N$ as $R_{u_0}$ or $-R_{u_0}$.

Note that $M$ and $N$ are complementary $\widetilde{H}(\phi^{[2]}_\chi)$-invariant subspaces and the latter restricts to $N$ with determinant $-1$ and $\det(\widetilde{H}(\phi^{[2]}_\chi))=1$. Hence, $\widetilde{H}(\phi^{[2]}_\chi)$ restricts to the $23$-dimensional $M$ with determinant $-1$ and so acts on $M$ by $-1$. Thus, $-R_{u_0}\circ \widetilde{H}(\phi^{[2]}_\chi)$ acts on $M$ by $1$ and on $N$ by $\pm 1$. Now, $M$ and $N$ are not orthogonal, and so $-R_{u_0}\circ \widetilde{H}(\phi^{[2]}_\chi)$ must act on $N$ by $1$ as well. Hence,

Denote by $\fS_{2,\Delta}$ the diagonal subgroup of $\fS_2\times\fS_2$.

The non-unit of $\fS_{2,\Delta}$ acts by $(x_1,x_2,x_3,x_4)\mapsto (x_2,x_1,x_4,x_3).$
We get the $\fS_{2,\Delta}$-equivariant object

\[
\P\boxtimes\P:=L\pi_{13}^*\P\otimes L\pi_{24}^*\P
\]

over $(S\times S)\times (S\times S)$, endowed with the linearization $\rho_{\boxtimes}$ acting by permuting the factors. The object $\P\boxtimes\P$  is represented by the complex

\begin{equation}
\label{eq-right-exact-sequence-of-beta}
\StructureSheaf{S^4}\RightArrowOf{\alpha}\StructureSheaf{\Delta_{13}}\oplus \StructureSheaf{\Delta_{24}}
\RightArrowOf{\beta} \StructureSheaf{\Delta_{13}\cap\Delta_{24}},
\end{equation}

where $\Delta_{ij}\subset S^4$ is the subvariety of quadruples $(x_1,x_2,x_3,x_4)$ satisfying $x_i=x_j$,
$\beta$ is the difference of the
natural homomorphisms from each direct summand, and $\alpha$ is the direct sum of the two natural homomorphisms.
Note that $\ker(\beta)$ is $\StructureSheaf{\Delta_{13}\cup\Delta_{24}}$ and so the above complex is quasi-isomorphic to the ideal sheaf $\Ideal{\Delta_{13}\cup\Delta_{24}}$.
The integral transform $\Phi_{(\P\boxtimes\P,\rho_\boxtimes)}:D^b_{\fS_2}(S\times S)\rightarrow D^b_{\fS_2}(S\times S)$ is an auto-equivalence, which is conjugated via the BKR equivalence to an auto-equivalence

\begin{equation}
\label{eq-P-[2]-as-a-composition}
\Phi_\P^{[2]}:= \Phi_{{(\StructureSheaf{\Gamma},\rho)}}\circ \Phi_{(\P\boxtimes\P,\rho_{\boxtimes})}\circ
\Phi_{(\omega_{\Gamma},\rho_\omega)}: D^b(S^{[2]})\rightarrow D^b(S^{[2]}),
\end{equation}

by <cit.>.
Composing $\Phi_{(\P\boxtimes\P,\rho_\boxtimes)}$ with the auto-equivalence $\Phi_\chi$ of tensoring with the sign character $\chi$ of $\fS_2$ we get the auto-equivalence $\Phi_{(\P\boxtimes\P,\chi\otimes\rho_\boxtimes)}$ and we denote its BKR-conjugate by
$\Phi_{\P,\chi}^{[2]}:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$.
The projections $\pi_i:(S\times S)\times (S\times S)\rightarrow (S\times S)$, $i=1,2$, are equivariant with respect to the natural isomorphism $\fS_{2,\Delta}\rightarrow\fS_2$, and we get  $\Phi_\chi\circ \Phi_{(\P\boxtimes\P,\rho_{\boxtimes})}=\Phi_{(\P\boxtimes\P,\rho_{\boxtimes})}\circ \Phi_\chi$.
We have

\[
\Phi_{\P,\chi}^{[2]}=
\Phi_\chi^{[2]}\circ\Phi_\P^{[2]}=\Phi_\P^{[2]}\circ\Phi_\chi^{[2]},
\]

by Equation (<ref>).

$\Phi^{[2]}_{\P.\chi}(\StructureSheaf{S^{[2]}})  \cong \StructureSheaf{S^{[2]}}(-\delta)[-4]$ and
$\Phi^{[2]}_{\P.\chi}(\StructureSheaf{S^{[2]}}(-\delta))  \cong \StructureSheaf{S^{[2]}}[-4].$

The isomorphism $\Phi_\P(\StructureSheaf{S})\cong \StructureSheaf{S}[-2]$ explains the third isomorphism below.

\begin{eqnarray*}
\Phi^{[2]}_{\P.\chi}(\StructureSheaf{S^{[2]}}) & \cong &
\chi\otimes \Phi_{(\P\boxtimes\P,\rho_\boxtimes)}(
\right)
\\
\chi\otimes \Phi_{(\P\boxtimes\P,\rho_\boxtimes)}(\StructureSheaf{S\times S},1)
\right)
\cong
\chi\otimes (\StructureSheaf{S\times S}[-4],1)
\right)
\\
\stackrel{\mbox{Lemma \ref{lemma-BKR-conjugate-of-tensorization-by-sign-character}}}{\cong}
\left(BKR(\StructureSheaf{S\times S}[4],1)\right)^\vee\otimes \StructureSheaf{S^{[2]}}(-\delta)
\stackrel{(\ref{eq-BKR-of-structure-sheaf})}{\cong} \StructureSheaf{S^{[2]}}(-\delta)[-4].
\end{eqnarray*}

The proof of the second isomorphism is similar.

\begin{eqnarray*}
\Phi^{[2]}_{\P.\chi}(\StructureSheaf{S^{[2]}}(-\delta))&\cong&
BKR\left(\chi\otimes \Phi_{(\P\boxtimes\P,\rho_\boxtimes)}(BKR^{-1}(\StructureSheaf{S^{[2]}}(-\delta)))
\right)
\\
&\cong& BKR\left(\chi\otimes \Phi_{(\P\boxtimes\P,\rho_\boxtimes)}(\StructureSheaf{S\times S},\chi))
\right)
\cong
BKR((\StructureSheaf{S\times S},1)[-4])\cong\StructureSheaf{S^{[2]}}[-4].
\end{eqnarray*}

§.§ A local coordinate description of the strict transforms of diagonals

We will need a local coordinates description of $\widehat{S\times S}\times \widehat{S\times S}$ and its exceptional divisors.
We replace $S$ by $\CC^2$ and let $(x_i,y_i)$ be coordinates on the $i$-th factor of $S$. Let $(s:t)$ be homogeneous coordinates on $\PP^1$.
The blow-up of $\CC^2$ at the origin is the hypersurface $V(xt-ys)$ in $\CC^2\times \PP^1$. Let $z=s/t$ denote the affine coordinate on $\PP^1\setminus\{(1:0)\}$.  Away from

the blow-up
has coordinates $y,z$ and $x=yz$. The exceptional divisor $\hat{D}$ is $V(y)$ in these coordinates.
Consider the map

\begin{eqnarray*}
\CC^2\times \CC^2 &\rightarrow & \CC^2\times \CC^2,
\\
((x_1,y_1),(x_2,y_2)) & \mapsto & ((x_1,y_1),(x_1-x_2,y_1-y_2)).
\end{eqnarray*}

The blow-up of the left copy of $\CC^2\times \CC^2$ along the diagonal corresponds
to the blow-up along $\CC^2\times \{0\}$ via the above change of coordinates.
Hence, $\widehat{S\times S}$ embeds as the subvariety

\[
\]

of $\CC^2\times \CC^2\times \PP^1$ and has coordinates $x_1,y_1,y_2,z$, and $x_1-x_2=(y_1-y_2)z$, away from the divisor
$\CC^2\times\CC^2\times \{(1:0)\}$.
The exceptional divisor is $V(y_1-y_2)$ in these coordinates.
Over $\widehat{S\times S}\times \widehat{S\times S}$ we get the coordinates

\[
\]

satisfying $x_1-x_2=(y_1-y_2)z_1$ and $x_3-x_4=(y_3-y_4)z_2$.
The strict transform $\hat{\Delta}_{13}$ of $\Delta_{13}$ in $\widehat{S\times S}\times \widehat{S\times S}$ corresponds to $V(x_1-x_3,y_1-y_3)$.
The strict transform $\hat{\Delta}_{24}$  of $\Delta_{24}$ corresponds to $V(y_2-y_4,x_2-x_4)$, where
$x_2=x_1-z_1(y_1-y_2)$ and $x_4=x_3-z_2(y_3-y_4)$.
The intersection $\hat{\Delta}_{13}\cap \hat{\Delta}_{24}$ is given by

\begin{eqnarray*}
\hat{\Delta}_{13}\cap \hat{\Delta}_{24}&=&V(x_1-x_3,y_1-y_3,y_2-y_4,(y_1-y_2)(z_1-z_2)), \mbox{where}
\\
\hat{D}\times_S\hat{D}&=& V(x_1-x_3,y_1-y_3,y_2-y_4,y_1-y_2)
\\
\Delta_{\widehat{S\times S}} &=& V(x_1-x_3,y_1-y_3,y_2-y_4,z_1-z_2).
\end{eqnarray*}

Note that $x_1-x_3,y_1-y_3,y_2-y_4,(y_1-y_2)(z_1-z_2)$ is a regular sequence, and so $\hat{\Delta}_{13}\cap \hat{\Delta}_{24}$
is a complete intersection. Consider the sections  $s_1:=(x_1-x_3,y_1-y_3)$ and $s_2:=(y_2-y_4,x_2-x_4)$  of
$\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}^{\oplus 2}$. The Koszul complex associated to $s_1$
is a locally free resolution of $\StructureSheaf{\hat{\Delta}_{13}}$ and the one associated to $s_2$ is a locally free resolution of $\StructureSheaf{\hat{\Delta}_{24}}$. The tensor product of these two Koszul complexes is the
Koszul complex associated to the section $(s_1,s_2)$ of
$\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}^{\oplus 4}$, which is exact, as the section  $(s_1,s_2)$ is regular.
Hence, the torsion sheaves $\Tor_i(\StructureSheaf{\hat{\Delta}_{13}},\StructureSheaf{\hat{\Delta}_{24}})$ vanish, for $i>0$.

