corIntrthmIntr lemthm lem corthm cor propthm prop questthm quest remthm rem
defnthm defn exthm ex

# On the local geometry of the moduli space of $(2,2)$-threefolds in
${\mathcal{A}}_{9}$

E. Colombo  Elisabetta Colombo
Università degli Studi di Pavia
Dipartimento di Matematica
Via Cesare Saldini 50
20133 Milano, Italy<EMAIL_ADDRESS>,  P. Frediani  Paola
Frediani
Università degli Studi di Pavia
Dipartimento di Matematica
Via Ferrata 1
27100 Pavia, Italy<EMAIL_ADDRESS>,  J.C. Naranjo1,2 Juan Carlos
Naranjo
1\. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran
Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
2\. Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, 08193
Bellaterra, Spain<EMAIL_ADDRESS>and  G.P. Pirola  Gian Pietro Pirola
Università degli Studi di Pavia
Dipartimento di Matematica
Via Ferrata 1
27100 Pavia, Italy<EMAIL_ADDRESS>
###### Abstract.

We study the local geometry of the moduli space of intermediate Jacobians of
$(2,2)$-threefolds in ${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$. More
precisely, we prove that a composition of the second fundamental form of the
Siegel metric in $\mathcal{A}_{9}$ restricted to this moduli space, with a
natural multiplication map is a nonzero holomorphic section of a vector
bundle. We also describe its kernel. We use the two conic bundle structures of
these threefolds, Prym theory, gaussian maps and Jacobian ideals.

E. Colombo, P. Frediani and G.P. Pirola are members of GNSAGA (INdAM) and are
partially supported by PRIN project Moduli spaces and Lie theory (2017) and by
PRIN project Moduli spaces and special varieties (2022). J.C. Naranjo is
partially supported by the Spanish MINECO research project
PID2019-104047GB-I00 and AGAUR project 2021 SGR 00697.

##  1\. Introduction

This paper deals with the local geometry of the moduli space of intermediate
Jacobians of $(2,2)$-threefolds in ${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$.
These threefolds have the remarkable property of having two conic bundle
structures. As pointed out by Verra in [verra], this translates into the fact
that the intermediate Jacobian has two interpretations as the Prym variety of
two admissible double covers of two plane sextics. In fact, using this, he
proves that the restriction of the Prym map to the locus of double covers of
plane sextics has degree two, thus giving a counterexample to the tetragonal
conjecture of Donagi ([donagi]). He also proves a generic Torelli theorem for
the period map of these threefolds in ${\mathcal{A}}_{9}$.

Our aim is to investigate the local geometry of this period map studying
properties of its second fundamental form. In fact, we consider
${\mathcal{A}}_{9}$ endowed with the orbifold Kähler metric induced by the
symmetric metric on the Siegel space and we study the second fundamental form
$II$ of the locus ${\mathcal{Q}}$ of intermediate Jacobians of these
threefolds in ${\mathcal{A}}_{9}$ with respect to this metric.

One of the main difficulties is that $II$ is non-holomorphic. However, the
composition of this map with a convenient multiplication map turns out to be
holomorphic and can be described. This has been done in the case of the
Jacobian and Prym loci in ${\mathcal{A}}_{g}$ (see [cpt], [cfg], [cf_prym]),
where this composition is given by the second gaussian map of either the
canonical bundle, or the Prym-canonical one. These gaussian maps are
generically of maximal rank. In contrast, an analogous composition of the
second fundamental form of the moduli space of intermediate Jacobians of cubic
threefolds in ${\mathcal{A}}_{5}$ with a natural multiplication map is shown
to be zero in [cfnp_cubic]. In the case of cubic threefolds, an important tool
was the relation between the Jacobian ring of the cubic threefold and that of
the family of plane quintics provided by the Prym construction.

This suggested to address a similar problem to other families of intermediate
Jacobians of Fano threefolds and ask if a similar behaviour occurs, in
particular for the ones that can be described in terms of Prym varieties of
coverings of plane curves.

In this paper, we prove that the composition of the second fundamental form of
the moduli space of intermediate Jacobians of $(2,2)$ threefolds in
${\mathcal{A}}_{9}$ with a natural multiplication map is nonzero. More
precisely, it gives a nonzero holomorphic section ${\mathcal{S}}$ of the
normal bundle $N_{\mathcal{Q}/\mathcal{A}_{9}}$. Again, Jacobian rings of
both, the threefold and the associated plane sextics, play a fundamental role.

More precisely, for a $(2,2)$ threefold $T$ in
${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$ there is a natural bigraded Jacobian
ring $\oplus R_{T}^{a,b}$ such that the differential of the period map can be
expressed in terms of this ring (see [green], [green_LNM]). In particular, the
dual of the the differential of the period map corresponds to the
multiplication map

$\nu:Sym^{2}R^{1,1}_{T}\longrightarrow R^{2,2}_{T}.$

Hence, there is a natural isomorphism:

$N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}=Ker(Sym^{2}R^{1,1}_{T}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}R_{T}^{2,2}).$

So, the second fundamental form:

$II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}\longrightarrow
Sym^{2}\Omega^{1}_{{\mathcal{Q}},JT}$

can be seen as a map:

$Ker(Sym^{2}R^{1,1}_{T}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}R_{T}^{2,2})\longrightarrow
Sym^{2}R^{2,2}_{T}.$

The main result of this paper is the following:

######  Theorem 1.1.

For a general $T$, the composition

$m\circ
II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}=Ker(Sym^{2}R^{1,1}_{T}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}R_{T}^{2,2})\longrightarrow
R^{4,4}_{T}\cong\mathbb{C},$

where $m:Sym^{2}R^{2,2}_{T}\longrightarrow R^{4,4}_{T}$ is the multiplication
map, is nonzero and gives a canonical non-trivial section ${\mathcal{S}}$ of
the normal bundle. As a consequence, the second fundamental form
$II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}\longrightarrow
Sym^{2}\Omega^{1}_{{\mathcal{Q}},JT}$ is nonzero.

Moreover, for a general $T$, the kernel of the composition $m\circ II$ is
explicitly described in terms of the geometry of $T$ (see Proposition 4, and
Remark 4).

We remark that the statement that the second fundamental form is different
from zero is also implied by the fact that the monodromy group of the family
of these threefolds is the whole symplectic group (see Remark 4).

