# Infinitesimal invariants of mixed Hodge structures

Rodolfo Aguilar, Mark Green and Phillip Griffiths  Department of Mathematics,
University of Miami, Coral Gables, FL 33146 E-mail address:
<EMAIL_ADDRESS>Department of Mathematics, University of California
at Los Angeles, Los Angeles, CA 90095 E-mail address<EMAIL_ADDRESS>Department of Mathematics, University of Miami, Coral Gables, FL 33146, and
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 E-mail
address<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of infinitesimal variations of mixed Hodge structures
and invariants associated to them. We describe these invariants in the case of
a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3
surface and in more detail in the case when $X$ is a cubic threefold. In this
last setting, we obtain a generic global Torelli theorem for pairs.

###### Contents

  1. 1 Introduction
  2. 2 Infinitesimal variations of mixed Hodge structures
    1. 2.1 Definitions
    2. 2.2 Mixed Hodge structures of length two
    3. 2.3 Delta Invariant for extensions of Hodge Structures
    4. 2.4 Fano n-folds
    5. 2.5 Infinitesimal Torelli for anti-canonical pairs
  3. 3 Fano-K3 pairs
    1. 3.1 Fano Threefolds and anti-canonical sections
      1. 3.1.1 Fano threefolds
      2. 3.1.2 Properties of $(X,Y)$
      3. 3.1.3 Basic exact sequences and interpretation
    2. 3.2 Cubic form
      1. 3.2.1 Cubic form for Fano threefolds
      2. 3.2.2 Cubic complex
      3. 3.2.3 Normal bundle to the period map fixing Y
      4. 3.2.4 Properties of cubics
    3. 3.3 Delta Invariant for Fano-K3 pairs
    4. 3.4 Period map in dimension three
    5. 3.5 Relation with Abel-Jacobi map
  4. 4 The Cubic 3-fold
    1. 4.1 Notation
    2. 4.2 Cubic form
      1. 4.2.1 Condition for triviality
      2. 4.2.2 Relation of C to Q and F
      3. 4.2.3 Formula for C
    3. 4.3 Associated quintic
    4. 4.4 Delta invariant for X fixed
      1. 4.4.1 Tangent space to deformations of Y
      2. 4.4.2 Tangent space to deformation space of pairs
      3. 4.4.3 Map from normal bundle of Y in X to its deformations
      4. 4.4.4 Proof of the Theorem
    5. 4.5 Generic Torelli Theorem for pairs
  5. A Macaulay2 code

##  1\. Introduction

Infinitesimal variations of Hodge structures, or IVHS for short, has been used
for some time to provide a tool to connect variational Hodge theory and
geometry. They were introduced and parts of the theory developped in [CGGH83,
GH83, Gri83]. For additional results and further applications see [PS83],
[Gre94], [Voi02, Partie VI], [CMSP17, Chapter 5].

In practice it provides connection

${\overset{\text{geometry}}{\bullet}}$${\overset{\text{Hodge
theory}}{\bullet}}$${\overset{algebra}{\bullet}}$

As such it is complementary to the use of Hodge theory in the cohomology of
algebraic varieties. Since IVHS uses only the complex vector space $V$ and not
the $\mathbb{Q}$-structure it may seem a bit surprising that it has proved as
useful as it has for geometric applications. Since it does not involve the
$\mathbb{Q}$-structure of $V$ it is not expected to be able to lead to
existence results. However it can be used for generic global uniqueness
results such as that of Donagi-Green [DG84] and Theorem 1.2 below.

Here, our first goal is to define the general notion of infinitesimal
variations of mixed Hodge structure, or IVMHS for short. We are particularly
interested in it as a method to study the geometry of smooth pairs $(X,Y)$,
that is, both $X$ and $Y$ are smooth and $Y\subset X$. In this case the mixed
Hodge structure on $H^{n}(X\setminus Y)$ can be seen as a two-step extension
of pure Hodge structures and the structure of the mixed period domain is
particularly simple [Usu84, Car87]. We can define an infinitesimal invariant
of this extension class by using the derivative of the period map; in spirit,
this is very similar to the infinitesimal invariant of a normal function
[Gre89]. To identify the basic properties of this invariant can be seen as the
second main goal of this note.

The third goal is to extensively illustrate the structure of the IVMHS in the
following setting: we specialize to the case when $X$ is a smooth projective
$n$-dimensional Fano and $Y$ is a smooth anti-canonical section. We obtain a
$n$-symmetric form out of this pair and we prove an infinitesimal Torelli
theorem for these anti-canonical pairs. This is done in section 2. Note that
an infinitesimal Torelli theorem for anti-canonical pairs $(X,D)$ with $X$ a
surface and $D$ a SNC-divisor is proved in [Fri16, Thm 3.16].

When $X$ is a Fano threefold and $Y$ an anticanonical K3 surface, the
$3$-symmetric form can be seen as an analog of the Yukawa coupling in the log
Calabi-Yau case. This form appeared in work of Markusevich, see [Mar08]. We
can then relate the infinitesimal invariant of the extension class of
$H^{3}(X\setminus Y)$ to the first derivative of this cubic form. Moreover, we
can relate this infinitesimal invariant with an Abel-Jacobi type map. This is
done in Section 3.

If we let $X$ be the cubic threefold, we can use the description of the Hodge
filtration of hypersurfaces as described in [CG80] or [Gri69] to get more
information about the cubic form and the infinitesimal invariant of the
extension class. In particular, we obtain conditions for the triviality of the
cubic form, for its non-degeneracy and smoothness. With the computer aid of
Macaulay2, we can prove the following theorem.

######  Theorem 1.1.

For $X$ a cubic threefold and $Y$ an anticanonical smooth section, both of
them general, the cubic $C$ is smooth.

Then we find that the dependence of the cubic form $C$ on the coefficients of
the polynomial $F$ defining $X$ is of very high degree. After finding explicit
formulas for the tangent space at several deformation spaces such as
deformation of pairs $(X,Y)$, deformations in the normal direction of $Y$, we
prove that the infinitesimal invariant for the MHS of $H^{3}(X\setminus Y)$ is
non-zero. This implies that this MHS is non-split.

We end section 4 with a proof of a generic Torelli theorem for pairs.

######  Theorem 1.2.

Let $X$ be the cubic $3$-fold and $Y$ an anticanonical $K3$ surface. Assume
that they are generic. Then the MHS on $H^{3}(X\setminus Y)$ determines the
pair $(X,Y)$.

In fact, our proof is construtive. It utilizes a second order invariant
involving the cubic form. Thus, we can rephrase the theorem as saying that we
can recover the pair $(X,Y)$ from the $2$-jet of the period maps for
$X,Y,X\setminus Y$ for $X,Y$ generic. To the best of our knowledge, the only
instance where some type of Torelli theorem for pairs was known, other than
infinitesimally, was the surface anti-canonical pairs as in [GHK15], see also
[Fri16].

There is a vast literature on the subject of Fano varieties, log-Calabi Yau
pairs and the geometry of cubic hypersurfaces, see for example [PS99, Huy23]
and references there-in. Here we have tried to illustrate the use of IVMHS in
some geometric questions concerning these varieties.

##  2\. Infinitesimal variations of mixed Hodge structures

###  2.1. Definitions

######  Definition 2.1.

A Hodge structure $(V,F^{\bullet})$ of weight $n$ consists of a finite-
dimensional $\mathbb{C}$-vector space $V$ and an decreasing filtration
$F^{\bullet}$ such that

$\left\\{\begin{array}[]{l}F^{n}\subset F^{n-1}\subset\ldots\subset
F^{1}\subset F^{0}=V\\\
F^{p}\oplus\bar{F}^{n-p+1}\overset{\sim}{\to}V.\end{array}\right.$

A polarization $Q:V\times V\to\mathbb{C}$ is a bilinear form satisfying the
Hodge bilinear relations.

By considering the set of filtrations $\\{F^{\prime\bullet}\\}$ of $V$ such
that the $\dim F^{\prime p}=\dim F^{p}$ and that satisfy the Hodge bilinear
relations with respect to $Q$, we obtain the so-called _period domain_ $D$.

When we have a variation of Hodge structures or VHS, see [CMSP17], over a
quasi-projective smooth base $S$, there is associated a _period map_

$\Phi:S\to\Gamma\backslash D.$

For $s\in S$, let $T:=T_{s}S$ be the tangent space at this point. We obtain a
map $\Phi_{s}:T\to T_{\Phi(s)}D$ that gives a map

$T\overset{\Phi}{\to}F^{-1}\operatorname{End}(V)/F^{0}\operatorname{End}(V).$

######  Definition 2.2.

An Infinitesimal Variation of Hodge Structures or IVHS is a triple
$(V,F^{\bullet},T)$ where $T$ is a complex vector space, $V$ is a
$\operatorname{Sym}T$-module, $F^{\bullet}$ is a decreasing filtration and
$T\otimes F^{p}\to F^{p-1}$.

We refer to [Gre94] for a gentle introduction to IVHS, applications and
relations with geometry.

######  Remark 2.1.

Usually $V$ is a $\mathbb{Q}$-vector space and we use
$V_{\mathbb{C}}=V\otimes\mathbb{C}$. For our purposes here we can take $V$ to
be a complex vector space.

######  Remark 2.2.

Setting $H^{p,q}=F^{p}/F^{p+1}$ with $p+q=n$ equal the weight, we have that
$\oplus^{p+q=n}H^{p,q}$ is a graded $\operatorname{Sym}T-$module. Frequently,
IVHS passes to this instead of using $V$. This is similar to taking the Higgs
bundle associated to a VHS. In principle, some information is lost.

######  Definition 2.3.

A mixed Hodge structure is a triple $(V,W_{\bullet},F^{\bullet})$ with
$W_{\bullet}$ an increasing filtration of $V$ and such that
$(\operatorname{Gr}_{k}^{W}(V),F^{\bullet}\cap W_{k}/W_{k+1})$ is a Hodge
structure of weight $k$.

######  Definition 2.4.

An IVMHS is $(V,W_{\bullet},F^{\bullet},T)$ where $T$ is a complex vector
space, $V$ is a $\operatorname{Sym}$ $T$-module, $W_{\bullet}$ is an
increasing filtration, $F^{\bullet}$ a decreasing filtration and $T\otimes
F^{p}\to F^{p-1}$, $T\otimes W_{k}\to W_{k}$ such that

(1) $(\operatorname{Gr}_{k}^{W}V,F^{\bullet}\cap W_{k}V/W_{k-1},T)\text{ is an
IVHS}.$

For both IVHS and IVMHS we have Koszul-type groups. For IVMHS, the filtration
$W_{\bullet}$ gives a spectral sequence whose $E_{1}$-term arises from (1).