§.§ The kernel of the BKR-conjugate of the square of a spherical twist by $\StructureSheaf{S}$

Let $I\subset S^{[2]}\times S^{[2]}$ be the incidence variety consisting of pairs of ideals of non-disjoint subschemes.
Let $\nu:\tilde{I}\rightarrow I$ be its normalization.
Let $\hat{\Delta}_{ij}$ be the strict transform of $\Delta_{ij}$ in $\widehat{S\times S}\times \widehat{S\times S}$.
Let $\hat{\delta}:\widehat{S\times S}\rightarrow \widehat{S\times S}\times \widehat{S\times S}$ be the diagonal embedding
and denote its image by $\Delta_{\widehat{S\times S}}$.
Let $\hat{D}\subset \widehat{S\times S}$ be the exceptional divisor of $b$.
Then $\hat{\Delta}_{13}\cap \hat{\Delta}_{24}=\Delta_{\widehat{S\times S}}\cup \hat{D}\times_S\hat{D}$   and each of
$\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ is naturally isomorphic to $\tilde{I}$. The diagonal involution interchanges $\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ leaving $\hat{D}\times_S\hat{D}$ pointwise invariant. Thus
$\hat{D}\times_S\hat{D}$ embeds naturally in $\tilde{I}$ and we denote its image by $\hat{D}\times_S\hat{D}$ as well.
Denote by $\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}$ the ideal sheaf of $\hat{D}\times_S\hat{D}$ in $\tilde{I}$.
The homomorphism $\StructureSheaf{\tilde{I}}\rightarrow\StructureSheaf{\Delta_{\widehat{S\times S}}}$
maps a local function vanishing to order $1$ along $\hat{D}\times_S\hat{D}$ to an anti-invariant function with respect to the diagonal involution.

We have the short exact sequence

\[
0\rightarrow \Ideal{I,D\times_SD\cup\Delta_{S^{[2]}}}\rightarrow \nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}
\rightarrow \StructureSheaf{\Delta_{S^{[2]}}}\otimes\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\rightarrow 0.
\]

The auto-equivalence
$\Phi_\P^{[2]}:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$ is the Fourier-Mukai transformation with kernel

\[
\P^{[2]} :=
\pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes \left[\left\{
\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\oplus \pi_2^*\StructureSheaf{S^{[2]}}(-\delta)
\right\}
%\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus \StructureSheaf{S^{[2]}\times S^{[2]}}(-\delta_1-\delta_2)
\RightArrowOf{a} \nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}
\right],
\]

where the homomorphism $a$ is surjective and is induced from the restriction homomorphism
$\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\rightarrow \StructureSheaf{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}}$ via push-forward via $q\times q$ and taking the $\fS_{2,\Delta}$-anti-invariant subsheaves.

The auto-equivalence
$\Phi_{\P,\chi}^{[2]}:D^b(S^{[2]})\rightarrow D^b(S^{[2]})$ is the Fourier-Mukai transformation with kernel

\[
\P^{[2]}_\chi := \pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes \left[
\left\{\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus
\left[\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\otimes \pi_2^*\StructureSheaf{S^{[2]}}(-\delta)\right]\right\}
%\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus \StructureSheaf{S^{[2]}\times S^{[2]}}(-\delta_1-\delta_2)
\RightArrowOf{a_\chi} \StructureSheaf{I}
\right],
\]

where the homomorphism $a_\chi$ is surjective and is induced from the restriction homomorphism
$\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\rightarrow \StructureSheaf{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}}$ via push-forward via $q\times q$ and taking the $\fS_{2,\Delta}$-invariant subsheaves.

(<ref>). Step 1:
The kernel of $\Phi_\P^{[2]}$, as defined in (<ref>) using <cit.>, is

isomorphic to
$\pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes R(q\times q)_*^{\fS_{2,\Delta}}\left( L(b\times b)^*(\P\boxtimes\P)\right),$
by the argument below using the following diagram.

\[
\xymatrix{
& \widehat{S\times S}\times \widehat{S\times S} \ar[ld]_{q\times q} \ar[d]_{(q,b)\times(b,q)} \ar[rd]^{b\times b}
\\
S^{[2]}\times S^{[2]} &
[S^{[2]}\times (S\times S)]\times [(S\times S)\times S^{[2]}] \ar[l]^-{\pi_{14}} \ar[r]_-{\pi_{23}} &
(S\times S)\times (S\times S).
\]

Above $\pi_{ij}$ is the projection onto the product of the $i$-th and $j$-th factors, where each $(S\times S)$ is regarded as a single factor of the middle bottom variety in the diagram.
Let $\hat{\pi}_i$ be the projection from $\widehat{S\times S}\times \widehat{S\times S}$ to the $i$-th factor, $i=1,2$.
The object $L\pi_{34}^*R(b,q)_*\StructureSheaf{\widehat{S\times S}}$ is isomorphic to the structure sheaf of
$[(S\times S)\times S^{[2]}]\times \Gamma$. Similarly, $L\pi_{12}^*R(q,b)_*\StructureSheaf{\widehat{S\times S}}$
is isomorphic to the structure sheaf of
$\Gamma\times [S^{[2]}\times (S\times S)]$. The transversality of these two subschemes yields the isomorphism

\begin{equation}
\label{eq-tensor-product-is-push-forward}
[L\pi_{34}^*R(b,q)_*\StructureSheaf{\widehat{S\times S}}]\otimes L\pi_{12}^*R(q,b)_*\StructureSheaf{\widehat{S\times S}}
\cong
R((q,b)\times (b,q))_*\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}.
\end{equation}

The kernel of $\Phi_\P^{[2]}$ is isomorphic to

\begin{eqnarray*}
[L\pi_{34}^*R(b,q)_*^{\fS_2}(\StructureSheaf{\widehat{S\times S}},1)]
\otimes
[L\pi_{12}^*R(q,b)_*^{\fS_2}(\omega_{\widehat{S\times S}},\rho_\omega)]
\otimes
\right\} &\cong&
\\
R((q,b)\times (b,q))_*^{\fS_{2,\Delta}}(L\hat{\pi}_1^*(\omega_{\widehat{S\times S}},\rho_\omega))\otimes L\pi_{23}^*(\P\boxtimes\P,\rho_\boxtimes)
\right\}&\cong&
\\
R((q,b)\times (b,q))_*^{\fS_{2,\Delta}}\left(L\hat{\pi}_1^*(\omega_{\widehat{S\times S}},\rho_\omega)\otimes L(b\times b)^*(\P\boxtimes\P,\rho_\boxtimes)\right)
\right\}&\cong&
\\
R(q\times q)_*^{\fS_{2,\Delta}}\left\{
L\hat{\pi}_1^*(\omega_{\widehat{S\times S}},\rho_\omega)\otimes L(b\times b)^*(\P\boxtimes\P,\rho_\boxtimes)
\right\}&\cong&
\\
\pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes R(q\times q)_*^{\fS_{2,\Delta}}\left( L(b\times b)^*(\P\boxtimes\P,\chi\otimes\rho_\boxtimes)\right),
\end{eqnarray*}

where the first isomorphism is due to (<ref>) and cohomology and base change,
the second isomorphism is due to the projection formula and the equality
$b\times b=\pi_{23}\circ ((q,b)\times (b,q))$, the third is due to the equality
$q\times q=\pi_{14}\circ ((q,b)\times (b,q))$, and the fourth is due to the projection formula, the equality
$\pi_1\circ(q\times q)=q\circ \hat{\pi}_1$, and the isomorphism $(\omega_{\widehat{S\times S}},\chi\otimes\rho_\omega)\cong Lq^*\StructureSheaf{S^{[2]}}(\delta)$. It remains to exhibit the isomorphism

\begin{equation}
\label{eq-isomorphism-of-invariant-subcomplex-of-push-forward-of-ideal-sheaf}
R(q\times q)_*^{\fS_{2,\Delta}}\left( L(b\times b)^*(\P\boxtimes\P,\chi\otimes \rho_\boxtimes)\right)
\cong
\left[
\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\oplus \pi_2^*\StructureSheaf{S^{[2]}}(-\delta)
%\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus \StructureSheaf{S^{[2]}\times S^{[2]}}(-\delta_1-\delta_2)
\RightArrowOf{a} \nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}
\right].
\end{equation}

Step 2: (Calculation of $L(b\times b)^*(\P\boxtimes\P,\chi\otimes \rho_\boxtimes)$).
Recall that $\hat{\Delta}_{ij}$ is the strict transform of $\Delta_{ij}$ in $\widehat{S\times S}\times \widehat{S\times S}$.
Let $\tilde{\Delta}_{13}$ and $\tilde{\Delta}_{23}$ be the strict transforms of $\Delta_{13}$ and $\Delta_{24}$  in
$\widehat{S\times S}\times (S\times S)$.
The subscheme $\Delta_{13}$ is
transversal to $\Delta_{12}$ and $\tilde{\Delta}_{13}$ is
transversal to  $\tilde{\Delta}_{34}$, by a computation in local coordinates,
and so $L(b\times b)^*(\StructureSheaf{\Delta_{13}})$ is represented by $\StructureSheaf{\hat{\Delta}_{13}}$.
Then $L(b\times b)^*(\P\boxtimes \P)$ is isomorphic to $\Ideal{\hat{\Delta}_{13}}\stackrel{L}{\otimes}\Ideal{\hat{\Delta}_{24}}$, which is isomorphic to

\[
[\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\rightarrow
\StructureSheaf{\hat{\Delta}_{13}}] \stackrel{L}{\otimes}
[\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\rightarrow
\StructureSheaf{\hat{\Delta}_{24}}],
\]

and hence to the complex

\begin{equation}
\label{eq-hat-alpha}
\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\RightArrowOf{\hat{\alpha}}
\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}} \RightArrowOf{\hat{\beta}}
\left(\StructureSheaf{\hat{\Delta}_{13}}\stackrel{L}{\otimes} \StructureSheaf{\hat{\Delta}_{24}}\right).
\end{equation}