One of the main ingredients in the proof is to relate the second fundamental
form $II$ with the restriction of the second fundamental form of the Prym map
${\mathcal{R}}_{10}\longrightarrow{\mathcal{A}}_{9}$ to the locus of double
coverings of plane sextics. In fact, the intermediate Jacobian of $T$ is
isomorphic, as principally polarized abelian variety, to the Prym variety of
the two double coverings of the two plane sextics determined by the two conic
bundle structures.

We show that the restriction of the composition $m\circ II$ to the space of
quadrics containing the Prym-canonical image of both the plane sextics is
nonzero. This is done using the fact that this restriction coincides with the
composition of the second gaussian map of the Prym-canonical bundle of the
plane sextics with a suitable projection.

Then we compute this composition in a specific example for the quadric
containing the Prym-canonical image of a sextic that is defined by the
equation of the threefold, and we show that it does not vanish. This implies
the main theorem.

Finally, we are also able to explicitly describe the kernel of this
composition by computing the second gaussian map on a set of quadrics of rank
4, constructed using some pencils on the plane sextics. To show that this set
of quadrics generically generates the kernel, we make an explicit computation
on an example using a MAGMA script.

This allows us to completely describe the kernel of $m\circ II$, since the
conormal bundle is generated by the two vector spaces of the quadrics
containing the Prym-canonical images of the two plane sextics.

The structure of the paper is as follows:

In section 2 we recall the properties of $(2,2)$ threefolds following [verra].
In section 3 we introduce the second fundamental form $II$, we recall the
theory of bigraded Jacobian rings and we state the main Theorem. In section 4
we give an interpretation of the second fundamental form in terms of Prym
theory, and we prove the main Theorem.

Acknowledgments: We warmly thank Bert van Geemen for the fundamental help in
the computation in the example in section 4 and for the MAGMA script.

##  2\. $(2,2)$-threefolds in $\mathbb{P}^{2}\times\mathbb{P}^{2}$

In this section we mainly recall the properties of $(2,2)$-threefolds proved
by Verra in [verra].

We consider a threefold $T$ in $\mathbb{P}^{2}\times\mathbb{P}^{2}$ given by a
bihomogeneous equation $F$ of degree $(2,2)$. More precisely:

$F=\sum_{0\leq i,j,k,l\leq 2}a_{ijkl}x_{i}x_{j}y_{k}y_{l}.$

Let $W$ be the image of the Segre embedding
$s:\mathbb{P}^{2}\times\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{8}$. Then:

$T\subset\mathbb{P}^{2}\times\mathbb{P}^{2}\cong W\subset\mathbb{P}^{8}.$

We denote $\alpha_{ik}:=x_{i}y_{k}$. These are natural coordinates in
$\mathbb{P}^{8}$, and $T$ can be seen as a complete intersection $T=W\cap Q$,
where $Q\subset\mathbb{P}^{8}$ is the quadric given by the equation:

$Q_{0}=\sum_{0\leq i,j,k,l\leq 2}a_{ijkl}\alpha_{ik}\alpha_{jl}.$

Remember that the quadratic equations
$\alpha_{ik}\alpha_{jl}-\alpha_{il}\alpha_{jk}$ generate the ideal $I_{W}$ of
$W$. Then the ideal of $T$ is $I_{T}=I_{W}+\langle Q_{0}\rangle.$

The projections on each factor $p_{i}:T\longrightarrow\mathbb{P}^{2}$ provide
two conic bundle structures on $T$. In other words, fixed a point
$(x_{0}:x_{1}:x_{2})$ in the first plane, the equation
$F(x_{0},x_{1},x_{2},y_{0},y_{1},y_{2})$ gives a conic whose matrix $A(x)$ has
degree $2$ entries in the $x_{i}^{\prime}s$,
$A(x)_{kl}=\sum_{i,j}a_{ijkl}x_{i}x_{j}$. Therefore, the determinant of this
matrix gives a sextic plane curve $C$ which parametrizes the degenerate conics
of the family. The lines contained in these degenerate conics define a curve
$\widetilde{C}$ contained in the corresponding Grassmannian and a natural
degree two map $\pi_{1}:\widetilde{C}\mapsto C$. It was proved by Beauville in
[be_jac_int] that $\pi_{1}$ is an admissible covering of degree $2$ and that
its Prym variety $P(\widetilde{C},C)$ is isomorphic (as principally polarized
abelian variety) to the intermediate Jacobian $JT$. Similarly, using the
second projection, there is another sextic plane curve $D$ and a covering
$\pi_{2}:\widetilde{D}\mapsto D$ with the same property. In particular
$P(\widetilde{C},C)\cong P(\widetilde{D},D)$. The main Theorem in [verra]
states that the Prym map has degree exactly $2$ when restricted to the locus
of unramified double coverings of plane sextics.

We will assume from now on that $T$, $C$ and $D$ are generic, in particular
all three are smooth. We are interested in the realization of $C$ and $D$ in
$\mathbb{P}^{8}$. Let $C^{\prime}$ (resp. $D^{\prime}$) be the set of double
points of the conics of the first (second) conic bundle structure on $T$.
Notice that $C^{\prime},D^{\prime}\subset T\subset W\subset\mathbb{P}^{8}$ and
that $C^{\prime}\cong C$, $D^{\prime}\cong D$. Moreover $C^{\prime}$ (resp.
$D^{\prime}$) is the locus of points of $T$ where the partial derivatives
$F_{y_{0}},F_{y_{1}},F_{y_{2}}$ (resp. $F_{x_{0}},F_{x_{1}},F_{x_{2}}$)
vanish. So the corresponding ideals are:

$I_{C^{\prime}}=I_{T}+\langle F_{y_{0}},F_{y_{1}},F_{y_{2}}\rangle,\qquad
I_{D^{\prime}}=I_{T}+\langle F_{x_{0}},F_{x_{1}},F_{x_{2}}\rangle.$

In fact, $C^{\prime}$ is the Prym-canonical image of $C$: let $\eta\in JC$ be
the $2$-torsion point that defines the covering $\pi_{1}$. For a generic $C$
we have that $h^{0}(C,\eta(2))=3$ and the tensor product:

$H^{0}(C,\mathcal{O}_{C}(1))\otimes H^{0}(C,\eta(2))\mapsto
H^{0}(C,\omega_{C}\otimes\eta)$

is an isomorphism. Then the embedding $\phi:C\hookrightarrow{\mathbb{P}}^{8}$
is the composition:

$C\,\stackrel{{\scriptstyle\varphi_{1}\times\varphi_{2}}}{{\longrightarrow}}\,\mathbb{P}^{2}\times\mathbb{P}^{2}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}W\hookrightarrow\mathbb{P}^{8}\cong\mathbb{P}(H^{0}(C,\omega_{C}\otimes\eta)^{\vee})$

where $\varphi_{1},\varphi_{2}$ are the maps defined by the linear systems
$|\mathcal{O}_{C}(1)|$ and $|\eta(2)|$. Thus, by definition,
$C^{\prime}=(\varphi_{1}\times\varphi_{2})(C)$. The same holds for the cover
$\pi_{2}$ and we denote by $\eta^{\prime}\in JD$ the corresponding two-torsion
point.