###  2.2. Mixed Hodge structures of length two

Let $B$ be a mixed Hodge structure of length two

(2) $0\to A\to B\to C\to 0.$

We have that $A=W_{n}(B),C=\operatorname{Gr}_{n+1}^{W}(B)$. In this case, $B$
has a special structure for the period domains $D_{1},D_{2}$ of the graded
pieces $A,C$ in relation to the usual fibration

${D_{1}}$${D}$${D_{2}}$ given by ${B}$${\operatorname{Gr}_{n+1}^{W}(B)=C,}$

there is a map

${D}$${D_{1}\times
D_{2}}$${B}$${(W_{n}(B),\operatorname{Gr}_{n+1}^{W}(B)).}$$\scriptstyle{\in}$$\scriptstyle{\in}$

If we have a $\mathbb{Z}$-structure then we have that the set of equivalences
of MHS’s $B$ in (2) is

(3)
$\operatorname{Ext}^{1}_{\operatorname{MHS}}(C,A)=\frac{\operatorname{Hom}_{\mathbb{C}}(C,A)}{F^{0}\operatorname{Hom}_{\mathbb{C}}(C,A)+\operatorname{Hom}_{\mathbb{Z}}(C,A)}.$

See for example [Usu84, Car87].

This is a compact complex torus which contains a maximal abelian subvariety
$E_{a}\subset\operatorname{Ext}_{\operatorname{MHS}}^{1}(C,A)$ whose tangent
space lies in the $(0,-1)\oplus(-1,0)$ part of the tangent space to (3). If we
have a VMHS with parameter space $S$ and where $A,C$ are constant the period
mapping

(4) $\Phi:S\to e+E_{a}$

maps to a translate of $E_{a}$.

If $S$ is a quasi-projective variety then the period mapping (4) extends to
any smooth completion $\bar{S}$ and there is a factorization (omitting the $e$
in (4))

(5)
${S}$${E_{a}}$${\bar{S}}$${\operatorname{Alb}(\bar{S}).}$$\scriptstyle{\subset}$

Here we only treat the case when the weight of $C$ equals the weight of $A$
plus one. There are interesting cases when the weight of $C$ is bigger than
the weight of $A$ plus one.

######  Remark 2.3.

For a MHS $(V,W_{\bullet},F^{\bullet})$ we have the canonical Deligne
splitting $V=\oplus V^{p,q}$ where

$W_{k}=\oplus_{p+q\geq k}V^{p,q},\quad F^{p}=\oplus_{p^{\prime}\geq p}V^{p,q}$

and where $\bar{V}^{p,q}\cong V^{q,p}\mod W_{p+q-2}$. Thus length two MHS’s
have a canonical $V^{p,q}$ decomposition where $\bar{V}^{p,q}=V^{q,p}$.

###  2.3. Delta Invariant for extensions of Hodge Structures

Let $A$ and $B$ be pure Hodge structures. We have a MHS given as an extension
by the exact sequence

$0\to A\overset{f}{\to}B\overset{g}{\to}C\to 0.$

Now, suppose we vary these over a base $S$. The derivative of the period map
is

$d\Phi_{B}\in T_{S}^{*}\otimes
F^{-1}\operatorname{End}_{\mathbb{C}}(B)/F^{0}\operatorname{End}_{\mathbb{C}}(B).$

Then, $d\Phi_{B}$ has the form

$\left(\begin{array}[]{ll}d\Phi_{A}&\alpha\\\ 0&d\Phi_{C}\end{array}\right).$

To compute $\alpha$, let us have

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where $\mu_{F}\in
F^{0}\operatorname{Hom}_{\mathbb{C}}(C,B),\mu_{\mathbb{Z}}\in\operatorname{Hom}_{\mathbb{Z}}(C,B)$.
The extension class $e\in\operatorname{Ext}^{1}_{HS}(C,A)$ is defined by

$\mu_{F}-\mu_{\mathbb{Z}}=f\circ e.$

If $\mu_{F}^{\sharp},\mu_{\mathbb{Z}}^{\sharp}$ are two different choices of
sections, then

$\displaystyle\mu_{F}^{\sharp}=\mu_{F}+f\circ s_{F},$ $\displaystyle s_{F}\in
F^{0}\operatorname{Hom}_{\mathbb{C}}(C,A),$
$\displaystyle\mu_{\mathbb{Z}}^{\sharp}=\mu_{\mathbb{Z}}+f\circ
s_{\mathbb{Z}},$ $\displaystyle s_{\mathbb{Z}}\in
F^{0}\operatorname{Hom}_{\mathbb{Z}}(C,A).$

Now,

$F^{p}B=f\circ F^{p}A\oplus\mu_{F}(F^{p}C).$

Since $f,g$ are defined over $\mathbb{Z}$, they are constant under $\nabla$.
Hence,

$\nabla_{B}F^{p}B=f\circ\nabla_{A}F^{p}A\oplus\left(\mu_{F}\circ\nabla_{C}F^{p}C+\nabla\mu_{F}(F^{p}C)\right).$

Then,

$\alpha=\nabla\mu_{F}\in T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(C,A).$

The ambiguity that comes from the choice of $\mu_{F}$ is

$\nabla(f\circ s_{F})=f\circ\nabla s_{F}.$

Now, as $\mu_{F}-\mu_{\mathbb{Z}}=f\circ e$, we have
$\nabla\mu_{F}=f\circ\nabla e$ because $\nabla\mu_{\mathbb{Z}}=0$. Using
$\mu_{F}$ to decompose $B=A\oplus\mu_{F}(C)$,

$\alpha=\nabla e\in T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(C,A).$

Taking account of the ambiguity,

$\alpha=\delta e\in\frac{T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(C,A)}{\nabla
F^{0}\operatorname{Hom}_{\mathbb{C}}(C,A)}.$

Using $\nabla^{2}=0$, we have that $\delta e$ sits in the cohomology at the
middle of

(6)
$F^{0}\operatorname{Hom}_{\mathbb{C}}(C,A)\overset{\nabla}{\to}T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(C,A)\overset{\nabla}{\to}\wedge^{2}T_{S}^{*}\otimes
F^{-2}\operatorname{Hom}_{\mathbb{C}}(C,A)$

with $\nabla=\nabla_{\operatorname{Hom}(C,A)}$.

######  Definition 2.5.

The infinitesimal invariant $\delta e$ of the extension class $e$ is the class
$\nabla e$ at the middle cohomology in (6).

###  2.4. Fano n-folds

In this subsection we let $X$ be a Fano $n$-fold. By taking a global section
of $-(n-2)K_{X}$, we will construct a symmetric $n$-form in
$H^{1}(\Omega_{X}^{n-1})^{\ast}$. It will be further studied in sections 3 and
4 in the case when $n=3$. In this last case, the cubic form appeared in
[Mar08, Prop. 3.1], where Markusevich identifies it to a cubic form introduced
by Donagi-Markman in their study of algebraically completely integrable
Hamiltonian systems [DM96].

######  Lemma 2.4.

Let $X$ be a Fano $n$-fold. Then there exists a map

$H^{0}(-(n-2)K_{X})\to S^{n}H^{1}(\Omega_{X}^{n-1})^{*}$

###### Proof.

Recall that for a $n$-dimensional vector space $U$, we have that

$\bigwedge^{n-1}U\cong(\det U)\otimes U^{*},$

hence

$\displaystyle\bigwedge^{n}(\bigwedge^{n-1}U)$ $\displaystyle\cong(\det
U)^{n}\otimes\bigwedge^{n}U^{\ast}$ $\displaystyle\cong(\det U)^{n-1}.$

Globally, we have
$\wedge^{n}\Omega_{X}^{n-1}\overset{\sim}{\to}(\Omega_{X}^{n})^{\otimes(n-1)}$,
so

$S^{n}H^{1}(\Omega^{n-1})\to H^{n}(K_{X}^{n-1})\cong H^{0}(-K_{X}^{n-2})^{*},$

which is dual to

$H^{0}(-(n-2)K_{X})\to S^{n}H^{1}(\Omega_{X}^{n-1})^{*}.$

Recall that, by horizontality, we have that the period map sends

$H^{1}(T_{X})\to\operatorname{Hom}(H^{n-1,1},H^{n-2,2}),$

with $H^{i,j}:=H^{i,j}(X)$.

We have then that

$S^{n-2}H^{1}(T_{X})\to\operatorname{Hom}(H^{n-1,1},H^{1,n-1})\cong
S^{2}(H^{n-1,1})^{*},$

by the iterated period map. So, since

$H^{0}(-K_{X})\to H^{1}(T_{X})\otimes(H^{n-1,1}(X))^{*},$

we have that

${S^{n-2}H^{0}(-K_{X})}$${S^{n-2}H^{1}(T_{X})\otimes
S^{n-2}(H^{n-1,1})^{\ast}}$${S^{2}(H^{n-1,1})^{*}\otimes
S^{n-2}(H^{n-1,1})^{*}}$${S^{n}(H^{n-1,1})^{*},}$

and this factors through $H^{0}(-(n-2)K_{X})$. ∎

###  2.5. Infinitesimal Torelli for anti-canonical pairs

The content of this subsection is well-known but we use it to fix some
notation. The case when $X$ is of dimension two and $Y$ is a SNC-divisor is
treated in [Fri16, Thm 3.16].

Let $(X,Y)$ be a smooth anti-canonical pair, that is, $X$ and $Y$ are smooth
projective varieties and $Y\in\left|-K_{X}\right|$. If $\dim X=n$, we choose

(7) $\Omega\in H^{0}(\Omega^{n}(\log Y)),$

to have a fixed identification

(8) $K_{X}\otimes[Y]\cong\mathscr{O}_{X}.$

The classical proof of infinitesimal Torelli for smooth Calabi-Yau manifolds
extends directly for the map

(9) $\operatorname{Def}(X,Y)\to D_{(X,Y)}$

where $\operatorname{Def}(X,Y)$ denotes the moduli space of deformations of
the pair $(X,Y)$ and $D_{(X,Y)}$ denotes the period domain for the MHS on
$H^{n}(X\setminus Y)$.

In the case at hand, points of $D_{(X,Y)}$ are $(V,W_{\bullet},F^{\bullet})$
where the weight filtration is

$W_{n}\subset W_{n+1}=V$

and the Hodge filtration is

$F^{n}\subset F^{n-1}\subset\ldots\subset F^{0}=V_{\mathbb{C}}.$

The first piece of (9) is given by

(10) $(X,Y)\to F^{n}H^{n}(X\setminus Y)\subset F^{n-1}(X\setminus Y)$

######  Proposition 2.5.

The differential of (10) is injective.

###### Proof.

Using $H^{n}(X\setminus
Y,\mathbb{C})\cong\mathbb{H}^{n}(\Omega_{X}^{\bullet}(\log Y))$ and the
degeneration of the Hodge-de Rham spectral sequence gives

$F^{n}H^{n}(X\setminus Y)\cong H^{0}(\Omega_{X}^{n}(\log
Y))\cong\mathbb{C}\cdot\Omega$ $\operatorname{Gr}_{F}^{n-1}H^{n}(X\setminus
Y)\cong H^{1}(\Omega_{X}^{n-1}(\log Y))$

The differential of (10) is induced by

${H^{1}(T_{X}(-\log Y))}$${\operatorname{Hom}(H^{0}(\Omega_{X}^{n}(\log
Y)),H^{1}(\Omega_{X}^{n-1}(\log Y)))}$${H^{1}(\Omega_{X}^{n-1}(\log
Y))}$${H^{1}(\Omega_{X}^{n-1}(\log
Y))}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where the identifications are given by (8). The dashed arrow is then the
identity. ∎

For a diagram in the case of dimension 3, see (30).