Note that $\hat{\Delta}_{13}\cap\hat{\Delta}_{24}$ is the union of
the diagonal embedding of $\widehat{S\times S}$ and the fiber product $\hat{D}\times_S\hat{D}$, where $\hat{D}\subset \widehat{S\times S}$ is the exceptional divisor of $b$. Hence, the intersection of $\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ is not transversal along the diagonal embedding of $\hat{D}$.
Nevertheless, $\hat{\Delta}_{13}\cap\hat{\Delta}_{24}$ is a complete intersection and so
the derived tensor product $\StructureSheaf{\hat{\Delta}_{13}}\stackrel{L}{\otimes} \StructureSheaf{\hat{\Delta}_{24}}$
is represented by the usual tensor product $\StructureSheaf{\hat{\Delta}_{13}}\otimes \StructureSheaf{\hat{\Delta}_{24}}$, which is isomorphic to $\StructureSheaf{\hat{\Delta}_{13}\cap \hat{\Delta}_{24}}$, and the sheaves
$\Tor_i(\StructureSheaf{\hat{\Delta}_{13}},\StructureSheaf{\hat{\Delta}_{24}})$ vanish for $i>0$ (see Section <ref>).
The complex (<ref>)  thus becomes

\begin{equation}
\label{eq-right-exact-sequence-of-hat-beta}
\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\RightArrowOf{\hat{\alpha}}
\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}} \RightArrowOf{\hat{\beta}}
\StructureSheaf{\hat{\Delta}_{13}\cap \hat{\Delta}_{24}}
\end{equation}

in degrees $0$, $1$, and $2$ and with sheaf cohomology only in degree $0$, so it is a right exact sequence. The
homomorphisms $\hat{\alpha}$ and $\hat{\beta}$ are analogous to $\alpha$ and $\beta$ in (<ref>). Furthermore, the short exact sequence

\[
0\rightarrow \Ideal{\hat{\Delta}_{13}}\rightarrow \StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\rightarrow
\StructureSheaf{\hat{\Delta}_{13}}\rightarrow 0
\]

yields the isomorphism $\Tor_i(\Ideal{\hat{\Delta}_{13}},F)\cong\Tor_{i+1}(\StructureSheaf{\hat{\Delta}_{13}},F)$, for every coherent sheaf $F$ and for $i>0$, and similarly for $\Ideal{\hat{\Delta}_{24}}$. The symmetry of the functor $\Tor$ yields
$\Tor_i(\Ideal{\hat{\Delta}_{13}},\Ideal{\hat{\Delta}_{24}})\cong \Tor_{i+2}(\StructureSheaf{\hat{\Delta}_{13}},\StructureSheaf{\hat{\Delta}_{24}})=0,$ for all $i>0$. Hence, the complex (<ref>)
represents the sheaf $\Ideal{\hat{\Delta}_{13}}\otimes\Ideal{\hat{\Delta}_{24}}$ (as well as the object $\Ideal{\hat{\Delta}_{13}}\stackrel{L}{\otimes}\Ideal{\hat{\Delta}_{24}}$). Finally, $\Ideal{\hat{\Delta}_{13}}\otimes\Ideal{\hat{\Delta}_{24}}$ is isomorphic to
the kernel $\Ideal{\hat{\Delta}_{13}\cup \hat{\Delta}_{24}}$ of $\hat{\alpha},$ by the right exactness of (<ref>).

Step 3: (First step in the construction of the homomorphism $a$ in (<ref>)).
The quotient map $q\times q:\widehat{S\times S}\times \widehat{S\times S}\rightarrow S^{[2]}\times S^{[2]}$ restricts to
the complement of $\hat{\Delta}_{13}\cap\hat{\Delta}_{24}$ in
each of $\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ as an isomorphism onto $I\setminus[\Delta_{S^{[2]}}\cup D\times_S D]$,
but $I$ is singular along $\Delta_{S^{[2]}}$, as $q\times q:\Delta_{\widehat{S\times S}}\rightarrow \Delta_{S^{[2]}}$ has degree $2$, while it is smooth[
The kernel of $d(q\times q)$ at a point $p$ of $\widehat{S\times S}\times \widehat{S\times S}$ is the sum of non-trivial characters
of $\fS_2\times\fS_2$. Let $\sigma_{ij}\in \fS_2\times\fS_2$ be the transposition interchanging $x_i$ and $x_j$ in $(x_1,x_2,x_3,x_4)\in S^4$. At a point $p$ of
$\hat{\Delta}_{13}\cap\hat{\Delta}_{24}$ the trivial eigenspace in $T_p[\widehat{S\times S}\times \widehat{S\times S}]$ of each of $\sigma_{12}$ and $\sigma_{34}$ has codimension $1$, and the kernel of $d(q\times q)$ is the $2$-dimensional $-1$-eigenspace of the diagonal involution $\sigma_{12}\sigma_{34}$. The latter interchanges $T_p\hat{\Delta}_{13}$ and
$T_p\hat{\Delta}_{24}$, and so  if $\xi\in T_p\hat{\Delta}_{13}$ is a $-1$-eigenvector of $\sigma_{12}\sigma_{34}$,
then $\xi$ belongs to $T_p\hat{\Delta}_{13}\cap T_p\hat{\Delta}_{24}$. The computation in local coordinates in
Section <ref> shows that
$\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ intersect transversally along $\hat{D}\times_S \hat{D}\setminus [\Delta_{\widehat{S\times S}}]$. Hence,
at a point $p$ of the latter, $T_p\hat{\Delta}_{13}\cap T_p\hat{\Delta}_{24}=T_p[\hat{D}\times_S\hat{D}]$ is contained in the $1$-eigenspace of $\sigma_{12}\sigma_{34}$ and so $d(q\times q)$ restricts to
$T_p[\hat{D}\times_S\hat{D}]$ as an injective homomorphism.
along $[D\times_S D]\setminus \Delta_{S^{[2]}}$.
The diagonal involution interchanges $\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$.
We conclude that both $\hat{\Delta}_{13}$ and $\hat{\Delta}_{24}$ are isomorphic to the normalization $\tilde{I}$ of $I$ and  $q\times q$ restricts to each as the normalization morphism $\nu:\tilde{I}\rightarrow I$.

The linearization $\bar{\rho}_\boxtimes$  induced by $\rho_\boxtimes$ on
$\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}}$ is the usual one,

\[
\bar{\rho}_{\boxtimes,g}:\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}}\rightarrow g^*\left[\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}}\right]=\StructureSheaf{\hat{\Delta}_{24}}\oplus \StructureSheaf{\hat{\Delta}_{13}},
\]

for $g$ the involution in $\fS_{2,\Delta}$.
On the other hand, the linearization
on $\StructureSheaf{\hat{\Delta}_{13}\cap \hat{\Delta}_{24}}$ induced by $\rho_{\boxtimes}$ is the sign character $\chi$,  so that  the difference homomorphism $\hat{\beta}$ in (<ref>) is $\fS_{2,\Delta}$-equivariant.

\[
(q\times q)_*(\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}})\cong
\nu_*\StructureSheaf{\tilde{I}}\oplus (\nu_*\StructureSheaf{\tilde{I}}\otimes_\CC\chi)
\]

we get the isomorphisms

\begin{eqnarray}
\nonumber
\nu_*\StructureSheaf{\tilde{I}} &\cong &
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}},\bar{\rho}_\boxtimes),
\\
\label{eq-pushforward-of-structure-sheaf-of-normalization-isomorphic-toequivariant-pushforward-with-chi-turned-on}
\nu_*\StructureSheaf{\tilde{I}} &\cong &
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}},\chi\otimes \bar{\rho}_\boxtimes).
%R(q\times q)_*(\StructureSheaf{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})^{\fS_{2,\Delta}}.
\end{eqnarray}

Set $\widehat{DI}_1:=\hat{\Delta}_{13}\cap [\hat{D}\times \widehat{S\times S}]$ and  $\widehat{DI}_2:=\hat{\Delta}_{13}\cap  [\widehat{S\times S}\times \hat{D}]$. We have $\widehat{DI}_1\cap \widehat{DI}_2=\hat{D}\times_S\hat{D}$.
$(q\circ\hat{\pi}_i)^*\StructureSheaf{S^{[2]}}(-\delta)$ restricts to $\hat{\Delta}_{13}$ as
$\StructureSheaf{\hat{\Delta}_{13}}(-\widehat{DI}_i)$. Hence, the sum of the restriction homomorphisms yields the surjective homomorphisms

\begin{eqnarray*}
(q\circ\hat{\pi}_1)^*\StructureSheaf{S^{[2]}}(-\delta)\oplus (q\circ\hat{\pi}_2)^*\StructureSheaf{S^{[2]}}(-\delta)
&\rightarrow &\Ideal{\hat{\Delta}_{13},\hat{D}\times_S\hat{D}},
\\
(q\circ\hat{\pi}_1)^*\StructureSheaf{S^{[2]}}(-\delta)\oplus (q\circ\hat{\pi}_2)^*\StructureSheaf{S^{[2]}}(-\delta)
&\rightarrow &\Ideal{\hat{\Delta}_{24},\hat{D}\times_S\hat{D}}.
\end{eqnarray*}

Note the isomorphism

\[
(q\circ\hat{\pi}_1)^*\StructureSheaf{S^{[2]}}(-\delta)\oplus (q\circ\hat{\pi}_2)^*\StructureSheaf{S^{[2]}}(-\delta)\cong L(q\times q)^*(\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\oplus \pi_2^*\StructureSheaf{S^{[2]}}(-\delta)).
\]

The right hand side comes with a natural $\fS_{2,\Delta}$-linearization.
Evaluating the functor $R(q\times q)_*^{\fS_{2,\Delta}}$ on the homomorphism $\hat{\alpha}$ in (<ref>)$\otimes \chi$
we get

\begin{equation}
\label{eq-morphism-that-restricts-to-a}
R_{q\times q, *}^{\fS_{2,\Delta}}\left(\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}},\chi\right)
\rightarrow
\nu_*\StructureSheaf{\tilde{I}}.
\end{equation}

Step 4:
We show next that the above displayed homomorphism factors through a surjective homomorphism

\begin{equation}
\label{eq-bar-alpha-into-ideal}
\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\oplus \pi_2^*\StructureSheaf{S^{[2]}}(-\delta)
%R(q\times q)_*^{\fS_{2,\Delta}}\left(
%(q\circ\hat{\pi}_1)^*\StructureSheaf{S^{[2]}}(-\delta)\oplus (q\circ\hat{\pi}_2)^*\StructureSheaf{S^{[2]}}(-\delta)
\RightArrowOf{a} \nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}.
\end{equation}

The complex  $R_{q\times q, *}\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}$ is isomorphic to the
pushforward $(q\times q)_ *\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}$, which decomposes as the direct sum of four character line subbundles for the $\fS_2\times \fS_2$-action

\[
R_{q\times q, *}\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}\cong
\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus\pi_1^*\StructureSheaf{S^{[2]}}(-\delta) \oplus\pi_2^*\StructureSheaf{S^{[2]}}(-\delta)
\oplus[\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\otimes\pi_2^*\StructureSheaf{S^{[2]}}(-\delta)].
\]

Thus $R_{q\times q, *}^{\fS_{2,\Delta}}\left(\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}},\chi\right)$ is the rank two vector bundle
The homomorphism $\hat{\alpha}$ in (<ref>) restricts to

\[
R(q\times q)_*^{\fS_{2,\Delta}}(\hat{\alpha}):\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\oplus\pi_2^*\StructureSheaf{S^{[2]}}(-\delta)\rightarrow R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}},\chi\otimes \bar{\rho}_\boxtimes)\cong \nu_*\StructureSheaf{I},
\]

whose image is claimed to be $\nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}$.

The exactness of $R(q\times q)_*^{\fS_{2,\Delta}}$ implies that to verify the latter claim
it remains to prove that $R(q\times q)_*^{\fS_{2,\Delta}}(\ker(\hat{\beta})\otimes\chi)$ is isomorphic to $\nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}$, where $\hat{\beta}$ is the homomorphism in (<ref>)$\otimes\chi$.