##  3\. $2$nd fundamental form

Following [verra], we consider the following moduli spaces:

$\displaystyle\mathcal{Q}=\\{T\subset\mathbb{P}^{2}\times\mathbb{P}^{2}\mid
T\text{ is a }(2,2)\text{ smooth threefold}\\}/\cong,$
$\displaystyle\widetilde{\mathcal{Q}}=\\{(T,p_{i})\mid
T\in\mathcal{Q},\,p_{i}\text{ projection on }\mathbb{P}^{2}\\}/\cong,$
$\displaystyle\mathcal{P}_{6}=\\{\text{admissible double coverings
}\tilde{C}\to C,\text{ where }C\text{ is a plane sextic}\\}/\cong,$
$\displaystyle\mathcal{A}_{9}=\text{ moduli space of principally polarized
abelian varieties of dimension }9.$

Then, there is a commutative diagram:

(3.1)
$\textstyle{\widetilde{\mathcal{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{f}$$\textstyle{\mathcal{P}_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\mathcal{A}_{9},}$

where $d$ associates to the conic bundle
$p_{i}:T\longrightarrow\mathbb{P}^{2}$ the discriminant curve
$C\subset\mathbb{P}^{2}$ and the natural admissible covering
$\widetilde{C}\longrightarrow C$; $f$ is the forgetful map;
$j(T)=H^{1,2}(T)/H^{3}(T,\mathbb{Z})$ is the intermediate Jacobian of $T$; and
$p$ is the Prym map restricted to $\mathcal{P}_{6}$. Moreover
$\widetilde{\mathcal{Q}}$, $\mathcal{Q}$ and $\mathcal{P}_{6}$ are irreducible
of dimension $19$.

The main result in [verra] is the computation of the degrees of all these maps
that turn out to be generically finite on their images. He proves, based on
results in [be_det], that $d$ and $j$ have degree $1$ and that $f$ and $p$
have degree $2$.

Notice that (using that $d$ has degree $1$):

$\Omega^{1}_{\mathcal{Q},\bullet}\cong\Omega^{1}_{\widetilde{\mathcal{Q}},\bullet}\cong\Omega^{1}_{\mathcal{P}_{6},\bullet}\cong\Omega^{1}_{\mathcal{M}_{10}^{pl},\bullet},$

where we denote by $\mathcal{M}_{10}^{pl}$ the moduli space of smooth plane
sextics in $\mathcal{M}_{10}$, the moduli space of genus $10$ curves.

Recall that $\mathcal{A}_{9}$ has a natural orbifold Kähler form, which is the
one induced by the natural symmetric form on the Siegel space. Our aim is to
find some properties of the second fundamental form on
$\overline{j(\mathcal{Q})}=\overline{p(\mathcal{P}_{6})}$ associated with the
restriction of this Kähler metric via the embedding in $\mathcal{A}_{9}$. By
abuse of notation, we also denote by $\mathcal{Q}$ the closure of its image in
$\mathcal{A}_{9}$.

We have the cotangent exact sequence

$0\rightarrow{N_{{\mathcal{Q}}/\mathcal{A}_{9}}^{*}}\rightarrow{\Omega^{1}_{\mathcal{A}_{9}}}_{|{\mathcal{Q}}}\stackrel{{{}^{t}dj}}{{\rightarrow}}\Omega^{1}_{\mathcal{Q}}\rightarrow
0.$

Denote by $\nabla$ the Chern connection of the Siegel metric and consider the
second fundamental form

$II=(^{t}dj\otimes
Id)\circ\nabla_{|{N_{{\mathcal{Q}}/\mathcal{A}_{9}}^{*}}}:{N_{{\mathcal{Q}}/\mathcal{A}_{9}}^{*}}\rightarrow
Sym^{2}\Omega^{1}_{\mathcal{Q}}.$

Then, we want to study the map, in a generic point $j(T)=JT\in\mathcal{Q}$:

$N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}\stackrel{{\scriptstyle
II}}{{\longrightarrow}}Sym^{2}\Omega^{1}_{\mathcal{Q},JT}.$

The conormal sheaf of $\mathcal{Q}$ in $\mathcal{A}_{9}$ is the kernel of the
dual the differential of $j$:

${}^{t}dj:\Omega^{1}_{\mathcal{A}_{9}|\mathcal{Q},JT}\cong
Sym^{2}H^{2,1}(T)\longrightarrow\Omega_{\mathcal{Q},JT}^{1}.$

Recall that Griffiths studied the periods of hypersurfaces by means of the
Jacobian ring. Later, Green extended this theory to a more general setting,
covering in particular, the case of hypersurfaces in the product of two
projective spaces (see [green] and [green_LNM, Lecture 4]). More precisely, we
consider the bigraded Jacobian ring $\oplus R^{a,b}_{T}$ of $T$, which is the
quotient of the ring of bihomogeneous polynomials $\oplus S^{a,b}$ by the
bihomogeneous ideal $\oplus J^{a,b}_{T}$ generated by the partial derivatives
of $F$.

Using [green], $T_{\mathcal{Q},T}$ can be identified with $R^{2,2}_{T}$.
Moreover ${\mathcal{O}}_{\mathcal{Q},T}\cong R^{4,4}_{T}\cong\mathbb{C}$.
Then, there are isomorphisms: $H^{2,1}(T)\cong R^{1,1}_{T}$ and
$\Omega_{\mathcal{Q},T}^{1}\cong R^{2,2}_{T}$, and ${}^{t}dj$ identifies with
the multiplication map:

$\nu:Sym^{2}R^{1,1}_{T}\longrightarrow R^{2,2}_{T}.$

Hence there is a natural isomorphism:

$N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}=Ker(Sym^{2}R^{1,1}_{T}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}R_{T}^{2,2}).$

Then, we can compose $II$ with the multiplication map:

$m:Sym^{2}R^{2,2}_{T}\longrightarrow R^{4,4}_{T}\cong\mathbb{C},$

and $m\circ
II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9}}\longrightarrow{\mathcal{O}}_{\mathcal{Q}}$
gives a holomorphic section ${\mathcal{S}}$ of the normal bundle
$N_{\mathcal{Q}/\mathcal{A}_{9}}$.