##  3\. Fano-K3 pairs

###  3.1. Fano Threefolds and anti-canonical sections

####  3.1.1. Fano threefolds

In this section, we will consider the MHS’s on $H^{3}(X-Y)$ and dually on
$H^{3}(X,Y)$, where $X$ is smooth Fano $3$-fold and $Y\in\left|-K_{X}\right|$
is a smooth anti-canonical divisor. We set

$\displaystyle 2g-2=(-K_{X})^{3}\text{ with $g$ the genus of }X$
$\displaystyle R=\operatorname{Im}\\{H^{2}(X,\mathbb{Z})\to
H^{2}(Y,\mathbb{Z})\\}\subset\operatorname{Pic}(Y)$

and use the notation

$\displaystyle\mathcal{F}_{g}^{R}=\left\\{\begin{array}[]{l}\text{moduli stack
of pairs $(X,Y)$, with $X$ a Fano threefold of genus $g$ and}\\\ \text{$Y$ a
smooth surface in $\left|-K_{X}\right|$ with a lattice isomorphism
$R\cong\operatorname{Pic}(Y)$}\end{array}\right.$
$\displaystyle\mathcal{K}_{g}^{R}=\left\\{\begin{array}[]{l}\text{moduli space
of smooth polarized $K3$ surfaces of degree $2g-2$ with }\\\ \text{a marking
}R\subset\operatorname{Pic}(Y)\end{array}\right.$
$\displaystyle\mathcal{M}_{g}=\left\\{\begin{array}[]{l}\text{moduli space of
triples $(X,Y,f)$ where }X\in\mathcal{F}_{g}^{R},Y\in\mathcal{K}_{g}^{R}\text{
and $f$ corresponds}\\\ \text{to the graph $G_{f}\subset Y\times X$ of an
embedding $y\mapsto(y,f(y)),y\in Y$.}\end{array}\right.$

####  3.1.2. Properties of $(X,Y)$

We will use the following notation for the basic numerical invariants of the
pair $(X,Y)$:

(11) $\left\\{\begin{array}[]{ll}(i)&h^{2,1}(X)=\dim
H^{1}(\Omega_{X}^{2})=\dim J(X),\\\ &\text{where $J(X)$ is the intermediate
Jacobian of $X$;}\\\ (ii)&r=\dim{\operatorname{Im}(H^{2}(X)\to H^{2}(Y))};\\\
&\text{this map is injective by the weak Lefschetz theorem;}\\\
(iii)&\text{the \emph{genus} $g$ of $X$ is given as above by}\\\
&2g-2=(-K_{X})^{3}=(-K_{X}|_{Y})^{2};\end{array}\right.$

Since $Y\in\left|-K_{X}\right|$ and $X$ is Fano,

$N_{Y/X}=K_{X}^{-1}|_{Y}$

is an ample line bundle over a $K3$ surface. Thus

(12) $h^{i}(N_{Y/X})=0,\ i>0$

and by Riemann-Roch

(13) $h^{0}(N_{Y/X})=\frac{1}{2}N_{Y/X}^{2}+\chi(\mathscr{O}_{Y})=g+1.$

Since $-K_{X}\to X$ is ample

(14) $h^{i}(T_{X})=h^{i}(T_{X}\otimes K_{X}\otimes
K_{X}^{-1})=h^{i}(\Omega_{X}^{2}\otimes K_{X}^{-1})=0,\ i\geq 2$

where the last step is Akizuki-Nakano vanishing.

Since $X$ is Fano

(15) $h^{0}(\Omega_{X}^{p})=0,\text{ for }p>0.$

Finally, we will make the assumption

(16) $h^{0}(T_{X})=0.$

####  3.1.3. Basic exact sequences and interpretation

We will frequently use the following basic exact sequences

(17) $0\to T_{X}\otimes[Y]^{-1}\to T_{X}\to T_{X}|_{Y}\to 0.$

Note that $T_{X}\otimes[Y]^{-1}\cong\Omega_{X}^{2}$, where $[Y]$ denotes the
line bundle associated to $Y$.

(18) $0\to T_{Y}\to T_{X}|_{Y}\to N_{Y/X}\to 0$ (19) $0\to T_{X}(-\log Y)\to
T_{X}\to N_{Y/X}\to 0$ (20) $0\to T_{X}\otimes[Y]^{-1}\to T_{X}(-\log Y)\to
T_{Y}\to 0$

combining the cohomology of (19), (20) and using the assumption (16) together
with (12), (14), (15) gives the following commutative diagram:

(21)
${0}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g+1}}{H^{0}(N_{Y/X})}}$${0}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}h^{2,1}(X)}}{H^{1}(\Omega_{X}^{2})}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}20-r+h^{2,1}(X)}}{H^{1}(T_{X}(-\log
Y))}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}20}}{H^{1}(T_{Y})}}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}r}}{H^{2}(\Omega_{X}^{2})}}$${0}$${\underset{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}20-r+h^{2,1}(X)-g-1}}{H^{1}(T_{X})missing}}$${0}$$\scriptstyle{d\pi_{Y}}$$\scriptstyle{d\pi_{X}}$

The dimensions have been written in red below the relevant group. The
$d\pi_{Y}$ and $d\pi_{X}$ are the differentials of the mappings
$\pi_{Y},\pi_{X}$ in

(22)
${\operatorname{Def}(X,Y)}$${\operatorname{Def}(X)}$${\operatorname{Def}(Y)}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\pi_{Y}}$

Following [Bea04], we may refine the diagram by recalling that
$R=\operatorname{Im}\operatorname{Pic}(X)\to\operatorname{Pic}(Y)$. Then (22)
becomes

(23)
${\mathcal{F}_{g}^{R}}$${\operatorname{Def}(X)}$${\mathcal{K}_{g}^{R}}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\pi_{Y,R}}$

where $\pi_{X,R}$ is now a submersion onto an open set in a period domain
which is Hermitian symmetric of type $IV$.

###  3.2. Cubic form

####  3.2.1. Cubic form for Fano threefolds

Recall that from 2.4, we have a cubic form given by as follows. Fix a smooth
projective Fano threefold $X$. We have a map:

$H^{0}(-K_{X})\otimes H^{1}(\Omega_{X}^{2})\to H^{1}(T_{X}),$

equivalently:

$H^{0}(-K_{X})\to H^{1}(T_{X})\otimes H^{1}(\Omega_{X}^{2})^{*}.$

Now, the period map takes

$H^{1}(T_{X})\to S^{2}H^{1}(\Omega_{X}^{2})^{*},$

taken together, we have a map

${H^{0}(-K_{X})}$${S^{2}H^{1}(\Omega_{X}^{2})^{*}\otimes
H^{1}(\Omega_{X}^{2})^{*}}$${S^{3}H^{1}(\Omega_{X}^{2})^{*}}$

Thus, we have a map from $H^{0}(-K_{X})$ to cubic forms on
$H^{1}(\Omega_{X}^{2})^{*}$.

####  3.2.2. Cubic complex

######  Proposition 3.1.

There is a complex

$H^{0}(-K_{X})\to(H^{2,1}(X))^{*}\otimes
H^{1}(T_{X})\to\wedge^{2}(H^{2,1}(X))^{*}\otimes H^{1,2}(X)$

###### Proof.

Let $\tau\in H^{0}(-K_{X})$ and $\omega_{i}$ be a basis for $H^{2,1}(X)$,

$\tau\mapsto\sum\omega_{i}^{*}\otimes(\tau\rfloor\omega_{i})\mapsto\sum(\omega_{i}^{*}\wedge\omega_{j}^{*})(\tau\rfloor\omega_{i})\rfloor\omega_{j}.$

Now on the form level, the contraction commutes, but on the cohomology level,
it anti-commutes, so this is a complex. ∎

####  3.2.3. Normal bundle to the period map fixing Y

In this subsection, we will identify the normal bundle to the period map while
keeping $Y$ fixed.

Let $Y\in\left|-K_{X}\right|$ given by a section $s_{Y}\in H^{0}(X,-K_{X})$
and denote its associated cubic form by $C$.

Using the short exact sequence (20), let us denote

$H^{1}(T_{Y})_{R}:=\ker H^{1}(T_{Y})\to H^{2}(\Omega_{X}^{2}).$

######  Proposition 3.2.

We have a natural map

$H^{1}(T_{Y})_{\operatorname{R}}\to\frac{S^{2}H^{2,1}(X)^{*}}{J_{2}(C)}.$

Here $J_{2}(C)$ denotes the subspace of $S^{2}H^{2,1}(X)^{*}$ generated by the
first partials of $C$.

###### Proof.

Suppose that $Y$ is associated to a section $s_{Y}\in H^{0}(X,-K_{X})$ and
denote by $C$ its cubic form. We have that

${H^{2,1}(X)}$${H^{1}(T_{X})}$${S^{2}H^{2,1}(X)^{*}}$$\scriptstyle{s_{Y}}$$\scriptstyle{C}$$\scriptstyle{d\Phi_{X}}$

where $d\Phi_{X}$ denotes the derivative of the period map of $X$. The
diagonal map is given by first partials of $C$.

Using again (20)

${0}$${H^{1}(\Omega_{X}^{1})}$${H^{1}(T_{X}(-\log
Y))}$${H^{1}(T_{Y})_{R}}$${0}$${H^{1}(T_{X})}$${S^{2}H^{1}(\Omega_{X}^{2})^{*}}$$\scriptstyle{C}$

so $H^{1}(T_{Y})_{R}\to\frac{S^{2}H^{1}(\Omega_{X}^{2})^{*}}{J_{2}(C)}$. ∎

To interpret this, consider the following diagram

(24)
${0}$${T_{S_{Y}}}$${S^{2}H^{1,2}(X)}$${T_{\mathcal{F}_{g}^{R}}}$${H^{1}(T_{Y})_{\operatorname{R}}}$${\frac{S^{2}H^{1,2}(X)}{\operatorname{Im}T_{S_{Y}}}\cong\frac{S^{2}H^{1,2}(X)}{J_{2}(C)}}$${0}$

and
$\frac{S^{2}H^{1,2}(X)}{\operatorname{Im}T_{S_{Y}}}\cong\frac{S^{2}H^{1,2}(X)}{J_{2}(C)}$
represents the normal bundle to the period map while keeping $Y$ fixed.

####  3.2.4. Properties of cubics

Now, let us recall some general basic properties of cubics.