Note that
$R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{D}\times_S\hat{D}},\chi)=0$ and
$R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{D}\times_S\hat{D}},1)=\StructureSheaf{D\times_S D}$.
Similarly, $R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\Delta_{\widehat{S\times S}}\cap\hat{D}\times_S\hat{D}},\chi)=0$.
The short exact sequence

\[
0\rightarrow \StructureSheaf{\Delta_{\widehat{S\times S}}\cup\hat{D}\times_S\hat{D}}\rightarrow
\StructureSheaf{\Delta_{\widehat{S\times S}}}\oplus \StructureSheaf{\hat{D}\times_S\hat{D}} \rightarrow
\StructureSheaf{\Delta_{\widehat{S\times S}}\cap\hat{D}\times_S\hat{D}}\rightarrow 0
\]

yields the isomorphisms

\begin{eqnarray}
\label{eq-pushforward-of-structure-sheaf-of-intersection-linearized-by-chi}
\mbox{\hspace{2ex}}
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}},\chi)&\cong&
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\Delta_{\widehat{S\times S}}},\chi)\cong
\StructureSheaf{\Delta_{S^{[2]}}}\otimes\pi_1^*\StructureSheaf{S^{[2]}}(-\delta),
\\
\label{eq-pushforward-of-structure-sheaf-of-intersection-linearized-by-1}
\mbox{\hspace{2ex}} R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}},1) &\cong& \StructureSheaf{[\Delta_{S^{[2]}}\cup D\times_SD]}.
\end{eqnarray}

The kernel of

\[
\nu_*\StructureSheaf{\tilde{I}}
\stackrel{(\ref{eq-pushforward-of-structure-sheaf-of-normalization-isomorphic-toequivariant-pushforward-with-chi-turned-on})}{\cong}
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}},\chi\otimes\bar{\rho}_\boxtimes)
\RightArrowOf{R(q\times q)_*^{\fS_{2,\Delta}}(\hat{\beta})}
R(q\times q)_*^{\fS_{2,\Delta}}(\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}},1)\cong
\StructureSheaf{[\Delta_{S^{[2]}}\cup D\times_SD]}
\]

is equal to the image $\nu_*\Ideal{\tilde{I},\hat{D}\times_S\hat{D}}$ of $R(q\times q)_*^{\fS_{2,\Delta}}(\hat{\alpha})$. Indeed, if $f$ is a local function on $\hat{\Delta}_{13}$ then the local anti-invariant section
$(f,-\sigma_{12}^*\sigma_{34}^*(f))$ of $\StructureSheaf{\hat{\Delta}_{13}}\oplus \StructureSheaf{\hat{\Delta}_{24}}$
is mapped via the difference homomorphism $\hat{\beta}$ to the restriction of $f+\sigma_{12}^*\sigma_{34}^*(f)$ to $\hat{D}\times_S\hat{D}\cup \Delta_{\widehat{S\times S}}$. If, furthermore, $f$ is in the kernel of
$R(q\times q)_*(\hat{\beta})$ then the restriction of $f$ to $\Delta_{\widehat{S\times S}}$ is anti-invariant and its restriction to
$\hat{D}\times_S\hat{D}$ vanishes. This completes the proof that the morphism (<ref>) factors through the sheaf homomorphism $a$ in (<ref>).

The construction of the isomorphism (<ref>)
is thus complete.

Proof of part (<ref>):
Evaluating the functor $R(q\times q)_*^{\fS_{2,\Delta}}$ on the short exact sequence (<ref>)
we get

\[
\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus
\LongRightArrowOf{(q\times q)_ *^{\fS_{2,\Delta}}(\hat{\alpha})}
\nu_*\StructureSheaf{\tilde{I}}
\LongRightArrowOf{(q\times q)_ *^{\fS_{2,\Delta}}(\hat{\beta})}
\StructureSheaf{\Delta_{S^{[2]}}}\otimes\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)
\rightarrow 0,
%R_{q\times q, *}^{\fS_{2,\Delta}}\left(\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}},\chi\right)
\]

where we used the isomorphism

Now, $(q\times q)_ *^{\fS_{2,\Delta}}(\hat{\alpha})$ factors through the surjective homomorphism $a_\chi$ onto

* The kernel $\P^{[2]}_\chi$ in $D^b(S^{[2]}\times S^{[2]})$ of $\Phi_{\P,\chi}^{[2]}$ is the tensor product of $\pi_1^*\StructureSheaf{S^{[2]}}(\delta)$
with the pushforward to $S^{[2]}\times S^{[2]}$ of the ideal sheaf in
the quotient  $\widehat{S\times S}\times \widehat{S\times S}/\fS_{2,\Delta}$ of the quotient

* The sheaves $E_k$ and $F_k$ in Proposition
<ref> have the following geometric interpretation.
Set $X:=\widehat{S\times S}\times \widehat{S\times S}/\fS_{2,\Delta}$ and $I':=(\hat{\Delta}_{13}\cup\hat{\Delta}_{24})/\fS_{2,\Delta}$.
Let $\tilde{\pi}_i$ be the two projections from $X$ to $S^{[2]}$, $i=1,2$. Set $Z:=g(F(C))$.
Then $(\tilde{\pi}_1,\tilde{\pi}_2)_*\StructureSheaf{X}$ is $\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus
We have

\begin{eqnarray*}
\tilde{\pi}_{2,*}(\tilde{\pi}_1^*g_*\StructureSheaf{F(C)}(k))&\cong&
\pi_{2,*}\left[(\tilde{\pi}_1,\tilde{\pi}_2)_*\tilde{\pi}_1^*g_*\StructureSheaf{F(C)}(k)
\right]\cong
\pi_{2,*}\left[\pi_1^*g_*\StructureSheaf{F(C)}(k)\otimes
\left\{
\right\}\right]
\\
\pi_{2,*}\left[\pi_1^*g_*\StructureSheaf{F(C)}(k)\otimes
\left\{
\StructureSheaf{S^{[2]}\times S^{[2]}}\oplus
\right\}\right].
\end{eqnarray*}

Thus, the graded sheaf of algebras

\[
\A:=\bigoplus_{k\geq 0}\left\{
\oplus
\right\}
\]

over $S^{[2]}$
has relative Proj the subscheme $\tilde{\pi}_1^{-1}(Z)$ of $X$.
The $k+1$ direct summand of $\A$ appears in (<ref>) due to the isomorphism $\StructureSheaf{Z}(1)\cong\StructureSheaf{Z}(\delta)$ established in Lemma <ref> below.
The restriction of $\tilde{\pi}_1\times\tilde{\pi}_2$ to $I'$ is an isomorphism of the latter onto $I$. Hence, $(Z\times S^{[2]})\cap I$ is isomorphic to $\tilde{\pi}^{-1}(Z)\cap I'$.
Let $p_i:(Z\times S^{[2]})\cap I\rightarrow S^{[2]}$ be the restriction of $\pi_i$.  The line bundle $p_1^*\StructureSheaf{Z}(\delta)$ corresponds to the graded $\A$-module
$\oplus_{k>0}[p_{2,*}p_1^*\StructureSheaf{Z}(k)(\delta)]$, which is isomorphic to
$\oplus_{k\geq 0}F_{k+1}$, and $\oplus_{k\geq 0}E_{k+1}$ is a graded $\A$-module corresponding to the ideal of $(Z\times S^{[2]})\cap I\cong \tilde{\pi}^{-1}(Z)\cap I'$ in $\tilde{\pi}_1^{-1}(Z)$.

* $\Phi_\P$ induces an involution of the Mukai lattice $\tilde{H}(S,\Integers)$. Hence, both $\Phi_\P^{[2]}$
and $\Phi_{\P,\chi}^{[2]}$ induce involutions of
$\tilde{H}(S^{[2]},\QQ)$. This is equivalent to the equality $ch(\P^{[2]})=\tau^*(ch((\P^{[2]})^\vee))$ and
$ch(\P^{[2]}_\chi)=\tau^*(ch((\P^{[2]}_\chi)^\vee))$. In particular,

\[
ch(\P^{[2]}\oplus \P^{[2]}_\chi)=\tau^*(ch((\P^{[2]}\oplus \P^{[2]}_\chi)^\vee)).
\]

Let us verify the above equality.
The direct sum $\P^{[2]}\oplus \P^{[2]}_\chi$ is isomorphic to $\pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})$, and so the equality above is equivalent to

\[
\pi_1^*\StructureSheaf{S^{[2]}}(\delta)\otimes R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})
\right\}
\cong
\tau^*ch\left\{
\pi_1^*\StructureSheaf{S^{[2]}}(-\delta)\otimes \left[R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})\right]^\vee
\right\}.
\]

The $\tau$-invariance of $R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})$ implies that latter is equivalent to

\begin{equation}
\label{eq-involution-boils-down-to}
R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})\otimes \left\{\StructureSheaf{S^{[2]}}(\delta)\boxtimes \StructureSheaf{S^{[2]}}(\delta)\right\}
\right\}
\cong
\left[R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})\right]^\vee
\right\}.
\end{equation}

We have seen above that $\StructureSheaf{\hat{\Delta}_{13}}\cong L(b\times b)^*\StructureSheaf{\Delta_{13}}$ and
$\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}}\cong \StructureSheaf{\hat{\Delta}_{13}}\stackrel{L}{\otimes} \StructureSheaf{\hat{\Delta}_{24}}$. Hence, the derived dual of
each of $\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}}$, $\StructureSheaf{\hat{\Delta}_{13}}$, $\StructureSheaf{\hat{\Delta}_{24}}$, and $\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}}$ is isomorphic to an even shift of itself. Now,
$ch(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})=ch(\StructureSheaf{\widehat{S\times S}\times \widehat{S\times S}})-
ch(\StructureSheaf{\hat{\Delta}_{13}})-ch(\StructureSheaf{\hat{\Delta}_{24}})+ch(\StructureSheaf{\hat{\Delta}_{13}\cap\hat{\Delta}_{24}})$.  Thus, $ch(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}}^\vee)=ch(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})$.

We have $\omega_{q\times q}\cong (q\times q)^*\left(\StructureSheaf{S^{[2]}}(\delta)\boxtimes \StructureSheaf{S^{[2]}}(\delta)\right)$. Hence,

\[
\left[R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})\right]^\vee\cong
R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}}^\vee\otimes \omega_{q\times q})\cong
R(q\times q)_*(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}}^\vee)\otimes \{\StructureSheaf{S^{[2]}}(\delta)\boxtimes \StructureSheaf{S^{[2]}}(\delta)\},
\]

where the first isomorphism is due to Grothedieck-Verdier duality and the second due to the projection formula. The self duality of $ch(\Ideal{\hat{\Delta}_{13}\cup\hat{\Delta}_{24}})$   yields the desired Equality (<ref>).