As stated in Theorem 1.1, the main result of this paper is the following:

######  Theorem 3.1.

For a general $T$, the composition:

$m\circ
II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}=Ker(Sym^{2}R^{1,1}_{T}\stackrel{{\scriptstyle
m}}{{\longrightarrow}}R_{T}^{2,2})\longrightarrow
R^{4,4}_{T}\cong{\mathbb{C}}$

is nonzero, and it gives a non-trivial canonical holomorphic section of the
normal bundle $N_{\mathcal{Q}/\mathcal{A}_{9}}$.

In particular, the second fundamental form:

$II:N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}\longrightarrow
Sym^{2}\Omega^{1}_{{\mathcal{Q}},JT}$

is nonzero.

##  4\. Prym theory and proof of the main Theorem

The proof of Theorem 3.1 relies on the fact that $\mathcal{Q}$ can be seen as
the image of the Prym map restricted to $\mathcal{P}_{6}$. Let $C$ be a smooth
sextic plane curve, and let $R^{\bullet}_{C}$ be the Jacobian ring of $C$. So
the identification $\Omega^{1}_{\mathcal{M}_{10}^{pl},C}\cong R_{C}^{6}$
holds. Then, we use Prym theory to express the normal sheaf of $\mathcal{Q}$
in $\mathcal{A}_{9}$ in another way. Consider the restriction $p$ of the Prym
map $p_{10}$ to $\mathcal{P}_{6}$:

$\mathcal{P}_{6}\hookrightarrow\mathcal{R}_{10}\stackrel{{\scriptstyle
p_{10}}}{{\longrightarrow}}\mathcal{A}_{9}.$

Recall that the map $p$ is generically finite of degree 2. The transpose of
$dp_{10}$ at a point $(C,\eta)\in{\mathcal{R}}_{10}$ is the multiplication map
$Sym^{2}H^{0}(C,\omega_{C}\otimes\eta)\longrightarrow
H^{0}(C,\omega_{C}^{2})$. In a generic point $(C,\eta)\in\mathcal{P}_{6}$ it
is surjective by [verra, 3.6]. Thus, we have:

$N^{*}_{\mathcal{R}_{10}/\mathcal{A}_{9},(C,\eta)}\cong
I_{2}(\omega_{C}\otimes\eta),$

which is the vector space of equations of quadrics through $C^{\prime}$, the
Prym-canonical image of $C$. We have a short exact sequence of conormal
bundles:

$0\longrightarrow I_{2}(\omega_{C}\otimes\eta)\longrightarrow
N^{*}_{\mathcal{Q}/\mathcal{A}_{9},T}\longrightarrow
N^{*}_{\mathcal{P}_{6}/\mathcal{R}_{10},(C,\eta)}=J^{6}_{C}/\langle
C\rangle\longrightarrow 0.$

Here $J^{\bullet}_{C}$ means the jacobian ideal of $C$, and $\langle C\rangle$
the subideal generated by the equation of the sextic $C$.

We have the following diagram:

(4.1) $\textstyle{N^{*}_{\mathcal{R}_{10}/\mathcal{A}_{9},(C,\eta)}\cong
I_{2}(\omega_{C}\otimes\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{Sym^{2}\Omega^{1}_{\mathcal{R}_{10},(C,\eta)}\cong
Sym^{2}H^{0}(C,\omega_{C}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{*}_{\mathcal{Q}/\mathcal{A}_{9},P(C,\eta)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Sym^{2}\Omega^{1}_{\mathcal{P}_{6},(C,\eta)}\cong
Sym^{2}R^{6}_{C},}$

where $\rho$ is the second fundamental form of the Prym map $p_{10}$ and the
second horizontal row is another way of representing the map $II$ using Prym
theory. Notice that the first vertical arrow is the inclusion of the conormal
bundles, while the second vertical arrow is induced by projection of cotangent
bundles.

Recall that, by [cf_prym], the composition of $\rho$ with the multiplication
map $Sym^{2}H^{0}(C,\omega_{C}^{2})\longrightarrow H^{0}(C,\omega_{C}^{4})$ is
the second gaussian map $\mu_{2}:I_{2}(\omega_{C}\otimes\eta)\longrightarrow
H^{0}(C,\omega_{C}^{4})$.

Observe that $H^{0}(C,\omega_{C}^{2})\cong H^{0}(C,\mathcal{O}_{C}(6))$ and we
have a map $H^{0}(C,\mathcal{O}_{C}(6))\longrightarrow R^{6}_{C}$, induced by
the projection
$H^{0}({\mathbb{P}}^{2},\mathcal{O}_{{\mathbb{P}}^{2}}(6))\longrightarrow
R^{6}_{C}$, since the equation of $C$ belongs to the jacobian ideal of $C$.
Analogously, we have a map $\tau:H^{0}(C,\mathcal{O}_{C}(12))\longrightarrow
R^{12}_{C}$.

So we have the following commutative diagram:

(4.2)
$\textstyle{I_{2}(\omega_{C}\otimes\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\mu_{2}}$$\textstyle{Sym^{2}H^{0}(C,\omega_{C}^{2})=Sym^{2}H^{0}(C,\mathcal{O}_{C}(6))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
71.13188ptm}$$\textstyle{H^{0}(C,\mathcal{O}_{C}(12))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{N^{*}_{\mathcal{Q}/\mathcal{A}_{9},JT}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{II}$$\textstyle{Sym^{2}R^{6}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{R^{12}_{C}=\mathbb{C}.}$

We will prove that the composition $\tau\circ\mu_{2}$ is nonzero, showing on
an explicit example that $\tau\circ\mu_{2}(F)\neq 0$, where:

$F=\sum_{0\leq i,j,k,l\leq 2}a_{ijkl}x_{i}x_{j}y_{k}y_{l}=\sum_{0\leq
i,j,k,l\leq 2}a_{ijkl}\alpha_{ik}\alpha_{jl}\in I_{2}(\omega_{C}\otimes\eta)$

denotes as usual the equation of the threefold $T$.