######  Lemma 3.3.

The following conditions are equivalent:

  * -

The cubic $C$ is non-degenerate in $S^{3}V^{\ast}$,

  * -

there is no proper subspace $V_{0}\subset V$ such that $C\in
S^{3}V_{0}^{\ast}\subset S^{3}V^{*}$,

  * -

the partials $\frac{\partial C}{\partial z_{1}^{\ast}},\ldots,\frac{\partial
C}{\partial z_{5}^{\ast}}$ are linearly independent in $S^{2}V^{\ast}$.

Recall that a cubic form $C$ is called non-degenerate if the corresponding
cubic variety spans the ambient projective space.

There is also a similar statement for smoothness.

######  Lemma 3.4.

The following conditions are equivalent:

  * -

The cubic $C$ is smooth,

  * -

$\operatorname{Var}(\frac{\partial C}{\partial
z_{1}^{\ast}},\ldots,\frac{\partial C}{\partial z_{5}^{\ast}})=\varnothing$ in
$\mathbb{P}^{4}$,

  * -

$S^{6}V^{\ast}/J_{C}=0$.

Note that, for an abstract cubic, we have the following implications

$C\not=0\Leftarrow C\text{ non- degenerate }\Leftarrow C\text{ smooth. }$

For example, $z_{1}(z_{1}^{2}+\ldots+z_{5}^{2})$ is non-degenerate, but not
smooth.

###  3.3. Delta Invariant for Fano-K3 pairs

Let us compute now the invariant $\delta e$ of section 2.3 for a Fano-K3 pair.
We will see in Corollary 3.6 that its non-trivial part can be related to the
normal bundle to the period map while keeping $Y$ fixed.

Let $Y,X$ be as above. This is, smooth manifolds, with $X$ a Fano threefold
and $Y\in\left|-K_{X}\right|$ an anti-canonical section. Consider the exact
sequence:

(25) $0\to\frac{H^{2}(Y)}{H^{2}(X)}\to H^{3}(X,Y)\to\ker(H^{3}(X)\to
H^{3}(Y))\to 0.$

We have $H^{3}(Y)=0,H^{2}(X)=\operatorname{Hdg}^{1}(C),H^{3,0}(X)=0$.
Following [Bea04], let us denote by
$H^{2}(Y)_{R}=H^{2}(Y)/\operatorname{Im}(H^{2}(X))$. The extension class of
the MHS lies in

$e_{Y/X}\in\frac{\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})}{F^{0}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})+\operatorname{Hom}_{\mathbb{Z}}(H^{3}(X),H^{2}(Y)_{R})}.$

The infinitesimal invariant $\delta e_{Y/X}:=\delta e$ from section 2.3 lies
in the middle cohomology of

${{F^{0}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})}}$${T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})}$${\wedge^{2}T_{S}^{*}\otimes
F^{-2}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R}),}$$\scriptstyle{\nabla}$

and hence

${\begin{array}[]{c}H^{2,1}(X)^{*}\otimes H^{2,0}(Y)\\\ \oplus\\\
H^{1,2}(X)^{*}\otimes
H^{1,1}(Y)_{R}\end{array}}$${\begin{array}[]{c}T_{S}^{*}\otimes
H^{2,1}(X)^{*}\otimes H^{1,1}(Y)_{R}\\\ \oplus\\\ T_{S}^{*}\otimes
H^{1,2}(X)^{*}\otimes H^{0,2}(Y)\end{array}}$${\wedge^{2}T_{S}^{*}\otimes
H^{2,1}(X)^{*}\otimes
H^{0,2}(Y).}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$

Holding $Y$ constant, $\delta e_{Y/X}$ decomposes as

$\begin{array}[]{lc}\delta{e}_{a}\in&H^{0,2}(Y)\otimes\ker(T_{S}^{*}\otimes
H^{2,1}(X)\overset{\nabla_{X}}{\to}\wedge^{2}T_{S}^{*}\otimes H^{1,2}(X))\\\
\oplus&\oplus\\\
\delta{e}_{b}\in&H^{1,1}(Y)_{R}\otimes\operatorname{coker}(H^{2,1}(X)\overset{\nabla_{X}}{\to}T_{S}^{*}\otimes
H^{1,2}(X)).\end{array}$

Using (20), we obtain the following exact sequence

(26) ${0=H^{0}(T_{Y})}$${H^{1}(T_{X}(-[Y]))}$${H^{1}(T_{X}(-\log
Y))}$${H^{1}(T_{Y}),}$${H^{1}(\Omega_{X}^{2})}$${T_{\mathcal{F}_{g}^{R}}}$${T_{\mathcal{K}_{g}^{R}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

It follows then that $T_{S_{Y}}\cong H^{1}(\Omega_{X}^{2})$, with $S_{Y}$ the
fiber of $Y$ in $\mathcal{F}_{g}^{R}\to\mathcal{K}_{g}^{R}$. Now $S=S_{Y}$,
then above formulas for $\delta e_{a}$ and $\delta e_{b}$ are

$\begin{array}[]{lc}\delta e_{a}\in&H^{0,2}(Y)\otimes\ker(H^{1,2}(X)\otimes
H^{2,1}(X)\overset{\nabla_{X}}{\to}\wedge^{2}H^{1,2}(X)\otimes H^{1,2}(X))\\\
\oplus&\oplus\\\ \delta
e_{b}\in&H^{1,1}(Y)_{R}\otimes\operatorname{coker}(H^{2,1}(X)\overset{\nabla_{X}}{\to}H^{1,2}(X)\otimes
H^{1,2}(X)).\end{array}$

######  Proposition 3.5.

The map $\delta e_{a}$ is

$\operatorname{Id}\in{H^{2,1}}^{*}(X)\otimes H^{1,2}(X)\otimes H^{0,2}(Y).$

###### Proof.

Recall that from (26) that $T_{S_{Y}}=H^{1}(T_{X}(-[Y]))$. Consider now the
exact sequence,

$0\to\mathscr{O}_{X}(-Y)\to\mathscr{O}_{X}\to\mathscr{O}_{Y}\to 0.$

From this, as $H^{0,2}(X)=0$, then

$H^{0,2}(Y)\cong\ker(H^{3}(\mathscr{O}_{X}(-Y))\to H^{3}(\mathscr{O}_{X})=0).$

The map

$T_{S_{Y}}\otimes H^{1,2}(X)\to H^{0,2}(Y),$

i.e.

$H^{1}(T_{X}(-Y))\otimes H^{2}(\Omega_{X}^{1})\to H^{3}(\mathscr{O}_{X}(-Y)),$

is cup product, using $T_{X}(-Y)\otimes\Omega_{X}^{1}\to\mathscr{O}_{X}(-Y)$.
Now, in our case $Y\in\left|-K_{X}\right|$, this becomes

$H^{1}(\Omega_{X}^{2})\otimes H^{2}(\Omega_{X}^{1})\to
H^{3}(K_{X})\cong\mathbb{C},$

by duality. So $T_{S_{Y}}\to H^{2,1}(X)\otimes H^{0,2}(Y)$ is
$H^{2,1}(X)\overset{\operatorname{Id}}{\to}H^{2,1}(X)$, identifying
$H^{0,2}(Y)\cong\mathbb{C}$. ∎

We have therefore the following diagram:

${{\operatorname{Id}}\in H^{2,1}(X)\otimes
H^{1,2}(X)}$${\wedge^{2}H^{1,2}(X)\otimes H^{1,2}(X)}$${S^{2}H^{1,2}(X)\otimes
H^{1,2}(X)}$${S^{3}H^{1,2}(X)}$$\scriptstyle{\text{partials of
}C\otimes\operatorname{Id}}$$\scriptstyle{C}$

where $C\in S^{3}H^{1,2}(X)$ is the cubic form associated to $Y\in
H^{0}(-K_{X})$.

From Proposition 3.2, we obtain the following description of $\delta e_{b}$.

######  Corollary 3.6.

The map for $\delta e_{b}$ is $H^{2,1}(X)\to S^{2}H^{1,2}$ by first partials
of $C$. Hence

$\delta e_{b}\in H^{1,1}(Y)_{R}\otimes\frac{S^{2}H^{1,2}(X)}{J_{2}(C)}.$

Note that from (24), we know that $\frac{S^{2}H^{1,2}(X)}{J_{2}(C)}$ is the
normal bundle of the image of $TS_{Y}$, the tangent bundle of fibers of
$\mathcal{F}_{g}^{R}\to\mathcal{K}_{g}^{R}$.

###  3.4. Period map in dimension three

In the case of consideration, $(X,Y)$ where $X,Y$ are smooth, $X$ is a Fano
threefold, and $Y\in\left|-K_{X}\right|$, it is well known that the
deformation space of pairs $\operatorname{Def}(X,Y)$ is smooth with tangent
space $H^{1}(T_{X}(-\log Y))$.

The differential of the period mapping for the mixed Hodge structure on
$H=H^{3}(X\setminus Y)$, obtained from the exact sequence

(27) $0\to H^{3}(X)\to H^{3}(X\setminus Y)\to H^{2}(Y)(-1)\to H^{4}(X)\to 0,$

is:

(28) $H^{1}(T_{X}(-\log
Y))\to\operatorname{Hom}(\operatorname{Gr}_{F}^{3}H,\operatorname{Gr}_{F}^{2}H)\oplus\operatorname{Hom}(\operatorname{Gr}_{F}^{2}H,\operatorname{Gr}_{F}^{1}H).$

The proof of Proposition 2.5 can be rewritten as follows.

We have that

(29) $\begin{array}[]{l}\operatorname{Gr}_{F}^{3}H\cong
H^{0}(\Omega_{Y}^{2})(-1)=:H^{3,1},\\\ \operatorname{Gr}_{F}^{2}H\cong
H^{1}(\Omega_{Y}^{1})(-1)\oplus H^{1}(\Omega_{X}^{2})=:H^{2,2}\oplus
H^{2,1},\\\ \operatorname{Gr}_{F}^{1}H\cong H^{2}(\mathscr{O}_{Y})(-1)\oplus
H^{2}(\Omega_{Y}^{1}):=H^{1,3}\oplus H^{1,2}.\end{array}$

Here, we have use the canonical splitting as in remark 2.3.

The differential of the period mapping may be pictured by

(30) ${\overset{\ \ \ H^{3,1}}{\bullet}}$${\overset{\ \ \
H^{2,2}}{\bullet}}$${\overset{\ \ \ H^{1,3}}{\bullet}\hskip
15.0pt\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$${\underset{H^{2,1}}{\bullet}}$${\underset{H^{1,2}}{\bullet}\
\ \big{\\}}W_{3}\hskip 2.0pt\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(2)}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1^{*})}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(2)}$

$W_{4}$$F^{3}$$F^{2}$

Here we represent the tangent space to a family in $\operatorname{Def}(X,Y)$
by a bullet point $\bullet$ and the arrows represent its action on the graded
pieces.

Using a choice of $\Omega\in H^{0}(\Omega_{X}^{3}(\log Y))\cong\mathbb{C}$, we
have:

$\displaystyle H^{1}(T_{X}(-\log Y))\cong H^{1}(\Omega_{X}^{2}(\log Y))$
$\displaystyle\operatorname{Gr}_{F}^{3}H\cong\mathbb{C},\operatorname{Gr}_{F}^{2}\cong
H^{1}(\Omega_{X}^{2}(\log Y))$

and then
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\oplus\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)$
equals the identity.