§.§ Reflexivity of the Fourier-Mukai image of $\StructureSheaf{Z}(k)$

Let $\LB$ be a line bundle over $S^{[2]}$.

Let $\pi_i$ be the projection from $S^{[2]}\times S^{[2]}$ onto the $i$-th factor, $i=1,2$.
Let $Z$ be a two-dimensional non-singular subvariety of $S^{[2]}$.  Denote by $e:Z\rightarrow S^{[2]}$ the inclusion.

Assume that the following three conditions hold.

The projection $\pi_2$ restricts to the intersection $(Z\times S^{[2]})\cap I$ as a finite and surjective morphism onto $S^{[2]}$

\begin{equation}
\label{eq-restriction-of-pi-2-to-4-dimensional-intersection}
\pi_2:(Z\times S^{[2]})\cap I\rightarrow S^{[2]}.
\end{equation}

$H^i(Z,e^*\LB)=0$ and $H^i(Z,e^*(\LB(\delta)))=0$, for $i>0$.

Let $d$ be the maximal length of fibers of the finite morphism (<ref>).
The homomorphism $H^0(Z,e^*(\LB(\delta)))\rightarrow H^0(A,\restricted{\LB(\delta)}{A})$ is surjective, for every zero-dimensional subscheme $A$ of $Z$ of length $\leq d$.

Then $\Phi^{[2]}_{\P,\chi}(e_*e^*\LB)$  is represented by a torsion free reflexive sheaf.

Note that

is represented by the complex

\begin{equation}
\label{eq-complex-of-locally-free-sheaves-with-locally-free-kernel}
\oplus
\RightArrowOf{a_\chi} \pi_{2,*}\left(\pi_1^*(\LB(\delta))\otimes\StructureSheaf{(Z\times S^{[2]})\cap I}\right),
\end{equation}

by Lemma <ref> and
assumptions (<ref>) and (<ref>). Furthermore, the homomorphism $a_\chi$ is surjective, by assumption (<ref>), and so $\Phi^{[2]}_{\P,\chi}(e_*e^*\LB)$ is represented by the sheaf $\ker(a_\chi)$. Each of the irreducible components of $(Z\times S^{[2]})\cap I$ is $4$-dimensional, since it has dimension $\geq 4$ and is finite over $S^{[2]}$.  Hence, the sheaf $\pi_{2,*}\left(\pi_1^*(\LB(\delta))\otimes\StructureSheaf{(Z\times S^{[2]})\cap I}\right)$ is torsion free and so $\ker(a_\chi)$ is reflexive, being a saturated subsheaf of a locally free sheaf.

* The sheaf $\ker(a_\chi)$ is locally free away from a closed subscheme of $S^{[2]}$ of co-dimension
$\geq 3$ as it is reflexive. We claim that this closed subscheme of singularities of $\ker(a_\chi)$ is contained in $Z$. It suffices to show that the morphism
(<ref>) is flat over $S^{[2]}\setminus Z$, since then the sheaf
$F:=\pi_{2,*}\left(\pi_1^*(\LB(\delta))\otimes\StructureSheaf{(Z\times S^{[2]})\cap I}\right)$ is locally free over $S^{[2]}\setminus Z$ and the kernel of a surjective homomorphism of locally free sheaves is locally free as well.
The length of the fiber of (<ref>) over $y\in S^{[2]}$ is equal to
the length of the intersection subscheme $(Z\times S^{[2]})\cap I\cap (S^{[2]}\times\{y\})$ of $S^{[2]}\times S^{[2]}$, which is independent of $y$ as long as it is of the expected dimension $0$ and
the object $\StructureSheaf{Z\times S^{[2]}}\stackrel{L}{\otimes}\StructureSheaf{I}\stackrel{L}{\otimes}\StructureSheaf{S^{[2]}\times\{y\}}$ is represented by the tensor product sheaf, i.e., all the higher torsion sheaves vanish.
This is the case if the intersection  $(Z\times S^{[2]})\cap I\cap (S^{[2]}\times\{y\})$  is contained in the non-singular locus of $I$.
Indeed, in that case each of $Z\times S^{[2]}$, $I$, and $S^{[2]}\times\{y\}$ is non singular at every point $p$ of their common intersection and
their local equations at $p$ yield a regular sequence for the subscheme $(Z\times S^{[2]})\cap I\cap (S^{[2]}\times\{y\})$ locally around $p$. The Koszul complex associated to that regular sequence is exact on the one hand, and is isomorphic to the tensor product of the three Koszul complexes associated to the local equations of each of the three subvarieties. Hence, the higher torsion sheaves vanish, and the intersection multiplicity at each point $p$ is equal to the length of the intersection subscheme at $p$.
We have seen in Step 3 of the proof of Lemma <ref> that the singular locus of $I$ is the diagonal $\Delta_{S^{[2]}}$. Hence,
if $y$ does not belong to $Z$ the fiber over $y$ is contained in the non-singular locus of $I$.

The morphism (<ref>) is thus flat over $S^{[2]}\setminus Z$, by <cit.>.

* The sheaf $F$ is not locally free over $Z$, hence not reflexive. Indeed, if $y\in Z$ and $y$ is reduced, then the intersection $I\cap (S^{[2]}\times\{y\})$ consists of two smooth surfaces in $S^{[2]}\times\{y\}$ meeting at $(y,y)$, and so the intersection of $I\cap (S^{[2]}\times\{y\})$ with
$Z\times \{y\}$ is a subscheme with stalk at $(y,y)$ of length at least $3$ (while the intersection multiplicity is $2$ if $Z$ is transversal at $y$ to each of the two surfaces), by
<cit.>. Nevertheless, $\ker(a_\chi)$ is locally free over the generic point of $Z$, being reflexive.

The morphism $g$, given in (<ref>), is a closed immersion.
$Z:=g(F(C))$ is a lagrangian surface in $S^{[2]}$ and  the projection $\pi_2$ restricts to $(Z\times S^{[2]})\cap I$ as a finite and surjective morphism of degree $12$ onto $S^{[2]}$.

Let $q(x_0, \dots, x_4)$ and $c(x_0, \dots,x_4)$ be homogeneous quadratic and cubic forms with zero divisors $Q$ and $C$ respectively satisfying Assumption <ref>.

\[
d(x_0, \dots, x_4,x_5):=x_5q(x_0, \dots, x_4)+c(x_0, \dots,x_4)
\]

and let $X$ be the corresponding cubic fourfold in $\PP^5$.
Then $X$ has a single ordinary double point at $p:=(0\!:\!0\!:\!0\!:\!0\!:\!0\!:\!1)$.
The Fano variety $F(X)$ of lines on $X$ is birational to $S^{[2]}$, as observed in
Let $F_0(X)\subset F(X)$ be the open subset of lines not passing through $p$.
Let $A\subset S^{[2]}$ the the divisor of subschemes $y$, such that the line $\bar{\ell}_y$ in $\PP^4$ is contained in $Q$.
Note that $A$ admits a morphism of degree $3$ onto the Fano variety $F(Q)$ of lines on $Q$, and that the latter is isomorphic to $\PP^3$. Note also that $A$ admits a morphism

\begin{equation}
\label{eq-phi-from-A-to-S}
\phi:A\rightarrow S,
\end{equation}

by sending a subscheme $y\in A$ to the point of $\bar{\ell}_y\cap C$ complementary to $y$. In fact, $\phi:A\rightarrow S$ is a conic bundle, which is the restriction to $S$ of the following conic bundle $\tilde{\phi}:\tilde{A}\rightarrow Q$. Given a point $a\in Q$, let $a^\perp$ be the hyperplane in $\CC^5$ orthogonal to $a$ with respect to $q$ and denote by $q_a$ the non-degenerate quadratic form on $a^\perp/a$ induced by $q$.
Then the fiber of $\tilde{\phi}$ over $a\in Q$ is the conic in the plane $\PP(a^\perp/a)$ defined by $q_a$, as the latter conic parametrizes lines on $Q$ through $a$.
Let $S^{[2]}_0$ be the complement of $A$.

Let $\iota:S\rightarrow F(X)$ be the embedding of $S$ as the singular locus of $F(X)$, sending $q\in S$ to the line $\overline{pq}$ through $p$ and $q$.
The following is a slightly more detailed version of <cit.>.

There exists a canonical morphism $f:S^{[2]}\rightarrow F(X)$, which restricts to an
isomorphism $f:S^{[2]}_0\rightarrow F_0(X)$.
Furthermore, $f$ restricts to an isomorphism from $Z$ onto $F(C)$, which is the inverse of $g$, and
the following diagram is commutative

\[
\xymatrix{
S^{[2]} \ar[r]^{f} & F(X)
\\
A \ar[u]_{\cup} \ar[r]_{\phi} & S. \ar[u]_{\iota}
\]

Let $\hat{Q}$ and $\hat{C}$ be the cones over $Q$ and $C$ given by $q$ and $c$ considered as forms in $x_0$, …, $x_5$. A line on $X$ passes through $p$, if and only if it is contained in both $\hat{Q}$ and $\hat{C}$.
Indeed, if $\ell$ passes through $p$ then a general point of $\ell$ is of the form
$(sa_0\!:\!sa_1\!:\!sa_2\!:\!sa_3\!:\!sa_4\!:\!t)$, $(s\!:\!t)\in\PP^1$,
so that $d(sa_0,sa_1,sa_2,sa_3,sa_4,t)=ts^2q(a_0, \dots, a_4)+s^3c(a_0, \dots, a_4)$ vanishes for all $(s\!:\!t)$, so that
$a:=(a_0\!:\!\cdots \!:\!a_4)$ is a point of $S$. Conversely, if $\ell$ is contained in $\hat{Q}\cap \hat{C}$, then it must pass through $p$, since otherwise it would project to a line in $S$ contrary to our assumption that $S$ does not contain any lines.