######  Remark .

Observe that, for a general $T$, by symmetry, similar diagrams and maps exist
for the curve $D$. Notice that, by [verra, proof of Proposition 5.3] the
intersection of $I_{2}(\omega_{C}\otimes\eta)$ and
$I_{2}(\omega_{D}\otimes\eta^{\prime})$ is the ideal of the equations of the
quadrics containing the threefold $T$ which is $10$-dimensional. Therefore,
the sum $I_{2}(\omega_{C}\otimes\eta)+I_{2}(\omega_{D}\otimes\eta^{\prime})$
has dimension $26$ and hence it equals
$N^{*}_{\mathcal{Q}/\mathcal{A}_{9},T}$. So the map $m\circ II$ is
holomorphic, since its restriction to both $I_{2}(\omega_{C}\otimes\eta)$ and
$I_{2}(\omega_{D}\otimes\eta^{\prime})$ is holomorphic.

Denote by $A(x)$ the symmetric matrix given by
$A(x)_{kl}=\sum_{i,j}a_{ijkl}x_{i}x_{j}$, and by $\hat{A}_{ij}(x)$ the product
of the determinant of the matrix obtained from $A(x)$ by removing the $i$-th
row and the $j$-th column with $(-1)^{i+j}$.

######  Lemma .

We have the following commutative diagram:

(4.3)
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\textstyle{{\mathbb{P}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{2}}$$\textstyle{{\mathbb{P}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\mathbb{P}}^{5},}$

where $h(x)=(\hat{A}_{ij}(x))_{i,j}$, $v_{2}$ is the Veronese map, and
$\varphi_{1},\varphi_{2}$ are the maps defined at the end of $\S 2$.

###### Proof.

Since for any $x\in C$, $\phi(x)$ is the intersection point of the two lines
corresponding to the singular conic $A(x)$,
$\varphi_{2}(x)=(y_{0}:y_{1}:y_{2})$ is the point in ${\mathbb{P}}^{2}$ given
by $ker(A(x))$. So there exist nonzero constants $\lambda_{i}$ such that
$\lambda_{i}(y_{0},y_{1},y_{2})=((\hat{A}_{i0}(x)),(\hat{A}_{i1}(x)),(\hat{A}_{i2}(x)))$,
for all $i=1,2,3$. By the symmetry of $A(x)$ we can identify
$\lambda_{i}=y_{i}$ for all $i=0,1,2$. So the diagram is commutative and in
$C$ we have:

(4.4) $y_{i}y_{j}=\hat{A}_{ij}(x).$

∎

Now choose a local coordinate $z$ on $C$, a local frame $l$ for
${\mathcal{O}}_{C}(1)$ and a local frame $\sigma$ for $\eta$. Then locally we
write $x_{i}=f_{i}(z)l$, $y_{i}=g_{i}(z)l^{2}\sigma$ and
$\alpha_{ij}=f_{i}(z)g_{j}(z)l^{3}\sigma$. So the local expression for
$\mu_{2}(F)$ is given by:

$\mu_{2}(F)=-\sum_{i,j,k,l}a_{ijkl}(f_{i}g_{k})^{\prime\prime}(f_{j}g_{l})(dz)^{2}l^{6}=\sum_{i,j,k,l}a_{ijkl}(f_{i}g_{k})^{\prime}(f_{j}g_{l})^{\prime}(dz)^{2}l^{6}.$

First, notice that since $F\in I_{2}(\omega_{C}\otimes\eta)$, we have:

$\sum_{i,j,k,l}a_{ijkl}(f_{i}g_{k})(z)(f_{j}g_{l})(z)=0,$

hence taking the derivative with respect to $z$, we get:

$\displaystyle 0=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}(f_{i}g_{k})^{\prime}(f_{j}g_{l})=\sum_{i,j,k,l}a_{ijkl}(f^{\prime}_{i}g_{k}+f_{i}g^{\prime}_{k})(f_{j}g_{l})$
$\displaystyle=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g_{k}g_{l}+\sum_{i,j,k,l}a_{ijkl}f_{i}f_{j}g_{l}g^{\prime}_{k}$
$\displaystyle=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g_{k}g_{l}+\sum_{k}g^{\prime}_{k}(\sum_{i,j,l}a_{ijkl}f_{i}f_{j}g_{l})$
$\displaystyle=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g_{k}g_{l}+\sum_{k}g^{\prime}_{k}(\sum_{l}A_{kl}(z)g_{l}(z))=\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g_{k}g_{l},$

where $A_{kl}(z)=\sum_{i,j}a_{ijkl}f_{i}(z)f_{j}(z)$, and the last equality
holds since the point $(y_{0},y_{1},y_{2})^{t}$ is in the kernel of the matrix
$A(x)$, hence $\sum_{l}A_{kl}(z)g_{l}(z)=0$.

So, differentiating with respect to $z$ the polynomial equation:

(4.5) $\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g_{k}g_{l}=0$

we obtain:

(4.6)
$\sum_{i,j,k,l}a_{ijkl}f^{\prime\prime}_{i}f_{j}g_{k}g_{l}+\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f^{\prime}_{j}g_{k}g_{l}+2\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g^{\prime}_{k}g_{l}=0.$

Now

$\displaystyle\mu_{2}(F)=$
$\displaystyle-\sum_{i,j,k,l}a_{ijkl}(f_{i}g_{k})^{\prime\prime}(f_{j}g_{l})(dz)^{2}l^{6}$
$\displaystyle=$
$\displaystyle-\left(\sum_{i,j,k,l}a_{ijkl}f^{\prime\prime}_{i}f_{j}g_{k}g_{l}+2\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g^{\prime}_{k}g_{l}+\sum_{i,j,k,l}a_{ijkl}f_{i}f_{j}g^{\prime\prime}_{k}g_{l}\right)(dz)^{2}l^{6}$
$\displaystyle=$
$\displaystyle-\left(\sum_{i,j,k,l}a_{ijkl}f^{\prime\prime}_{i}f_{j}g_{k}g_{l}+2\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f_{j}g^{\prime}_{k}g_{l}\right)(dz)^{2}l^{6}$
$\displaystyle=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f^{\prime}_{j}g_{k}g_{l}(dz)^{2}l^{6},$

where the last equality follows from (4.6) and we used that:

$\sum_{i,j,k,l}a_{ijkl}f_{i}f_{j}g^{\prime\prime}_{k}g_{l}=\sum_{k}g^{\prime\prime}_{k}\sum_{l}A_{kl}(z)g_{l}(z)=0.$

So we have found the following local expression for $\mu_{2}(F)$:

(4.7)
$\mu_{2}(F)=\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f^{\prime}_{j}g_{k}g_{l}(dz)^{2}l^{6}.$

######  Proposition .

The map $\tau\circ\mu_{2}$ is nonzero.