######  Remark 3.7.

  1. (1)

In (29) and in the diagram (30), we have split the weight filtration. Thus,
e.g., the mapping
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)$ is only
intrinsically defined in the kernel of
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)$. In
more detail, for a given $\eta\in H^{1}(T_{X}(-\log Y))$, there is a
derivative of the period map

${0}$${H^{2,1}}$${\mathbb{C}\cong H^{3,1}\cong F^{3}H^{3}(X\setminus
Y)}$${\frac{F^{2}H^{3}(X\setminus Y)}{F^{3}H^{3}(X\setminus
Y)}}$${H^{2,2}}$${0}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)}$

using the notation of (22), we have that:  (1) depends only on $\pi_{Y}(\eta)$
and therefore the kernel of  (1) on $T\operatorname{Def}(X,Y)$ equals the
kernel of $\pi_{Y}$. On the other hand,  (1) is only well-defined on
$\ker\pi_{Y}$, and is an isomorphism from $\ker\pi_{Y}\subset
T\operatorname{Def}(X,Y)$ to $H^{2,1}$.

  2. (2)

Thus, not only is infinitesimal Torelli true for pairs $(X,Y)$ as above, the
differential of the period mapping is injective on the first piece in (28) of
that mapping.

The top row in (30) is intrinsically defined; i.e., no splitting of
filtrations is involved. Moreover, there is a natural mapping

(31) $\operatorname{Def}(X,Y)\to\operatorname{Def}(Y)$

and the top row may be identified with the differential of the period mapping
of the polarized $K3$ surface $Y$. Moreover,
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1^{*})$ is
the dual of
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)$.

######  Remark 3.8.

By what was just said the kernel of the top row in (30) defines the tangent
space to the fibres of (31). We may think of such a fiber as a deformation
$X_{t}$ of $X$ where all $X_{t}$ contain a fixed $Y$ (an anchored deformation
of $(X,Y)$). On such a fiber the differential of the period mapping of the
$X_{t}$ is given by
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(2)$ in the
bottom row of the diagram (30). In principle, then we have:

On the kernel of
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)$ the
mapping $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(2)$
is determined by the mapping
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)$.

######  Question 1.

How can we express
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(2)$ in
terms of
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1)$ $+$
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(1)$?

We have a dual sequence to (27),

(27*) $0\to H^{2}(X)\to H^{2}(Y)\to H^{3}(X,Y)\to H^{3}(X)\to 0.$

This induces a MHS on $H^{3}(X,Y)$ which is dual to that of $H^{3}(X\setminus
Y)$. Let us denote $H^{1,1}(Y)/H^{1,1}(X)$ by $H^{1,1}(Y)_{\operatorname{R}}$.
The dual of (30) is:

(30*)
${\overset{H^{2,1}(X)}{\bullet}}$${\overset{H^{1,2}(X)}{\bullet}}$${\overset{H^{2,0}(Y)}{\bullet}}$${\overset{H^{1,1}(Y)_{\operatorname{R}}}{\bullet}}$${\overset{H^{0,2}(Y)}{\bullet}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\check{(2)}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\check{(2)}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\check{(1)}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\check{(1)}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\check{1}^{*})}$

###  3.5. Relation with Abel-Jacobi map

Consider $f\in\mathcal{M}_{g}$, this is, $f:Y\to X$ and $Y\subset X$ as above.
Denote by $G_{f}\in Y\times X$ its graph. We have

$\displaystyle[G_{f}]\in H^{6}(Y\times X)\cong$ $\displaystyle(H^{0}(Y)\otimes
H^{6}(X))\oplus(H^{2}(Y)\otimes H^{4}(X))$
$\displaystyle\oplus(H^{4}(Y)\otimes H^{2}(X))$ $\displaystyle\cong$
$\displaystyle\operatorname{Hom}(H^{0}(X),H^{0}(Y))\oplus\operatorname{Hom}(H^{2}(X),H^{2}(Y))$
$\displaystyle\oplus\operatorname{Hom}(H^{4}(X),H^{4}(Y)).$

Now, consider $p_{0}\in Y$ and

$\displaystyle d_{f}=p_{0}\times f(Y),\ [d_{f}]\in H^{4}(Y)\otimes
H^{2}(X)\cong\operatorname{Hom}(H^{4}(X),H^{4}(Y))$ $\displaystyle
c_{f}=Y\times f(p_{0}),\ [c_{f}]\in H^{0}(Y)\otimes
H^{6}(X)\cong\operatorname{Hom}(H^{0}(X),H^{0}(Y))$

Now $[G_{f}-d_{f}-c_{f}]\in H^{2}(Y)\otimes
H^{4}(X)\cong\operatorname{Hom}(H^{2}(X),H^{2}(Y))$. Recall that
$H^{2}(X)\overset{f^{*}}{\to}H^{2}(Y)_{\operatorname{R}}$ is $0$. Note that
$H^{2}(Y)=f^{*}H^{2}(X)\oplus H^{2}(Y)_{\operatorname{R}}$ is a direct sum of
HS.

Define $Z_{f}=G_{f}-d_{f}-c_{f}$. It projects to zero in $H^{6}(Y\times X)\mod
f^{*}H^{2}(X)\otimes H^{4}(X)$. Thus

$\operatorname{AJ}(Z_{f})_{\operatorname{R}}\in\frac{H^{5}(Y\times
X)}{F^{3}H^{5}(Y\times X)+H_{\mathbb{Z}}^{5}(Y\times
X)}\mod\operatorname{Im}(f^{*}H^{2}(X)\otimes H^{3}(X))$

is well-defined, because $[Z_{f}]_{R}=0$.

######  Theorem 3.9.

We have that

$\delta e_{Y/X}=d\operatorname{AJ}(Z_{f})_{\operatorname{R}}$

after appropriate identification.

###### Proof.

The identification is

$\displaystyle\operatorname{Hom}(H^{3}(X),H^{2}(Y)_{\operatorname{R}})$
$\displaystyle\cong H^{3}(X)\otimes H^{2}(Y)_{\operatorname{R}}(-3)$
$\displaystyle F^{p}\operatorname{Hom}(H^{3}(X),H^{2}(Y)_{\operatorname{R}})$
$\displaystyle\cong F^{p+3}(H^{3}(X)\otimes H^{2}(Y)_{\operatorname{R}})$

so

(32)
$\frac{F^{-1}\operatorname{Hom}(H^{3}(X),H^{2}(Y)_{\operatorname{R}})}{F^{0}\operatorname{Hom}(H^{3}(X),H^{2}(Y)_{\operatorname{R}})}\cong\frac{F^{2}(H^{3}(X)\otimes
H^{2}(Y)_{\operatorname{R}})}{F^{3}(H^{3}(X)\otimes
H^{2}(Y)_{\operatorname{R}})}$

and then $\delta e_{Y/X}$ and $d\operatorname{AJ}(Z_{f})_{\operatorname{R}}$
belong to a complex whose middle term is the tensor product of $T^{*}$ and
either term of (32). For example, for $\delta e_{Y/X}$ we have as in 3.3

${{F^{0}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})}}$${T_{S}^{*}\otimes
F^{-1}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R})}$${\wedge^{2}T_{S}^{*}\otimes
F^{-2}\operatorname{Hom}_{\mathbb{C}}(H^{3}(X),H^{2}(Y)_{R}).}$$\scriptstyle{\nabla}$

Now for the tangent space to $\mathcal{M}_{g}$ at $f$ we have that

$T_{\mathcal{M}_{g}}(f)\cong H^{0}(Y,N_{f(Y)/Y\times X})\cong
H^{0}(Y,f^{*}T_{X}),$

and that $d\operatorname{AJ}:H^{0}(Y,f^{*}T_{X})\to F^{2}H^{3}(Y\times
X)/F^{3}(Y\times X)$. This map can be described as follows: let

$\alpha\in H^{p,q}(X),\eta\in H^{0}(T_{X}|_{Y}).$

Now $\alpha|_{Y}\lrcorner\eta$ gives an element of $H^{p-1,q}(Y)$. We have
that

$H^{p,q}(X)^{*}\otimes H^{p-1,q}(Y)\cong H^{3-p,3-q}(X)\otimes
H^{p-1,q}(Y)\subset H^{2,3}(X\times Y)=\frac{F^{2}H^{5}(Y\times
X)}{F^{3}H^{5}(Y\times X)}.$

This induces the description of the element $d\operatorname{AJ}(Z_{f})_{R}$ by
passing to the quotient.

Now for the extension class, we have that from (17) and (20), it follows that
$H^{0}(T_{X}|_{Y})\subset H^{1}(\Omega_{X}^{2})$ consists of those elements
that let fixed $X$ and $Y$ in moduli. It induces two maps

$H^{0}(T_{X}|_{Y})\to H^{0,2}(Y)\otimes H^{2,1}(X)\cong H^{2,1}(X)$

and

$H^{0}(T_{X}|_{Y})\to H^{1,1}(Y)_{R}\otimes H^{1,2}(X).$

These correspond to $\delta e_{a}$ and $\delta e_{b}$ from Proposition 3.5 and
Corollary 3.6 respectively. ∎

We have also a classical Abel-Jacobi map associated to the pair $(X,Y)$ and
two embeddings $f,f_{0}:Y\to X$. Let $\lambda\in\operatorname{Pic}(Y)$, the
algebraic $1$-cycle $f(\lambda)-f_{0}(\lambda)$ is homologous to zero in $X$.
Thus

$\operatorname{AJ}_{X}(f(\lambda)-f_{0}(\lambda))\in J(X)$

is well-defined.

##  4\. The Cubic 3-fold

###  4.1. Notation

Let $X\subset\mathbb{P}V^{\ast}\cong\mathbb{P}^{4}$ be a cubic threefold and
fix equations: $X=\\{F=0\\}$, take $Y\in\left|-K_{X}\right|$ with
$Y=\\{F=0,Q=0\\}$ and suppose that both $X,Y$ are smooth. Let us denote by
$J_{F}$ the ideal in $S^{\bullet}V=\operatorname{Sym}^{\bullet}V$ generated by
the first partials of $F$ and denote $J_{F}\cap S^{k}V$ by $J_{F,k}$.

Note that

$F\in S^{3}V,\ Q\in S^{2}V,$ $H^{2,1}(X)=V,\ H^{1,2}=S^{4}V/J_{F,4},\
H^{3,3}(X)\cong S^{5}V/J_{F,5}\cong\mathbb{C}$

Note that in this last isomorphism with $\mathbb{C}$, a choice of an
isomorphism $K_{\mathbb{P}V^{*}}\cong\mathscr{O}_{\mathbb{P}V^{\ast}}(-5)$ is
made.

There is a map $H^{0}(-K_{X})\cong S^{2}V\to S^{3}(V^{\ast})\cong
S^{3}H^{2,1}(X)^{\ast}$ given by multiplication

$S^{2}V\otimes S^{3}V\to S^{5}V/J_{F}\cong\mathbb{C}.$

Let $C$ be the cubic form constructed in Section 3.2.