Let $H$ be the hyperplane $x_5=0$.
Given a point $y$ of $S^{[2]}$, we get the line $\bar{\ell}_y$ in $H$ intersecting $S$ along a subscheme containing $y$,[The scheme $\bar{\ell}_y\cap S$ is equal to $y$, if and only if $y$ does not belong to $A$. If $y$ belongs to $A$, then $y$ is a proper subscheme of $\bar{\ell}_y\cap S$, as the latter is equal to $\bar{\ell}_y\cap C$ and so has length $3$.] hence we get the plane $P_y$ in $\PP^5$ spanned by $p$ and $\bar{\ell}_y$. $P_y$ is not contained in $X$, since otherwise it would be contained in $\hat{Q}\cap\hat{C}$, by the above discussion, and so project to a line in $S$.
If $y$ consists of two distinct points $a$ and $b$ of $S$, then $P_y$ intersects $X$ along three lines, two of them are $\overline{pa}$ and $\overline{pb}$, and we set $f(y)$ to be the third line.
If $y$ is non-reduced supported set theoretically on a point $a$ of $S$,
then $P_y$ intersects $X$ along $2\overline{pa}+f(y)$ for a line $f(y)$ in $P_y$.
We claim that $f(y)$ passes through $p$, if and only if $y$ belongs to $A$. Indeed, let
$a=(a_0\!:\!a_1\!:\!a_2\!:\!a_3\!:\!a_4\!:\!0)$, $b=(b_0\!:\!b_1\!:\!b_2\!:\!b_3\!:\!b_4\!:\!0)$, and choose coordinates $(s\!:\!t\!:\!u)$ for $P_y$ via

\[
\eta(s\!:\!t\!:\!u)= s(a_0,a_1,a_2,a_3,a_4,0)+t(b_0,b_1,b_2,b_3,b_4,0)+(0,0,0,0,0,u).
\]

Then the cubic form $d$ pulls back to

\[
\]

where the symmetric bilinear form $q(\bullet,\bullet)$ is the polarization of $q$ and the symmetric trilinear form $c(\bullet,\bullet,\bullet)$ is that of $c$.
The morphism $\eta$ maps
the line $s=0$ to $\overline{bp}$, the line $t=0$  to $\overline{ap}$, and
the line cut out by $2q(a,b)u+3c(a,a,b)s+3c(a,b,b)t$ to $f(y)$. Note that $q(a,b)= 0$ if and only if the line $\ell_y$ is contained in $Q$. Now, $\eta(0\!:\!0\!:\!1)=p$, so $f(y)$ passes through $p$ if and only if $q(a,b)= 0$.
A similar argument shows that $f(y)$ passes through $p$ if and only if $y$ belongs to $A$ also in the case where $y$ is non-reduced. The commutativity of the diagram in the statement of the claim follows.

If a line $\ell$ on $X$ is contained in $\hat{Q}$, then it is contained in $\hat{C}$ and so passes through $p$. Hence, if a line $\ell$ on $X$ does not pass through $p$, then it intersects $\hat{Q}$ along a length $2$ subscheme.
Thus, the projection $\bar{\ell}$ of $\ell$ from $p$ intersects $Q$ along a length $2$ subscheme $\tilde{g}(\ell)\in S_0^{[2]}$.
We get a morphism $\tilde{g}:F_0(X)\rightarrow S_0^{[2]}$, which is clearly the inverse of the restriction of $f$ to $S_0^{[2]}$.
We conclude that $f:S_0^{[2]}\rightarrow F_0(X)$ is an isomorphism. This completes the proof of Claim <ref>.

The Fano variety of lines $F(C)$  is identified with $F(X\cap H)$.
The point $p$ does not belong to $X\cap H$, and so $F(X\cap H)$ is contained in $F_0(X)$.
The  subvariety $F(X\cap H)$ of $F(X)$ is lagrangian, by [83]. Hence, $Z:=\tilde{g}(F(X\cap H))$
is a lagrangian surface in $S^{[2]}$.
Through a generic point of $C$ pass $6$ lines lying on $C$, by [17], and the set of lines on $C$ through every point of $C$ is finite, by Assumption <ref>.
The fiber of the restriction of $\pi_2$ to $Z\times S^{[2]}\cap I$ over a point $y\in S^{[2]}$
consists of length $2$ subschemes of $S$ lying on a line passing through one of the two points of $y$ and is hence finite. If $y$ consists of two distinct points $a$, $b$ through each of which pass exactly $6$ lines on $C$, and such that the line $\overline{ab}$ does not lie on $C$,
the fiber of

\begin{equation}
\label{eq-finite-flat-projective-morphism}
\pi_2:(Z\times S^{[2]})\cap I\rightarrow S^{[2]}
\end{equation}

over $y$ consists of precisely $12$ points, as
none of the $12$  lines on $C$ through $a$ or $b$  lies on $Q$, since $S$ does not contain a line.
The morphism (<ref>) is  projective and has finite fibers, hence is a finite morphism (see <cit.>). The morphism (<ref>) is  clearly surjective.
This completes the proof of Lemma <ref>.

§.§ Proof of Proposition <ref>

Let $h=c_1(\StructureSheaf{S}(1))$ be the class of a hyperplane section of $S$ in $\PP^4$.
Let $\tilde{h}$ be the nef class of $\Pic(S^{[2]})$ corresponding to $h$ via (<ref>).
Let $\delta\in \Pic(S^{[2]})$ be half the class of the divisor of non-reduced subschemes.

The class of the divisor $A$ is $[A]=\tilde{h}-2\delta$.

The pullback via $f$ of
$\StructureSheaf{F(X)}(1)$ is $\lambda:=2\tilde{h}-3\delta$.

If $\Pic(S)=\Integers h$, then
the nef cone of $S^{[2]}$ is $\langle\lambda,\tilde{h}\rangle.$

The restrictions of $\lambda$ and $\delta$ to $F(C)$ are equal.

We may and do assume that $\Pic(S)$ is cyclic generated by $h$, as the parts of the lemma not assuming it follow from this case by continuity.
The  contraction $f:S^{[2]}\rightarrow F(X)$ of the divisor $A$ in Claim <ref> admits the following modular interpretation. The divisor $A$ consists precisely of those  subschemes $y$, such that $I_y(1)$ is not generated by global sections. Indeed, $y$ belongs to $A$, if and only if $\bar{\ell}_y$ intersects $S$ in a length $3$ subscheme. Every hyperplane section of $S$ containing $y$ contains $S\cap\bar{\ell}_y$, so
$H^0(S,I_y(1))$ generates the subsheaf $I_{\bar{\ell}_y\cap S}(1)$ of $I_y(1)$.
Let $M(2,h,1)$ be the moduli space of $h$-stable sheaves on $S$ with Mukai vector $(2,h,1)$.
There is a natural isomophism $\eta:M(1,h,2)\rightarrow M(2,h,1)$, by
<cit.> and the fact that
the moduli space $M(2,h,3)$ is empty. $M(1,h,2)$ parametrizes sheaves of the form $I_y(1)$, where $y$ is a length $2$ subscheme of $S$, and is thus naturally isomorphic to $S^{[2]}$.
The divisor $\eta(A)$ is precisely the divisor of non-locally-free sheaves in
$M(2,h,1)$, by construction of $\eta$. Explicitly, given $y\in A$, let $ev:H^0(I_y(1))\otimes_\CC\StructureSheaf{S}\rightarrow I_y(1)$ be the evaluation homomorphism. Then $\ker(ev)$ belongs to $M(2,-h,2)$, which is zero dimensional, so $\ker(ev)$ is isomorphic to the unique rigid and stable vector bundle $E$ with Mukai vector $(2,-h,2)$. The torsion free sheaf
$\eta(I_y(1))$ is a subsheaf of $E^*$ and the quotient $E^*/\eta(I_y(1))$ is supported on the point $\phi(y)$ of $S$ complementary to the subscheme $y$ in $\bar{\ell}_y\cap S$. We get the commutative diagram

\[
\xymatrix{
A\ar[r]^{\restricted{\eta}{A}} \ar[d]_\phi &\PP(E^*)\ar[d]
\\
\]

where $\phi$ is given in (<ref>) and the right vertical arrow is the natural projection.

The contraction $f:S^{[2]}\rightarrow F(X)$ is identified with the
Gieseker-Li morphism from $M(2,h,1)$ to the normalization of the
Uhlenbeck-Yau compactification [42].
Set $v:=(1,h,2)$ and $w=(2,h,1)$.
The divisor $A$ corresponds to the image of the Mukai vector $(2,h,2)$
in $H^2(M(w),\Integers)$ via the Mukai Hodge isometry $\theta_w:w^\perp\rightarrow H^2(M(w),\Integers)$
between the latter and the sublattice $w^\perp$ of the Mukai lattice of $S$ orthogonal to $w$, by <cit.>.
The isometry $\eta_*:H^2(M(v),\Integers)\rightarrow H^2(M(w),\Integers)$
is the image $\monrep(\gamma)$ via the parallel-transport representation $\monrep$ in <cit.>
of the isometry $\gamma:=-D\circ R_{(1,0,1)}$ of the Mukai lattice, where $D(r,c,s)=(r,-c,s)$ and $R_{(1,0,1)}$ is the reflection with respect to the sublattice orthogonal to $(1,0,1)$ <cit.>.
Now, $\monrep(\gamma)$
corresponds, via the Mukai isometries $\theta_v$ and $\theta_w$, to the isometry $-\gamma_|:v^\perp\rightarrow w^\perp$, which is minus
the restriction of
the isometry $\gamma$.
The isometry $\gamma$ sends $v$ to $w$ and sends the class $(2,h,2)$ to itself. Hence, the class of $A$ in
$H^2(M(v),\Integers)$ is $[A]=-\theta_v((2,h,2))$.

The class of the divisor  of non-reduced subschemes in $M(v)$ is  $2\delta$, where $\delta:=\theta_v((1,h,4))$.
The nef class $\tilde{h}$ orthogonal to $\delta$, corresponding to the class $h\in H^2(S,\Integers)$,  is
$\tilde{h}=\theta_v((0,h,6))$. The equality $[A]=\tilde{h}-2\delta$ follows.

(<ref>), (<ref>)
The extremal rays associated to the two divisorial contractions correspond via the isomorphism $H^2(M(v),\QQ)\rightarrow H_2(M(v),\QQ)$, $\alpha\mapsto (\alpha,\bullet)$, to the rays spanned by the classes $[A]$ and $\delta$ (see <cit.>).
The nef cone of $M(v)$ is thus dual to the cone generated by the classes  $[A]$ and $\delta$. $F(X)$ is a subvariety of $Gr(2,6)$
and $\StructureSheaf{Gr(2,6)}(1)$ pulls back via $f$ to a nef class $\lambda$ satisfying $(\lambda,\lambda)=6$ and $\lambda$ is orthogonal to $[A]$.
Hence $\lambda$ corresponds to $-\theta_v((3,h,0))$. The equality $\lambda=2\tilde{h}-3\delta$ follows.
The nef cone is thus $\langle
\lambda,\tilde{h}
\rangle$.

The restriction of $\lambda$ and of $\delta$ to $F(C)$
are equal, as $\lambda=2[A]+\delta$ and $A$ is disjoint from $F(C)$.

We are ready to prove Proposition <ref>.

The surface $g(F(C))$ is smooth and lagrangian, by Lemma <ref>. Let $L$ be the line bundle over $S^{[2]}$ with class $\lambda$, so that $L=f^*\StructureSheaf{F(X)}(1)$, by Lemma <ref>. The ampleness of the restriction of $L$ to $g(F(C))$ and
Lemma <ref> imply that
the object  $\Phi^{[2]}_\P(L^k\otimes g_*\StructureSheaf{F(C)})$
satisfies the three conditions of Lemma <ref>,
with $\LB=L^k$, for all $k$ sufficiently large,  and
is hence isomorphic to a torsion free reflexive sheaf.