###### Proof.

We show the proposition by exhibiting an explicit example where the map is
nonzero.

Consider the following smooth $(2,2)$-threefold $X=\\{F=0\\}$, where:

(4.8) $\displaystyle F$ $\displaystyle=6x_{0}^{2}y_{0}^{2}+2\lambda
x_{2}^{2}y_{0}y_{1}+2\lambda x_{1}^{2}y_{0}y_{2}+6x_{1}^{2}y_{1}^{2}+2\lambda
x_{0}^{2}y_{1}y_{2}+6x_{2}^{2}y_{2}^{2}=$
$\displaystyle=6\alpha_{00}^{2}+2\lambda\alpha_{20}\alpha_{21}+2\lambda\alpha_{10}\alpha_{12}+6\alpha_{11}^{2}+2\lambda\alpha_{01}\alpha_{02}+6\alpha_{22}^{2},$

with $\lambda^{3}=-108$. Thus, the equation of the plane sextic attached to
the first projection $p_{1}$ is the determinant of the following symmetric
matrix:

$A={\small\left(\begin{array}[]{ccc}6x_{0}^{2}&\lambda x_{2}^{2}&\lambda
x_{1}^{2}\\\ \lambda x_{2}^{2}&6x_{1}^{2}&\lambda x_{0}^{2}\\\ \lambda
x_{1}^{2}&\lambda x_{0}^{2}&6x_{2}^{2}\\\ \end{array}\right),}$
$det(A)=-6\lambda^{2}(x_{0}^{6}+x_{1}^{6}+x_{2}^{6}).$

So the plane curve $C$ is the Fermat sextic with equation
$G:=x_{0}^{6}+x_{1}^{6}+x_{2}^{6}=0$. We will show that
$\tau\circ\mu_{2}(F)\neq 0$. Consider the affine chart $x_{0}\neq 0$, set
$w_{i}=x_{i}/x_{0}$, $i=1,2$,
$h(w_{1},w_{2}):=G(1,w_{1},w_{2})=1+w_{1}^{6}+w_{2}^{6}.$ Assume that
$\partial h/\partial w_{2}=6w_{2}^{5}\neq 0,$ so $z:=w_{1}$ is a local
coordinate, hence $w_{2}=f_{2}(z)$. In this local setting we have $z=w_{1}$,
$l=x_{0}$, $x_{1}=w_{1}x_{0}=zl$, $x_{2}=w_{2}x_{0}=f_{2}(z)l$. So in the
above notation we have $f_{0}(z)=1$, $f_{1}(z)=z$, $f_{2}(z)=w_{2}(z)$, and
since $h(z,f_{2}(z))=0$, we have:

$\partial h/\partial w_{1}+f^{\prime}_{2}(z)\partial h/\partial w_{2}=0,$

so $f^{\prime}_{2}(z)=-h_{w_{1}}/h_{w_{2}}$ (where $h_{w_{i}}=\partial
h/\partial w_{i}$), while clearly $f^{\prime}_{0}=0$, $f^{\prime}_{1}=1$.
Then, by (4.7), we have:

$\displaystyle\mu_{2}(F)=$
$\displaystyle\sum_{i,j,k,l}a_{ijkl}f^{\prime}_{i}f^{\prime}_{j}g_{k}g_{l}(dz)^{2}l^{6}$
$\displaystyle=$
$\displaystyle\sum_{k,l}(g_{k}g_{l})l^{4}(a_{11kl}(f^{\prime}_{1})^{2}+2a_{12kl}f^{\prime}_{1}f^{\prime}_{2}+a_{22kl}(f^{\prime}_{2})^{2})(dz)^{2}l^{2}$
$\displaystyle=$
$\displaystyle\sum_{k,l}(g_{k}g_{l})l^{4}(a_{11kl}-2a_{12kl}(h_{w_{1}}/h_{w_{2}})+a_{22kl}(h_{w_{1}}/h_{w_{2}})^{2})(dz)^{2}l^{2}.$

Recall that by (4.3) we have $\hat{A}_{kl}(z)=(g_{k}g_{l})l^{4}$, thus:

$\displaystyle\mu_{2}(F)=$
$\displaystyle\sum_{k,l}\hat{A}_{kl}(z)(a_{11kl}-2a_{12kl}(h_{w_{1}}/h_{w_{2}})+a_{22kl}(h_{w_{1}}/h_{w_{2}})^{2})(dz)^{2}l^{2}$
$\displaystyle=$
$\displaystyle\sum_{k,l}\hat{A}_{kl}(z)(a_{11kl}-2a_{12kl}(h_{w_{1}}/h_{w_{2}})+a_{22kl}(h_{w_{1}}/h_{w_{2}})^{2})(h_{w_{2}})^{2}l^{8},$

since, by adjunction, we can write $dz=h_{w_{2}}l^{3}$. So we have:

$\displaystyle\mu_{2}(F)=$
$\displaystyle\sum_{k,l}\hat{A}_{kl}(a_{11kl}-2a_{12kl}(G_{x_{1}}/G_{x_{2}})+a_{22kl}(G_{x_{1}}/G_{x_{2}})^{2})\frac{(G_{x_{2}})^{2}}{x_{0}^{10}}x_{0}^{8}$
$\displaystyle=$
$\displaystyle\sum_{k,l}\hat{A}_{kl}(a_{11kl}(G_{x_{2}})^{2}-2a_{12kl}G_{x_{1}}G_{x_{2}}+a_{22kl}(G_{x_{1}})^{2})\frac{1}{x_{0}^{2}}.$

Now, using our equation and computing the minors of the matrix $A$, we obtain:

$\displaystyle\mu_{2}(F)=$
$\displaystyle\frac{36}{x_{0}^{2}}(x_{1}^{10}(2\lambda\hat{A}_{01}+6\hat{A}_{22})+x_{2}^{10}(2\lambda\hat{A}_{02}+6\hat{A}_{11}))$
$\displaystyle=$
$\displaystyle-18\lambda^{2}\frac{36}{x_{0}^{2}}(x_{1}^{10}x_{2}^{4}+x_{2}^{10}x_{1}^{4})$
$\displaystyle=$
$\displaystyle-18\lambda^{2}\frac{36}{x_{0}^{2}}x_{1}^{4}x_{2}^{4}(x_{1}^{6}+x_{2}^{6}).$

Since we are on the curve $C$, we have $x_{1}^{6}+x_{2}^{6}=-x_{0}^{6}$, so
finally we get:

$\mu_{2}(F)=18\lambda^{2}\frac{36}{x_{0}^{2}}x_{1}^{4}x_{2}^{4}x_{0}^{6}=(18\cdot
36)\lambda^{2}x_{0}^{4}x_{1}^{4}x_{2}^{4},$

which is not in the Jacobian ideal of $C$ that is generated by
$x_{0}^{5},x_{1}^{5},x_{2}^{5}$. Notice that if $\partial h/\partial w_{1}\neq
0$, then $w_{2}$ is a local coordinate, so in this case we have $z=w_{2}$,
$w_{1}=f_{1}(z)$, $f_{2}(z)=z$, $f_{0}(z)=1$, $dw_{2}=-(\partial h/\partial
w_{1})x_{0}^{3}$, $f^{\prime}_{1}(z)=-h_{w_{2}}/h_{w_{1}}$ and the computation
is the same. By symmetry, an analogous computation holds in the other affine
charts. So $\mu_{2}(F)$ is the restriction of the polynomial $(18\cdot
36)\lambda^{2}x_{0}^{4}x_{1}^{4}x_{2}^{4}\in
H^{0}({\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(12))$ to $C$ and hence
$\tau\circ\mu_{2}(F)\neq 0$. ∎

Recall that by (3.1) we have an identification:

$R_{T}^{2,2}\cong\Omega^{1}_{\mathcal{Q}}\cong\Omega^{1}_{{\mathcal{P}}_{6}}\cong
R^{6}_{C}.$

These identifications can be also seen as induced by the pullback via the map
$\phi=\varphi_{1}\times\varphi_{2}$:

$\phi^{*}:H^{0}({\mathbb{P}}^{2}\times{\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}\times{\mathbb{P}}^{2}}(2,2))\longrightarrow
H^{0}(C,{\mathcal{O}}_{C}(6)).$ In fact, we show the following:

######  Lemma .

The space $J^{2,2}_{T}$ maps to $J^{6}_{C}/\langle C\rangle$ via
$\varphi^{*}$, therefore there is an isomorphism $R_{T}^{2,2}\cong R^{6}_{C}$
and a commutative diagram:

(4.9)
$\textstyle{Sym^{2}R_{T}^{2,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{m}$$\textstyle{R_{T}^{4,4}\cong{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{Sym^{2}R^{6}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{R^{12}_{C}\cong\mathbb{C}.}$

###### Proof.

Clearly all the elements of the form $y_{j}F_{y_{i}}$ map to zero, since they
are in the ideal of $C^{\prime}=\phi(C)$. So we have to show that all the
elements $x_{j}F_{x_{i}}$ map to $J^{6}_{C}/\langle C\rangle$. In fact we show
that

$\phi^{*}(F_{x_{i}})={\partial}_{x_{i}}(detA(x))\in J^{6}_{C}/\langle
C\rangle,$

where the coefficients of the symmetric matrix
$(A(x))_{kl}=\sum_{i,j}a_{ijkl}x_{i}x_{j}$ are quadrics in the coordinates
$x_{i}$.

Using (4.3), we have:

$\phi^{*}(F_{x_{i}})=\phi^{*}({\partial}_{x_{i}}(\sum_{ij}A(x)_{ij}y_{i}y_{j}))={\partial}_{x_{i}}(\sum_{ij}A(x)_{ij}\hat{A}_{ij}(x))={\partial}_{x_{i}}(detA(x)),$

In the same way, one immediately sees that $\phi^{*}$ induces an isomorphism
$R_{T}^{4,4}\cong R^{12}_{C}$ and we have a commutative diagram as (4.9). ∎

Proof of the main Theorem 3.1: Using Proposition 4 and Lemma 4 we immediately
obtain that the restriction of $m\circ II$ to $I_{2}(\omega_{C}\otimes\eta)$
is nonzero for a generic $T$. So by Remark (4) $m\circ II$ gives a non-trivial
holomorphic section ${\mathcal{S}}$ of the normal bundle. ∎

We observe that the fact that the second fundamental form is nonzero is also
implied by the following:

######  Remark .

The monodromy group of the family ${\mathcal{Q}}$ is the symplectic group.
Hence the special Mumford-Tate group of a generic $(2,2)$-threefold is the
whole symplectic group.

###### Proof.

We prove that the monodromy group of ${\mathcal{P}}_{6}$ is the full
symplectic group $Sp(18,{\mathbb{Z}})$.

Recall that the monodromy map of plane sextics:

$\pi_{1}({\mathcal{M}}_{10}^{pl},C)\longrightarrow Sp(H^{1}(C,{\mathbb{Z}}))$

is surjective (see [beauville_mono, Theoreme 4]). In particular, reducing
coefficients modulo 2, this implies that ${\mathcal{P}}_{6}$ is irreducible.

For any symplectic basis
$\\{\alpha_{1},...,\alpha_{g},\beta_{1},...,\beta_{g}\\}$ of
$H^{1}(C,{\mathbb{Z}})$, where $\eta=\alpha_{g}$ $(\text{mod \ 2})$, there is
a natural symplectic basis in $H^{1}(P(C,\eta),{\mathbb{Z}})$ (see e.g.
[bir_lange, Prop. 12.4.2]).

Consider a Lefschetz pencil around a Prym semi-abelian variety attached to a
covering $\tilde{C}_{0}\longrightarrow C_{0}$ such that $C_{0}$ has only one
node and the two preimages of the node are two nodes of $\tilde{C}_{0}$. Then,
writing explicitly the Picard Lefschetz transformation in terms of the basis
defined above, one easily checks that the associated monodromy generates the
whole symplectic group. ∎

We will now give a complete description of the kernel of $m\circ II$. First,
we describe the kernel of
$\tau\circ\mu_{2}:I_{2}(\omega_{C}\otimes\eta)\longrightarrow R^{12}_{C}$.

Denote by $L:={\mathcal{O}}_{C}(1)$, $M:={\mathcal{O}}_{C}(2)\otimes\eta$, so
that $L\otimes M\cong\omega_{C}\otimes\eta$. Take a point $p\in C$ and
consider the two line bundles
$L(-p)={\mathcal{O}}_{C}(1)\otimes{\mathcal{O}}_{C}(-p)$,
$M(p)={\mathcal{O}}_{C}(2)\otimes\eta\otimes{\mathcal{O}}_{C}(p)$, then
clearly $L(-p)\otimes M(p)\cong\omega_{C}\otimes\eta.$ Since $L$ is base point
free, we have $h^{0}(C,L(-p))=2$, and we fix a basis $\\{s_{1},s_{2}\\}$ of
$H^{0}(C,L(-p))$. We have a map:

$\bigwedge^{2}H^{0}(M(p))\rightarrow I_{2}(\omega_{C}\otimes\eta),$ (4.10)
$(t_{1}\wedge t_{2})\mapsto Q(t_{1},t_{2}):=s_{1}t_{1}\odot
s_{2}t_{2}-s_{1}t_{2}\odot s_{2}t_{1},$

where $\alpha\odot\beta:=\alpha\otimes\beta+\beta\otimes\alpha$.