######  Lemma 4.1.

The map $H^{0}(-K_{X})\cong S^{2}V\to S^{3}(V^{\ast})\cong
S^{3}H^{2,1}(X)^{\ast}$ is induced by the cubic form $C$.

###### Proof.

Recall that the cubic form is defined by

$H^{1}(\Omega_{X}^{2})\otimes H^{0}(-K_{X})\to H^{1}(T_{X})$

and the derivative of the period map

$H^{1}(T_{X})\to S^{2}H^{1}(\Omega_{X}^{2})^{*}.$

So $H^{0}(-K_{X})\to H^{1}(\Omega_{X}^{2})^{*}\otimes
S^{2}H^{1}(\Omega_{X}^{2})^{*}$.

We have also a natural map

$\wedge^{3}T_{X}\to-K_{X}.$

Since

$\Omega_{X}^{2}\cong T_{X}\otimes K_{X},$

we get

$\wedge^{3}\Omega_{X}^{2}\to 2K_{X}.$

Now,

$S^{3}H^{1}(\Omega_{X}^{2})\to H^{3}(\wedge^{3}\Omega_{X}^{2})\to
H^{3}(2_{K})\cong H^{0}(-K_{X})^{*},$

so $S^{3}H^{1}(\Omega_{X}^{2})\to H^{3}(K_{X}^{2})\cong H^{0}(-K_{X})^{*}$.

Dually, $H^{0}(-K_{X})\to S^{3}H^{1}(\Omega_{X}^{2})^{*}$. This is the cubic
form. ∎

We have the pairing $\langle,\rangle:S^{k}V^{\ast}\otimes
S^{k}V\to\mathbb{C}$.

###  4.2. Cubic form

Here we will use the standard notation for $I,J$ ideals of a ring $R$

$(I:J):=\\{r\in R\mid rJ\subseteq I\\}$

We call it _the ideal quotient_.

####  4.2.1. Condition for triviality

Let $C$ be the cubic form constructed in Section 3.2 corresponding to
$Q\in\left|-K_{X}\right|$.

######  Lemma 4.2.

We have that $C=0$ if and only if $Q\in J_{F,2}$.

###### Proof.

We have that

$\frac{S^{2}V}{J_{F}}\otimes\frac{S^{3}V}{J_{F}}\to\frac{S^{5}V}{J_{F}}\cong\mathbb{C}$

is a non-degenerate pairing, so

$\frac{S^{2}V}{J_{F}}\to(\frac{S^{3}V}{J_{F}})^{*}\subseteq S^{3}V^{*}$

is injective. ∎

It can happen that $C=0$. For example, take $F=\sum_{i=0}^{4}z_{i}^{3}$ and
$Q=\sum_{i=0}^{4}z_{i}^{2}$. Note that $X=\\{F=0\\}$ and $Y=\\{F=Q=0\\}$ are
smooth.

####  4.2.2. Relation of C to Q and F

######  Lemma 4.3.

Let $m=(z_{1},\ldots,z_{5})$. Then

$(J_{F,k}:m)=J_{F,k-1}\text{ for }k\geq 3.$

###### Proof.

Note that $S^{a}V/J_{F}\otimes S^{b}V/J_{F}\to S^{a+b}V/J_{F}$ is non-
degenerate in each factor if $a+b\leq 5$. Now,

$\displaystyle P\in(J_{F,k}:m^{l})$ $\displaystyle\iff z^{I}P\in J_{F,k}\
\forall\left|I\right|=l$ $\displaystyle\iff z^{I}P=0\text{ in }S^{k}V/J_{F}\
\forall\left|I\right|=l$ $\displaystyle\iff P=0\text{ in }S^{k-l}V/J_{F}\text{
if }k\leq 5$ $\displaystyle\iff P\in J_{F,{k-l}}\text{ if }k\leq 5$

∎

######  Proposition 4.4.

The following equality holds

$J_{C,2}^{\perp}=(J_{F,4}:Q).$

###### Proof.

Recall that $<,>$ denote the pairing between $S^{k}V$ and $S^{k}V^{*}$ for any
$k$. Then for $P\in S^{3}V$,

$\displaystyle<P,C>=0$ $\displaystyle\iff PQ\in J_{F,5}$ by Lemma 4.2
$\displaystyle\iff P\in(J_{F,5}:Q).$

Thus $C$ is a basis for $(J_{F,5}:Q)^{\perp}\subset S^{3}V^{*}$. If
$z_{1},\ldots,z_{5}$ is a basis for $V$, then

$<z_{i}P,R>=<P,\frac{\partial}{\partial z_{i}}R>\quad\text{ for all}P\in
S^{k}V,R\in S^{k+1}V^{*}.$

So for $P\in S^{2}V$,

$<z_{i}P,C>=<P,\frac{\partial C}{\partial z_{i}}>.$

Thus,

$\displaystyle P\in J_{C,2}^{\perp}$ $\displaystyle\iff mP\in C^{\perp}$
$\displaystyle\iff mP\in(J_{F,5}:Q)$ by Lemma 4.2 $\displaystyle\iff mQP\in
J_{F,5}$ $\displaystyle\iff QP\in J_{F,4}$ by Lemma 4.3 $\displaystyle\iff
P\in(J_{F,4}:Q).$

Thus $J_{C,2}^{\perp}=(J_{F,4}:Q)$. ∎

Let

(33) $S^{2}V\overset{m_{Q}}{\to}S^{4}V/J_{F,4}\cong\mathbb{C}^{5}$

######  Corollary 4.5.

$C$ is non-degenerate if and only if $m_{Q}$ is surjective.

###### Proof.

We have that

$\displaystyle C\text{ is non degenerate}$ $\displaystyle\iff\frac{\partial
C}{\partial z_{1}},\ldots,\frac{\partial C}{\partial z_{5}}\text{ are linearly
independent}$ by Lemma 3.3 $\displaystyle\iff\dim J_{C,2}=5$
$\displaystyle\iff\dim J_{C,2}^{\perp}=10$
$\displaystyle\iff\dim(J_{F,4}:Q)=10$ by Prop 4.4 $\displaystyle\iff
S^{2}V\overset{m_{Q}}{\to}\frac{S^{4}V}{J_{F,4}}\text{ has rank }5$
$\displaystyle\iff m_{Q}\text{ is surjective}.$

∎

The following proposition follows from Lemma 3.4, which is well-known.

######  Proposition 4.6.

$C$ is smooth if and only if

$\frac{S^{k}V^{*}}{J_{C}}=0,$

for $k\geq 6$.

This allowed us to check via Macaulay2, see Appendix A, that

######  Theorem 4.7.

For generic $X$ and $Y$, the cubic $C$ is smooth.

####  4.2.3. Formula for C

Recall from Proposition 4.4 that
$\left(\frac{S^{2}V^{*}}{J_{C}}\right)^{*}\cong(J_{F,4}:Q)$, i.e.
$J_{C}=(J_{F,4}:Q)^{\perp}$. Now,

$\left\langle\frac{\partial C}{\partial
z_{i}^{*}},P\right\rangle=0\iff\left\langle C,z_{i}P\right\rangle=0$

so $C=\left(m\cdot(J_{F,4}:Q)\right)^{\perp}$. Note that
$m\cdot(J_{F,4}:Q)\subseteq(J_{F,5}:Q)$, but it may not be equal.

On the other hand, $J_{F,5}$ has codimension one in $S^{5}V$, so

$S^{3}V\overset{Q}{\to}\frac{S^{5}V}{J_{F,5}}$

has codimension one or zero in $S^{3}V$, depending on whether $Q\in J_{F,2}$.
We thus have proved the following Lemma.

######  Lemma 4.8.

If $Q\not\in J_{F,2}$, then

$span\ C=(J_{F,5}:Q)^{\perp}$

Let $V$ be a $\mathbb{C}$-vector space with $m=\dim V$. Take $F\in S^{d}V$
smooth. Then $S^{m(d-2)}V/J_{F}\cong\mathbb{C}$. We want to get information
about the formula for the cubic form. Let us digress a moment and consider the
following example.

######  Example 4.1.

Let $m=2$ and fix $d$ arbitrary. Consider

$F=A_{0}x^{d}+\ldots+A_{d}y^{d}$

and the $(2d-3)\times(2d-3)$ matrix

$\displaystyle\quad\quad{x^{*}}^{2d-4}\cdots{y^{*}}^{2d-4}$
$\displaystyle\left(\quad\begin{array}[]{c}x^{d-3}F_{x}\\\ x^{d-4}yF_{x}\\\
\vdots\\\ xy^{d-4}F_{x}\\\ y^{d-3}F_{x}\\\ y^{d-3}F_{y}\\\ \vdots\\\
x^{d-4}yF_{y}\\\ x^{d-4}F_{y}\end{array}\quad\quad\quad\right)$

This has degree $2(d-2)$ in the coefficients of $F$, and

$S^{2(d-2)}(S^{d}V)\to S^{d}V^{*},$

as

$S^{2(d-2)}(S^{d}V)\to(\det V)^{(d+1)(d-2)}\otimes S^{2}(d-2)V^{*}$

we obtain

######  Proposition 4.9.

The following conditions are equivalent:

  * •

The polynomial in $S^{2d-4}V^{*}$ we get is zero,

  * •

There is a non-trivial solution to $PF_{x}+QF_{y}=0$ with $\deg P=\deg Q=d-3$

  * •

$F_{x},F_{y}$ have $2$ common roots, counting multiplicities,

  * •

$F$ has $2$ singular points or one triple point.

###### Proof.

We will use Koszul complexes, see [Gre84].

Suppose that the singular locus $Z$ of $F$ is a smooth subvariety of $F$ of
dimension $0$.

We have

$0\to\mathscr{O}_{\mathbb{P}^{1}}(-2(d-1))[Z]\to\oplus_{2}\mathscr{O}_{\mathbb{P}^{1}}(-(d-1))\to\mathscr{O}_{\mathbb{P}^{1}}\to\mathscr{O}_{Z}\to
0$

If $F_{1}=\mu_{1}\cdot s_{Z},F_{2}=\mu_{2}\cdot s_{Z}$ near $Z$, where $s_{Z}$
defines $Z$, then the relations among $F_{1},F_{2}$ are $\mu_{2}\cdot
F_{1}-\mu_{1}F_{2}=0$, and this accounts for the $[Z]$ on the left.