The pair  $(S^{[2]},g_*\StructureSheaf{F(C)})$ is the limit of a flat family of pairs $(F(X_t),\StructureSheaf{F(X_t\cap H_t)})$, where $X_t$ is a smooth cubic fourfold and $X_t\cap H_t$ is a smooth hyperplane section, as $S^{[2]}$ is the minimal resolution of an $A_1$ singularity along the embedding of $S$ as the singular locus of $F(X)$, by Claim <ref>. Such a flat family is obtained from a flat deformation of $F(X)$ after passage to a double cover of the base of that family, see [53]. (This was the main point of Mukai's <cit.>, as remarked in <cit.>).
The sheaf $g_*\StructureSheaf{F(C)}$ thus  has a rank $1$ cohomological obstruction map,

by Lemma <ref> and Example <ref>.
$E_k$  has a rank $1$ cohomological obstruction map

as well, as the property is invariant under derived equivalences.

The exact sequence (<ref>) is obtained from
(<ref>), with $\LB=L^k$,
by adding the kernel of the homomorphism $a_\chi$ and using the equality of the restrictions  to $Z=g(F(C))$ of $\delta$ and of $L$

established in Lemma <ref>.

The rank of $F_k$ is 12, by Lemma <ref>.
The rank of $E_k$, for $k$ sufficiently large,  is thus

\[
-12+\dim H^0(Z,\StructureSheaf{Z}(k+1))+\dim H^0(Z,\StructureSheaf{Z}(k)).
\]

The multiplication table (<ref>) and
Lemma <ref>, with $(\lambda,\lambda)=6$, $c=5/8$, and $t=1$ as in Example <ref>, yields the third equality below.

\begin{eqnarray*}
\chi(\StructureSheaf{Z}(k))&=&
\int_{S^{[2]}}ch(\StructureSheaf{Z})\exp(k\lambda)td_X
=\chi(\StructureSheaf{Z})+ \frac{k^2}{2}\int_{S^{[2]}} [Z]\lambda^2+k\int_{S^{[2]}} ch_3(\StructureSheaf{Z}))\lambda
\\
&=& 6 +\frac{45k^2}{2}-\frac{45k}{2}=6+45\frac{k(k-1)}{2}.
\end{eqnarray*}

Hence, the rank of $E_k$, for $k$ sufficiently large, is $45k^2$.

It remains to calculate $\ell(E_k)$ and $c_1(E_k)$. The latter is determined by $\rank(E_k)$ and the former, by Theorem <ref>(<ref>).
by Lemma <ref>,
$\ell(\StructureSheaf{Z}(k))=B_{k\lambda}(0,\lambda,-3)=(0,\lambda,6k-3)$, and
$\ell(E_k)$ is the image via $\Phi^{[2]}_{\P,\chi}=\Phi^{[2]}_\chi\circ \Phi^{[2]}_\P$ of $\ell(\StructureSheaf{Z}(k))$.
$\Phi_\P$ acts on the Mukai lattice of $S$ via $\phi_\P(r,\alpha,s)= (s,-\alpha,r)$.
According to <cit.> $\Phi_\P^{[2]}$ acts on the LLV lattice of $S^{[2]}$ by

\[
\widetilde{H}(\phi_\P^{[2]})=\det(\phi_\P)(B_{-\delta/2}\circ \iota(\phi_\P)\circ B_{\delta/2}),
\]

where $\iota(\phi_\P)$ leaves the class $\delta\in H^2(S^{[2]})$ invariant and acts of $\delta^\perp$ by $\phi_\P$ via the natural identification
of $\delta^\perp$ with the LLV lattice of $S$. Note that $\det(\phi_\P)=-1$, $\lambda=2\tilde{h}-3\delta$, $(\delta/2,\lambda)=3$,
$B_{\delta/2}(0,\lambda,6k-3)=(0,\lambda,6k)$, and
$\iota(\phi_\P)(0,\lambda,6k)=(6k,-2\tilde{h}-3\delta,0)$. Hence,

\[
\widetilde{H}(\phi_\P^{[2]})(0,\lambda,6k-3)=-B_{-\delta/2}(6k,-2\tilde{h}-3\delta,0)=-(6k,-2\tilde{h}-(3+3k)\delta,-3-(3/2)k).
\]

Let $\phi^{[2]}_\chi:H^*(S^{[2]},\QQ)\rightarrow H^*(S^{[2]},\QQ)$ be the automorphism induced by $\Phi^{[2]}_\chi$.
Let $\widetilde{H}(\phi^{[2]}_\chi):\widetilde{H}(S^{[2]},\QQ)\rightarrow \widetilde{H}(S^{[2]},\QQ)$ be the induced isometry via
(<ref>), which appeared in Lemma <ref>.
Lemma <ref> yields
the equality $\widetilde{H}(\phi_\chi^{[2]}\circ\phi_\P^{[2]})=-R_{(0,\delta,1)}\circ \widetilde{H}(\phi_\P^{[2]})$,  and so

\[
\widetilde{H}(\phi_\chi^{[2]}\circ\phi_\P^{[2]})(0,\lambda,6k-1)))=\left(6k,-2\tilde{h}-(3k-3)\delta,3-\frac{3}{2}k\right).
\]

Thus, $c_1(E_k)=-15k\tilde{h}-\frac{45k(k-1)}{2}\delta$, by Theorem <ref>(<ref>).

Let $a'_k$ be the restriction of $a_k$ to the first direct summand in (<ref>) and let $a_k''$ be its restriction to the second direct summand.
Then $G'_k:=\ker(a'_k)$ and $G''_k:=\ker(a''_k)$ are saturated subsheaves of $E_k$ and $G''_k\cong G'_{k-1}(-\delta)$. Let us determine the open subcone of the ample cone of $S^{[2]}$, where $G'_k$ and $G''_k$ do not slope-destabilize $E_k$, when
$\Pic(S)$ is cyclic. In that case the nef cone of $S^{[2]}$ is $\langle\tilde{h},2\tilde{h}-3\delta\rangle$, by Lemma <ref>. Hence, the class $\kappa:=j\tilde{h}-\delta$, $j\in\QQ$, is ample precisely when $j>2/3$ and every rational ample class is a multiple of such a $\kappa$.
The normalized $\kappa$-slope of a sheaf $F$ on $S^{[2]}$ of positive rank $r$ is
$\mu_\kappa(F):=(\kappa,c_1(F))/r$. The equality

\[
\int_{S^{[2]}}\kappa^3c_1(F)=3(\kappa,\kappa)(\kappa,c_1(F))
\]

implies that the  slope of $F$ is $3(\kappa,\kappa)\mu_\kappa(F)$, and is hence proportional to $\mu_\kappa(F)$ by a positive constant independent of $F$.
The normalized $\kappa$-slopes with respect to $\kappa$ of the three sheaves are as follows.

\[
\begin{array}{cccc}
\underline{\mbox{sheaf}} & \underline{\mbox{rank}} & \underline{\hspace{10ex}c_1\hspace{10ex}} & \underline{\hspace{8ex}\mu_\kappa\hspace{8ex}}
\\
E_k & 45k^2 & -15 k\tilde{h}-\frac{45k(k-1)}{2}\delta & -\left(\frac{2j+k-1}{k}\right)
\\
G'_k & -6+\frac{45k(k+1)}{2} & -15k\tilde{h}+6\delta & \frac{-60kj+8}{15k(k+1)-4}
\\
G''_k & -6+\frac{45k(k-1)}{2} & -15(k-1)\tilde{h}+\left(12-\frac{45k(k-1)}{2}\right)\delta &
\frac{16-(k-1)[30k+60j]}{15k(k-1)-4}
\end{array}
\]

The inequality $\mu_\kappa(G'_k)>\mu_\kappa(G''_k)$ holds, for $k\geq 2$.
Hence, for

and $k\geq 2$ the normalized $\kappa$-slopes $\mu_\kappa(G'_k)$ and $\mu_\kappa(G''_k)$ are smaller than $\mu_\kappa(E_k)$.

We have the short exact sequence

\begin{equation}
\label{eq-exact-triangle}
0\rightarrow G'_k\rightarrow E_k\RightArrowOf{\eta'} H^0(Z,\StructureSheaf{Z}(k))\otimes_\CC\StructureSheaf{S^{[2]}}(-\delta)\rightarrow 0,
\end{equation}

where the homomorphism $\eta'$

is the natural one arising from the isomorphism
$H^0(Z,\StructureSheaf{Z}(k))\cong \Hom(E_k,\StructureSheaf{S^{[2]}}(-\delta))^*$.
The sheaf $G'_k$ is isomorphic to the image via $\Phi^{[2]}_{\P.\chi}$ of the object

\begin{equation}
\label{eq-preimage-of-G'-k}
\StructureSheaf{Z}(k)\rightarrow \StructureSheaf{S^{[2]}}[4]\otimes_\CC
\Hom\left(\StructureSheaf{Z}(k),\StructureSheaf{S^{[2]}}[4]\right)^*
\end{equation}

in degrees $0$ and $1$.

The exact sequence (<ref>)
follows from the Snake Lemma and the commutative diagram with short exact rows the second of which is (<ref>)

\[
\xymatrix{
0 \ar[r] &  G'_k \ar[r] \ar[d] & H^0(Z,\StructureSheaf{Z}(k+1))\otimes_\CC\StructureSheaf{S^{[2]}} \ar[r]^{a_k'} \ar[d] & F_k \ar[r] \ar[d]_{id}
\\
0\ar[r] & E_k \ar[r] &
\left[
H^0(Z,\StructureSheaf{Z}(k+1))\otimes_\CC \StructureSheaf{S^{[2]}}
\right]
\oplus
\left[
H^0(Z,\StructureSheaf{Z}(k))\otimes_\CC \StructureSheaf{S^{[2]}}(-\delta)
\right]
\ar[r]_-{a_k} & F_k \ar[r] &0.
\]

The exactness of both rows follows from Proposition <ref>.

$E_k$ is isomorphic to
$\Phi^{[2]}_{\P.\chi}(\StructureSheaf{Z}(k))$ and $\StructureSheaf{S^{[2]}}(-\delta)$ is isomorphic to
by Lemma <ref>.
The morphism $\eta'$ is the image via $\Phi^{[2]}_{\P.\chi}$ of the natural morphism in
where we identify $H^0(Z,\StructureSheaf{Z}(k))$ and $\Hom\left(\StructureSheaf{Z}(k),\StructureSheaf{S^{[2]}}[4]\right)^*$
via Serre's duality and the triviality of the canonical line bundle of $S^{[2]}$.