######  Proposition .

For a general $T$, the quadrics defined in (4.10) generate the kernel of
$\tau\circ\mu_{2}$.

###### Proof.

A direct computation (see e.g. [cf_michigan, Lemma 2.2]) shows that we have:

$\mu_{2}:I_{2}(\omega_{C}\otimes\eta)\rightarrow H^{0}(C,\omega_{C}^{\otimes
4})\cong H^{0}(C,{\mathcal{O}}_{C}(12)),$
$\mu_{2}(Q(t_{1},t_{2}))=\mu_{1,L(-p)}(s_{1}\wedge
s_{2})\cdot\mu_{1,M(p)}(t_{1}\wedge t_{2}),$

where:

$\mu_{1,L(-p)}:\bigwedge^{2}H^{0}(C,L(-p))\cong\langle s_{1}\wedge
s_{2}\rangle\rightarrow H^{0}(C,L^{\otimes 2}(-2p)\otimes\omega_{C})\cong
H^{0}(C,{\mathcal{O}}_{C}(5)(-2p)),$
$\mu_{1,M(p)}:\bigwedge^{2}H^{0}(C,M(p))\rightarrow H^{0}(C,M^{\otimes
2}(2p)\otimes\omega_{C})\cong H^{0}(C,{\mathcal{O}}_{C}(7)(2p))$

denote the first gaussian maps of the line bundles $L(-p)$ and $M(p)$.

In local coordinates, if $s_{i}=f_{i}(z)l$, where $l$ is a local frame for
$L(-p)$, we have:

$\mu_{1,L(-p)}(s_{1}\wedge
s_{2})=(f^{\prime}_{1}f_{2}-f_{1}f^{\prime}_{2})l^{2}dz.$

Then, clearly we obtain:

$div(\mu_{1,L(-p)}(s_{1}\wedge s_{2}))=div(Pol_{p}(C)_{|C})-2p,$

where $Pol_{p}(C)$ denotes the polar of the plane sextic $C$ with respect to
$p$.

Choose now $t_{1}\in H^{0}(C,M(-2p))\subset H^{0}(C,M(p))$, and assume that in
local coordinates we have $t_{i}=g_{i}(z)\sigma$, where $\sigma$ is a local
frame for $M(p)$. Then we have:

$\mu_{1,M(p)}(t_{1}\wedge
t_{2})=(g^{\prime}_{1}g_{2}-g_{1}g^{\prime}_{2})\sigma^{2}dz.$

Since $ord_{p}g_{1}=3$, clearly:

$div(\mu_{1,M(p)}(t_{1}\wedge t_{2}))=2p+E,$

where ${\mathcal{O}}_{C}(E)\cong{\mathcal{O}}_{C}(7)$. Since the restriction
to $C$ gives a surjective map:

$H^{0}({\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(7))\rightarrow
H^{0}(C,{\mathcal{O}}_{C}(7)),$

there exists a polynomial $G\in
H^{0}({\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(7))$, such that
$div(G_{|C})=E$. Then:

$div(\mu_{2}(Q(t_{1},t_{2})))=div(Pol_{p}(C)_{|C})-2p+2p+E=div((Pol_{p}(C)\cdot
G)_{|C}),$

hence $\tau\circ\mu_{2}(Q(t_{1},t_{2}))=0$, since $Pol_{p}(C)\cdot G\in
J^{12}_{C}$.

It remains to show that varying $p\in C$, and choosing $t_{1}\in
H^{0}(C,M(-2p))\subset H^{0}(C,M(p))$ (which is 1 dimensional for $p$ general
in $C$), and varying $t_{2}\in H^{0}(C,M(p))$, the quadrics
$\\{Q(t_{1},t_{2})\\}$ generate a 17-dimensional subspace of
$I_{2}(\omega_{C}\otimes\eta).$

We will show it in the example (4.8), and this is enough, since it is an open
property. To do the computation, first observe that the choice of $s_{1}$
corresponds to the choice of a line $l_{1}$ thought $p$, hence we have:

$div(s_{1})=p+D_{5},$

where $D_{5}$ is an effective divisor of degree $5$. The same holds for
$s_{2}$, so:

$div(s_{2})=p+D^{\prime}_{5}.$

Thus, by our choice of $t_{1}$, $s_{1}t_{1}$ is a generator of the one
dimensional vector space $H^{0}(C,\omega_{C}\otimes\eta(-D_{5}-3p)),$ and
$s_{2}t_{1}$ is a generator of
$H^{0}(C,\omega_{C}\otimes\eta(-D^{\prime}_{5}-3p)).$ To choose the forms
$s_{1}t_{2}$ and $s_{2}t_{2}$, namely to choose $t_{2}\in H^{0}(C,M(p))$
general, it is equivalent to choose three general points $q_{1},q_{2},q_{3}\in
C$ so that:

$\langle
s_{1}t_{2}\rangle=H^{0}(C,\omega_{C}\otimes\eta(-D_{5}-q_{1}-q_{2}-q_{3})),$
$\langle
s_{2}t_{2}\rangle=H^{0}(C,\omega_{C}\otimes\eta(-D^{\prime}_{5}-q_{1}-q_{2}-q_{3})).$

By a MAGMA script one can check that varying a point $p$ on $C$, choosing two
lines through $p$, and three points $q_{1},q_{2},q_{3}$ on $C$, one obtains a
linear space of quadrics constructed as above of dimension 17. This shows that
for a generic $(C,\eta)$ the quadrics we considered generate the kernel of
$\tau\circ\mu_{2}$. ∎

######  Remark .

By Remark (4) we know that
$I_{2}(\omega_{C}\otimes\eta)+I_{2}(\omega_{D}\otimes\eta^{\prime})=N^{*}_{\mathcal{Q}/\mathcal{A}_{9},T}$.

Hence the kernel of $m\circ II$ is generated by the quadrics of
$I_{2}(\omega_{C}\otimes\eta)$ and $I_{2}(\omega_{D}\otimes\eta)$ that we have
just described.

## References