Put another way, the Koszul complex

$\displaystyle 0\to$ $\displaystyle\mathscr{O}_{\mathbb{P}^{1}}(-2(d-1))$
$\displaystyle\to\oplus_{2}\mathscr{O}_{\mathbb{P}^{1}}(-(d-1))$
$\displaystyle\to\mathscr{O}_{\mathbb{P}^{1}}\to 0$ has homology
$\displaystyle\quad 0$ $\displaystyle\mathscr{O}_{Z}$
$\displaystyle\mathscr{O}_{Z}$

Let

$\displaystyle R_{k}=\frac{S^{k}V}{J_{F,k}},\quad e=2(d-2)$ $\displaystyle
K_{k}=\ker(\oplus_{2}S^{k}V\to S^{k+d-1}V)$

The hypercohomology spectral sequence has $E_{2}$ terms

$\begin{array}[]{lll}R_{e-k}^{*}&K_{e-r-(d-1)}^{*}&0\\\
0&K_{k-(d-1)}&R_{k}\end{array}$

It abuts to

$\begin{array}[]{lll}0&0&0\\\
0&\mathbb{C}^{\left|Z\right|}&\mathbb{C}^{\left|Z\right|}\end{array}$

Thus we get an exact sequence

$0\to
K_{k-(d-1)}\to\mathbb{C}^{\left|Z\right|}\overset{e_{Z}^{*}}{\to}R_{e-k}^{*}\to
R_{k}\overset{e_{Z}}{\to}\mathbb{C}^{\left|Z\right|}\to K_{e-k-(d-1)}^{*}\to
0$

where the map

$R_{k}\overset{e_{Z}}{\to}\mathbb{C}^{\left|Z\right|}\text{ is
}H^{0}(\mathscr{O}_{\mathbb{P}^{1}}(k))\to H^{0}(\mathscr{O}_{Z}(k)).$

For $k=0$, $e_{Z}$ has rank one if $Z\not=\varnothing$. So

$0\to K_{-(d-1)}\to\mathbb{C}^{\left|Z\right|}\to R_{e}^{*}\to
R_{0}\to\mathbb{C}^{\left|Z\right|}\to K_{e-(d-1)}^{*}\to 0$

Thus

$\displaystyle\left|Z\right|=1\Rightarrow K_{e-(d-1)}=0,\text{ i.e.
}K_{d-3}=0$ $\displaystyle\left|Z\right|\geq 2\Rightarrow K_{d-3}\not=0$

Taking $k=-1$, we have

$0=R_{-1}\to\mathbb{C}^{\left|Z\right|}\to K_{e+1-(d-1)}^{*}=K_{d-2}\to 0$

So $Z\not=\varnothing\iff K_{d-2}\not=0$. ∎

Once the number of variables $m\geq 3$, we have the following generalization.

######  Proposition 4.10.

$J_{F,m(d-2)}^{\bot}$ is given by the condition that $\sum P_{i}F_{i}=0$,
$\deg P_{i}=d-3$, has a solution not generated by Koszul relation, and this is
a polynomial of degree

$\left(\begin{array}[]{l}m(d-2)+m-1\\\ m-1\end{array}\right)-1$

in the coefficients of $F$.

######  Example 4.2.

Let us return to our main case when $m=5$ and $d=3$, so $m(d-2)=5$. There are
${5\choose 2}5=50$ Koszul relations. Hence, the space of possible polynomials
$P_{i}$ is $5{7\choose 3}-30=125$. Now, there are ${9\choose 4}=126$ monomials
of degree $5$. Hence, we get a $126\times 126$ matrix whose first row is given
by the monomials of degree $5$ in $z_{1},\ldots,z_{5}$ and the others are
given with by $z^{J}F_{i}$ with $\left|J\right|=3$.

$\left(\begin{matrix}z_{1}^{*5}&\cdots&z_{5}^{*5}\\\ &z^{J}F_{i}&\\\
&&\ddots\end{matrix}\right)$

The determinant of this matrix $M_{F}$ is the polynomial in $S^{5}V^{*}$ that
we search. Let $Q=\sum w_{ij}z_{i}z_{j}$ then

$P\in C^{\perp}\iff QP\in J_{F,5}\iff\left\langle
QP,M_{F}\right\rangle=0\iff\left\langle P,\sum
w_{ij}\frac{\partial^{2}}{\partial z_{i}^{*}\partial
z_{j}^{*}}M_{F}\right\rangle=0$

Hence $C=\sum w_{ij}\frac{\partial^{2}}{\partial z_{i}^{*}\partial
z_{j}^{*}}M_{F}$. This is linear in the coefficients of $Q$, and of degree
$125$ in the coefficients of $F$.

###  4.3. Associated quintic

######  Proposition 4.11.

If $C$ is smooth then $\frac{S^{5}V^{*}}{J_{C}}\cong\mathbb{C}$. This
isomorphism implies that $J_{C,5}^{\perp}=\\{P\in S^{5}V\mid
d^{3}P\subset(J_{F,4:Q})\\}$ has dimension one.

This assigns a quintic in $S^{5}V$ to the pair $(X,Y)$. We will not study this
quintic further here, but we expect it to have interesting geometry in the
setting of the pairs $(X,Y)$ as above.

###  4.4. Delta invariant for X fixed

The purpose of this subsection is to prove Theorem 4.18. It will say that the
infinitesimal invariant $\delta e_{Y/X}$ is non-zero when we held $X$
constant. In particular it will imply that the extension class $e_{Y/X}$ is
non-zero.

In the proof we will use the description of the geometric maps via different
subquotients of $S^{k}V$. For this we will first prove some formulas that will
be needed.

####  4.4.1. Tangent space to deformations of Y

Recall that, following [Bea04], we denote
$R=\operatorname{Im}\\{\operatorname{Pic}(X)\hookrightarrow\operatorname{Pic}(Y)\\}$
and

$H^{1,1}(Y)_{\operatorname{R}}=\frac{H^{1,1}(Y)}{H^{1,1}(X)}\cong\frac{\operatorname{Pic}(Y)_{\mathbb{C}}}{\operatorname{Pic}(X)_{\mathbb{C}}}=R_{\mathbb{C}}^{\perp}.$

######  Proposition 4.12.

We have an isomorphism

$H^{1,1}(Y)_{\operatorname{R}}\cong\frac{H^{0}(\mathscr{O}_{Y}(2))\oplus
H^{0}(\mathscr{O}_{Y}(3))}{\\{\sum A_{i}Q_{i}\oplus\sum A_{i}F_{i}\\}},$

where the $A_{i}$ are the same on both terms.

###### Proof.

We have a diagram

${0}$${\mathscr{O}_{Y}}$${\oplus_{5}\mathscr{O}_{Y}(1)}$${0}$${T_{Y}}$${T_{\mathbb{P}^{4}}|_{Y}}$${N_{Y/\mathbb{P}^{4}}}$${0}$${0}$

and an isomorphism
$N_{Y/\mathbb{P}^{4}}\cong\mathscr{O}_{Y}(2)\oplus\mathscr{O}_{Y}(3)$. Now

$\oplus_{5}H^{0}(\mathscr{O}_{Y}(1))\to H^{0}(T_{\mathbb{P}^{4}}|_{Y})\to
H^{1}(\mathscr{O}_{Y})=0$ $0=\oplus_{5}H^{1}(\mathscr{O}_{Y}(1))\to
H^{1}(T_{\mathbb{P}^{4}}|_{Y})\to H^{2}(\mathscr{O}_{Y})\to\oplus
H^{2}(\mathscr{O}_{Y}(1))=0$

and $H^{2}(\mathscr{O}_{Y})\cong\mathbb{C}$. So

$\frac{H^{0}(\mathscr{O}_{Y}(2)\oplus
H^{0}(\mathscr{O}_{Y}(3))}{\operatorname{Im}(\oplus_{5}H^{0}(\mathscr{O}_{Y}(1)))}\cong
H^{1,1}(Y)_{\operatorname{R}}$

i.e.

$H^{1,1}(Y)_{\operatorname{R}}\cong\frac{H^{0}(\mathscr{O}_{Y}(2))\oplus
H^{0}(\mathscr{O}_{Y}(3))}{\\{\sum A_{i}Q_{i}\oplus\sum A_{i}F_{i}\\}}.$

∎

Let us denote by $dQ\wedge dF$ the ideal in
$\oplus_{k}H^{0}(\mathscr{O}_{Y}(k))$ generated by the $Q_{i}F_{j}-Q_{j}F_{i}$
with $Q_{i}$, respectively $F_{i}$, the $i$-th partial derivative of $Q$,
respectively $F$. Similarly, denote by $\widetilde{dQ\wedge dF}$ the ideal in
$\oplus_{i}H^{0}(\mathscr{O}_{\mathbb{P}^{4}}(k))$ generated by $dQ\wedge
dF,Q$ and $F$.

######  Proposition 4.13.

We have an isomorphism

$H^{1,1}(Y)_{\operatorname{R}}\cong\frac{S^{3}V}{\widetilde{dQ\wedge dF}}.$

####  4.4.2. Tangent space to deformation space of pairs

######  Proposition 4.14.

We have an isomorphism

$T\operatorname{Def}(X,Y)\cong\frac{S^{3}V}{(dQ\wedge dF,F)}$

###### Proof.

Recall that for $Z\subset\mathbb{P}^{n}$ smooth, there is a prolongation
bundle $\Sigma_{Z}$ with extension class $c_{1}(H)\in H^{1}(\Omega_{Z}^{1})$:

$0\to\mathscr{O}_{Z}\to\Sigma_{Z}\to T_{Z}\to 0.$

We have that
$\Sigma_{\mathbb{P}^{4}}\cong\oplus_{5}\mathscr{O}_{\mathbb{P}^{4}}(1)$ and

$0\to\Sigma_{Z}\to\Sigma_{\mathbb{P}^{n}}|_{Z}\to N_{Z/\mathbb{P}^{n}}\to 0.$

In the following diagram, we are searching for an explicit description of the
bundle $E$.

(34)
${0}$${0}$${E}$${\Sigma_{Y}}$${0}$${\Sigma_{\mathbb{P}^{4}}|_{X}}$${\Sigma_{\mathbb{P}^{4}}|_{Y}}$${0}$${\mathscr{O}_{X}(2)\oplus\mathscr{O}_{X}(3)}$${N_{Y/\mathbb{P}^{4}}}$${0}$${0}$${0}$

Now we have

$0\to\Sigma_{X}\to\Sigma_{\mathbb{P}^{4}}|_{X}\to\mathscr{O}_{X}(3)\to 0,$

so

$0\to E\to\Sigma_{X}\to\mathscr{O}_{X}(2)\to 0.$

Now $\Sigma_{X}$ acts as section of $\mathscr{O}_{X}(2)$ by differentiation,

$\Sigma_{X}\otimes\mathscr{O}_{X}(2)\to\mathscr{O}_{X}(2)$

and we are acting on $Q$, so

${0}$${0}$${\mathscr{O}_{X}}$${Q\cdot\mathscr{O}_{X}}$${0}$${E}$${\Sigma_{X}}$${\mathscr{O}_{X}(2)}$${0}$${0}$${T_{X}(-\log
Y)}$${T_{X}}$${\mathscr{O}_{Y}(2)}$${0}$$\scriptstyle{\cong}$$\scriptstyle{Q,dQ}$$\scriptstyle{dQ}$

We have then that $E\cong T_{X}(-\log Y)$. Thus, since
$H^{1}(\Sigma_{\mathbb{P}^{4}|_{X}})=0$ and using the diagram (34),we conclude
that

$H^{1}(T_{X}(-\log Y))\cong\frac{H^{0}(\mathscr{O}_{X}(2))\oplus
H^{0}(\mathscr{O}_{X}(3))}{\oplus_{5}H^{0}(\mathscr{O}_{X}(1))}\cong\frac{S^{2}V\oplus
S^{3}V/F}{\\{\sum A_{i}Q_{i}\oplus\sum A_{i}F_{i}\\}}.$

∎

Using the prolongation bundle, we can give another proof of Proposition 4.12.