We have the exact sequence

\[
0\rightarrow G''_k\rightarrow E_k \RightArrowOf{\eta''} H^0(Z,\StructureSheaf{Z}(k+1))\otimes_\CC\StructureSheaf{S^{[2]}}
\rightarrow F_k/F_{k-1}\otimes\StructureSheaf{S^{[2]}}(-\delta)\rightarrow 0.
\]

The homomorphism $\eta''$ is the natural one under the identification of $H^0(Z,\StructureSheaf{Z}(k+1))$
with $\Hom(E_k,\StructureSheaf{S^{[2]}})^*$.

The exact sequence is obtained via the Snake Lemma after replacing the first row in the diagram in Lemma <ref> by

\[
0\rightarrow G''_k\rightarrow H^0(Z,\StructureSheaf{Z}(k))\otimes_\CC\StructureSheaf{S^{[2]}}(-\delta)
\rightarrow F_{k-1}\otimes\StructureSheaf{S^{[2]}}(-\delta)\rightarrow 0.
\]

The trivial line bundle is isomorphic to
by Lemma <ref>.
The morphism $\eta''$ is the image via $\Phi^{[2]}_{\P.\chi}$ of the natural homomorphism

\[
\StructureSheaf{Z}(k)\rightarrow \StructureSheaf{S^{[2]}}(-\delta)[4]\otimes_\CC
\Hom(\StructureSheaf{Z}(k),\StructureSheaf{S^{[2]}}(-\delta)[4])^*,
\]

where we identify $H^0(Z,\StructureSheaf{Z}(k+1))$ with $\Hom(\StructureSheaf{Z}(k),\StructureSheaf{S^{[2]}}(-\delta)[4])^*$
via Serre's duality, the triviality of the canonical line bundle of $S^{[2]}$ and Lemma <ref>(<ref>).

§ APPENDIX: NON RIGIDITY OF TENSOR PRODUCTS OF RIGID MODULAR VECTOR BUNDLES

Let $X=S^{[2]}$, for a $K3$ surface $S$. Let $G_i$ be a spherical vector bundle on $S$, $i=1,2$.
Set $F_i=G_i[2]^\pm$, $i=1,2$ and let $r_i$ be the rank of $F_i$.

$\Ext^1(F_1\otimes F_2,F_1\otimes F_2)$ does not vanish, if $r_1>0$ and $r_2>0$.

Let us calculate the Mukai pairing
$(v(F_1\otimes F_2),v(F_1\otimes F_2))$.

We have

\[
\int_Xc_2(X)^2=828, \ \ \ td_X=1+\frac{1}{12}c_2(X)+3[pt],
\ \ \ \sqrt{td_X}=1+\frac{1}{24}c_2(X)+\frac{25}{32}[pt].
\]

If $\kappa(F_1)=r_1+a_1c_2(X)+b_1[pt]$ and $\kappa(F_2)=r_2+a_2c_2(X)+b_2[pt]$, then

\begin{eqnarray*}
w&:=&\kappa(F_1\otimes F_2)\sqrt{td_X}
\\
&=& (r_1r_2+[r_1a_2+r_2a_1]c_2(X)+\left[r_1b_2+r_2b_1+828a_1a_2\right][pt])\sqrt{td_X}
\\
&=& r_1r_2+\left(r_1a_2+r_2a_1+\frac{r_1r_2}{24}\right)c_2(X)+\left([r_1a_2+r_2a_1]\frac{69}{2}+[r_1b_2+r_2b_1+828a_1a_2]+\frac{25}{32}r_1r_2\right)[pt],
\end{eqnarray*}

and, since $\kappa(F)=\kappa(F^*)$ if $\kappa(F)$ is $\bar{\LieAlg{g}}$-invariant (so that $\kappa_3(F)$ vanishes since $H^6(X,\RationalNumbers)$ is an irreducible $\bar{\LieAlg{g}}$-module),

\begin{eqnarray*}
(w,w)&=&-\int_X\kappa(F_1\otimes F_2)^2td_X
\\
\end{eqnarray*}

We have

\[
3=\int_X\kappa(F_i)^2td_X=(2r_ib_i+828a_i^2)+138 r_ia_i+3r_i^2.
\]

Hence, $b_i=\frac{3}{2r_i}\left(1-276a_i^2-46a_ir_i-r_i^2\right)$.
Set $e_i=a_i/r_i$. Then

\[
\frac{b_i}{r_i}=\frac{3}{2}\left(\frac{1}{r_i^2}-276e_i^2-46e_i-1\right).
\]

We get

\begin{equation}
\label{eq-non-rigidity-n=2}
(v(F_1\otimes F_2),v(F_1\otimes F_2))=(w,w)=3r_1^2r_2^2\left(1-\frac{1}{r_1^2}-\frac{1}{r_2^2}\right),
\end{equation}

which is positive if  $r_1>1$ and $r_2>1$.
Serre's duality yields

\[
(w,w)=-2\dim\Hom(F_1\otimes F_2,F_1\otimes F_2)+2\dim\Ext^1(F_1\otimes F_2,F_1\otimes F_2)-\dim\Ext^2(F_1\otimes F_2,F_1\otimes F_2).
\]

Hence, $\Ext^1(F_1\otimes F_2,F_1\otimes F_2)$ does not vanish if $r_1>1$ and $r_2>1$.

Assume that $F_1\otimes F_2$ is simple, i.e., $\dim \End(F_1\otimes F_2)=1$.
A choice of a hyperkähler structure on $X$ determines an action of the algebra $\HH$ of quaternions on $\Ext^1(F_1\otimes F_2,F_1\otimes F_2)$, by <cit.>. In particular, it is even dimensional. Hence,
$(w,w)+2+\dim \Ext^2(F_1\otimes F_2,F_1\otimes F_2)$ is divisible by $4$. Now $r_i=\rho_i^2$, so
0 & \mbox{if} & r_1 \ \mbox{and}\ r_2 \ \mbox{are even}
\\
1 & \mbox{if} & r_1 \ \mbox{or}\ r_2 \ \mbox{is odd}
\end{array}\right.$ (mod $4$), by Equation (<ref>).

\[
\dim \Ext^2(F_1\otimes F_2,F_1\otimes F_2)
\equiv\left\{\begin{array}{ccl}
2 & \mbox{if} & r_1 \ \mbox{and}\ r_2 \ \mbox{are even}
\\
1 & \mbox{if} & r_1 \ \mbox{or}\ r_2 \ \mbox{is odd}
\end{array}\right. \ \ (\mbox{mod} \ 4).
\]

Assume there exists an elliptic fibration $\pi:S\rightarrow \PP^1$ and let  $p:S^{[2]}\rightarrow \PP^2$ the associated lagrangian fibration. Denote by $C_x$ the fiber of $\pi$ over $x\in\PP^1$. Let $x$ and $y$ be two distinct points in $\PP^1$ with $C_y$ and $C_y$ smooth and denote by $x+y$ the  point of $\PP^2$ corresponding to the length two subscheme of $\PP^1$. The restriction of $F_i$ to the fiber $C_x\times C_y$ pf $p$ is $G_{\restricted{i}{C_x}}\boxtimes G_{\restricted{i}{C_y}}$.
The restriction of $F_1\otimes F_2$ is thus $(G_1\otimes G_2\restricted{)}{C_x}\boxtimes (G_1\otimes G_2\restricted{)}{C_y}$. Now, if $\gcd(\rho_1,\rho_2)$ divides the rank and degree of $(G_1\otimes G_1\restricted{)}{C_x}$. Hence,
if $\rho_1$ and $\rho_2$ are both even, then $(G_1\otimes G_1\restricted{)}{C_x}$ has even rank and even degree, and is thus polystable but unstable.

The Lie algebra $\bar{\LieAlg{g}}$ acts on $H^*(X,\QQ)$ preserving the grading and leaving the cup product infinitesimally invariant, i.e.,
$\xi(\alpha\beta)=\xi(\alpha)\beta+\alpha\xi(\beta)$, for all $\alpha,\beta\in H^*(X,\QQ)$ and all $\xi\in\bar{\LieAlg{g}}$. If $F$ and $G$ are objects in $D^b(X)$ of non-zero rank with $\bar{\LieAlg{g}}$-invariant $\kappa(F)$ and $\kappa(G)$, then so is $\kappa(F\otimes G)=\kappa(F)\kappa(G)$. Hence, if $F$ and $G$ are locally free modular sheaves, then so is $F\otimes G$.
The latter need not be rigid, even if $F$ and $G$ are. If, for example, $F$ and $G$ are spherical locally free sheaves on a $K3$ surface $X$ with $v(F)=r_1\alpha+\lambda_1+s_1\beta$ and $v(G)=r_1\alpha+\lambda_1+s_1\beta$, where
$(v(F),v(F))=(v(G),v(G))=-2$, so that $(\lambda_i,\lambda_i)=2r_is_i-2$,  then
$v(F\otimes G)=r_1r_2\alpha+(r_1\lambda_2+r_2\lambda_1)+(r_1s_2+r_2s_1+(\lambda_1,\lambda_2)-r_1r_2)\beta$ and

\begin{equation}
\label{eq-non-rigidity-n=1}
(v(F\otimes G),v(F\otimes G))=2[(r_1^2-1)(r_2^2-1)-1]=2r_1^2r_2^2\left(1-\frac{1}{r_1^2}-\frac{1}{r_2^2}\right),
\end{equation}

which is larger than  $-2$, whenever $r_1>1$ and $r_2>1$. Hence, $\dim \Ext^1(F\otimes G,F\otimes G)>0$, if $r_1>1$ and $r_2>1$.

Consider, for example, a $K3$ surface $X$ with a cyclic Picard group generated by an ample class $\lambda$ with $(\lambda,\lambda)=2$. Write $(r,t\lambda,s)$ for $r\alpha+t\lambda+s\beta$.
Let $F$ and $G$ be $\lambda$-stable sheaves with $v(F)=(2,\lambda,1)$ and $v(G)=(5,2\lambda,1)$. Then both are spherical, hence locally free, and the Mukai vector of the $\lambda$-polystable vector bundle $F\otimes G$ is $v(F\otimes G)=(10,9\lambda,1)$, which is primitive as well, and so $F\otimes G$ is $\lambda$-stable with
$(v(F\otimes G),v(F\otimes G))=142$. Hence, $F\otimes G$ belongs to a $144$-dimensional moduli space.

Equations (<ref>) and (<ref>) suggest the following question. Do
the vector bundles $F_i=G_i[n]^\pm$ of rank $r_i$
over the Hilbert scheme $S^{[n]}$ of a $K3$ surface $S$,  associated to spherical vector bundles $G_i$ over $S$, $i=1,2$, satisfy the following equation?

\[
-\chi\left([F_1\otimes F_2]^\vee\otimes F_1\otimes F_2\right)=
\]

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