####  4.4.3. Map from normal bundle of Y in X to its deformations

The map

$\alpha:H^{0}(N_{Y/X})\to H^{1,1}(Y)_{\operatorname{R}}$

is given by

$\displaystyle H^{0}(\mathscr{O}_{Y}(2))$
$\displaystyle\to\frac{H^{0}(\mathscr{O}_{Y}(2))\oplus
H^{0}(\mathscr{O}_{Y}(3))}{\\{\sum A_{i}Q_{i}\oplus\sum A_{i}F_{i}\\}}.$
$\displaystyle P$ $\displaystyle\mapsto P\oplus 0$

Now $P=\sum A_{i}Q_{i}$ for some $A_{i}$, and

$P\mapsto-\sum A_{i}F_{i}\in\frac{S^{3}V}{\widetilde{dQ\wedge dF}}.$

######  Lemma 4.15.

In particular

$\displaystyle\operatorname{Im}(\alpha)\cong\frac{J_{F,3}}{J_{F,3}\cap(\widetilde{dQ\wedge
dF})},$
$\displaystyle\operatorname{coker}(\alpha)\cong\frac{S^{3}V}{(J_{F,3},Q)}\cong\frac{H^{1,1}(Y)_{\operatorname{R}}}{\operatorname{Im}H^{0}(N_{Y/X})}.$

Now

$P\in\ker(\alpha)\iff-\sum A_{i}F_{i}\in(dQ\wedge dF,F,Q),$

we can choose the $A_{i}$ in this case so that $\sum A_{i}F_{i}\in(Q)$, i.e.
$\sum A_{i}F_{i}\in J_{F,3}\cap(Q)$.

######  Lemma 4.16.

Thus

$\ker\alpha\cong J_{F,3}\cap Q\cong(J_{F,5}:Q)$

###### Proof.

This follows by writing the diagram (21) for the cubic $3$-fold. By
Propositions 4.12 and 4.14, we have that

${0}$${\frac{S^{2}V}{Q}}$${0}$${V}$${\frac{S^{3}V}{dQ\wedge
dF,F}}$${\frac{S^{3}V}{dQ\wedge
dF,F,Q}}$${0}$${\frac{S^{3}V}{J_{F}}}$${0}$$\scriptstyle{\sum
A_{i}Q_{i}\mapsto\sum A_{i}F_{i}}$$\scriptstyle{\cdot Q}$

which can be completed as

${0}$${\frac{S^{2}V}{Q}}$${\frac{S^{2}V}{Q}}$${0}$${V}$${\frac{S^{3}V}{dQ\wedge
dF,F}}$${\frac{S^{3}V}{dQ\wedge
dF,F,Q}}$${V}$${\frac{S^{3}V}{J_{F}}}$${\frac{S^{3}V}{J_{F},Q}}$${0}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{Q}$

∎

Using the first part of the Macaulay2 code we obtain.

######  Proposition 4.17.

For generic $F,Q$, we have $(J_{F,3}:Q)=0$ and $\alpha$ is injective.

####  4.4.4. Proof of the Theorem

Let us explain what is $\delta e_{Y/X}$ when $X$ is held constant.

We have two complexes

(35) ${H^{2,1}(X)^{*}\otimes H^{2,0}(Y)}$${H^{0}(N_{Y/X})^{*}\otimes
H^{2,1}(X)^{*}\otimes H^{1,1}(Y)_{\operatorname{R}}}$${\wedge
H^{0}(N_{Y/X})^{*}\otimes H^{2,1}(X)^{*}\otimes
H^{0,2}(Y)}$$\scriptstyle{\nabla}$ (36) $H^{1,2}(X)^{*}\otimes
H^{1,1}(Y)_{\operatorname{R}}\to H^{0}(N_{Y/X})^{*}\otimes
H^{1,2}(X)^{*}\otimes H^{0,2}(Y)\to 0$

The cohomology at the middle term of (36) is
$H^{1,2}(X)\otimes(\ker\alpha)^{*}$, i.e. a map

${\ker\alpha}$${H^{2,1}(X)}$${(J_{F,3}:Q)}$${V}$$\scriptstyle{\cong}$$\scriptstyle{=}$

By Proposition 4.17, we have that for generic $F$ and $Q$, this piece has no
information.

For (35), we need an element of

$H^{2,1}(X)^{*}\otimes\left(\frac{H^{0}(N_{Y/X})^{*}\otimes
H^{1,1}(X)_{\operatorname{R}}}{\mathbb{C}\alpha}\right).$

Now this maps, losing some information, to a map

${H^{2,1}(X)}$${H^{0}(N_{Y/X})^{*}}$${\mkern-18.0mu\mkern-18.0mu\mkern-18.0mu\otimes\left(\frac{H^{1,1}(X)_{\operatorname{R}}}{\operatorname{Im}\alpha}\right)}$${V}$${\left(\frac{S^{2}V}{Q}\right)}$${\frac{S^{3}V}{(J_{F,3},Q)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

This map is induced by multiplication

${V\otimes\frac{S^{2}V}{Q}}$${\frac{S^{3}V}{Q}}$${\frac{S^{3}V}{(J_{F,3},Q)}}$

We get $\delta e_{Y/X}\not=0$ for $X$ fixed provided $(J_{F,3},Q)\subsetneq
S^{3}V$, i.e. $V\overset{\alpha}{\to}\frac{S^{3}V}{J_{F,3}}$ is not
surjective. But $\dim(\frac{S^{3}V}{J_{F,3}})=10,\dim V=5$. We have proved.

######  Theorem 4.18.

For $X$ a smooth cubic threefold fixed and $Y\in\left|-K_{X}\right|$ smooth,
we have that $\delta e_{Y/X}\not=0$. More precisely, the part of $\delta
e_{Y/X}$ in (36) is non-zero.

###  4.5. Generic Torelli Theorem for pairs

######  Theorem 4.19.

Let $(X,Y)$ be a generic pair in $\mathcal{F}_{g}^{R}$, then the MHS on
$H^{3}(X\setminus Y)$ determines the pair $(X,Y)$.

###### Proof.

From the sequence (25) we can recover $H^{3}(X)$,
$H^{2}(Y)_{\operatorname{R}}$ and their polarizations from the MHS on
$H^{3}(X,Y)$.

We therefore can recover the maps $T_{\mathcal{F}_{g}^{R}}\to T_{D_{X}}$ and
$T_{\mathcal{F}_{g}^{R}}\to T_{D_{Y}}$ where $D_{X}$ and $D_{Y}$ are the
period domains for $X$ and $Y$ respectively. It is enough to know
infinitesimal Torelli for $X$ and for $Y$ to conclude that the kernels of the
maps above are the kernels of $T_{\mathcal{F}_{g}^{R}}\to
T_{\operatorname{Def}(X)}$ and $T_{\mathcal{F}_{g}^{R}}\to
T_{\mathcal{K}_{g}^{R}}$. Recall that
$H^{2,1}(X)=\ker(T_{\mathcal{F}_{g}^{R}}\to T_{D_{Y}})$ by (20).

The period map for $X$ gives $H^{1}(T_{X})\to S^{2}H^{2,1}(X)^{*}$. To get the
cubic form $C$, we need

$H^{1}(\Omega_{X}^{2})\overset{s_{Y}}{\to}H^{1}(T_{X})$

with $s_{Y}\in H^{0}(-K_{X})$.

We have

${H^{1}(T_{X}(-Y))\cong\ker(T_{\mathcal{F}_{g}^{R}}\to
T_{Kg})}$${T_{\mathcal{F}_{g}^{R}}}$${T_{\operatorname{Def}(X)}}$

We know that $H^{1}(T_{X}(-Y))\cong H^{1}(\Omega_{X}^{2})$, with the
isomorphism determined up to a constant. So we have

${H^{1}(\Omega_{X}^{2})}$${H^{1}(T_{X})}$${S^{2}H^{1}(\Omega_{X}^{2})^{*}}$

is determined up to a constant, and hence: we can determine $C\in
S^{3}H^{1}(\Omega_{X}^{2})^{*}$ up to a constant. Now $C^{\perp}\subset
S^{3}H^{2}(\Omega_{X}^{1}),C^{\perp}=(J_{F,5}:Q)\subset S^{3}V$ by Lemma 4.8.

Recall that for $X,Y$ generic, we have that $H^{0}(T_{X}|_{Y})=0$ by
Proposition 4.17 and that $\ker(T_{\mathcal{F}_{g}^{R}}\to
T_{D_{X}})\cong\frac{H^{0}(N_{Y/X})}{H^{0}(T_{X}|_{Y})}$. Therefore, by using
infinitesimal Torelli for $X$ we can recover $H^{0}(N_{Y/X})$.

We now are going to use second order information:

For $\nu\in H^{0}(N_{Y/X})$, we want to look at

$\frac{\partial}{\partial\nu}(C^{\perp})=\frac{\partial}{\partial\nu}((J_{F,5}:Q))$

Now, because $X$ is staying fixed, we may regard $H^{2,1}(X)$ as a fixed
space, and hence the isomorphism of $H^{2,1}(X)$ with $V$ is fixed.

$\frac{\partial}{\partial\nu}((J_{F,5}:Q))\in\operatorname{Hom}((J_{F,5}:Q),\frac{S^{3}V}{(J_{F,5}:Q)})$

######  Lemma 4.20.

The following equality holds

$\\{P\in(J_{F,5}:Q)\mid\frac{\partial P}{\partial\nu}=0\text{ for all }\nu\in
H^{0}(N_{Y/X})\\}=(J_{F,5}:m^{2}).$

###### Proof.

Suppose that $QP=\sum A_{i}F_{i}$, so $\frac{\partial
Q}{\partial\nu}P=\sum_{i}\frac{\partial A_{i}}{\partial\nu}F_{i}$ this implies
that $\frac{\partial Q}{\partial\nu}P\in J_{F,5}$. Now we know

$H^{0}(N_{Y/X})\cong H^{0}(\mathscr{O}_{Y}(2))\cong\frac{S^{2}V}{Q}$

and hence

$\frac{\partial P}{\partial\nu}=0\text{ for all }\nu\in H^{0}(N_{Y/X})\iff
P\in(J_{F,5}:m^{2})$

∎

Recall that

$S^{2}V\otimes\frac{S^{3}V}{J_{F,3}}\to\frac{S^{5}V}{J_{F,5}}$

is non-degenerate in $\frac{S^{3}V}{J_{F,3}}$, so

$(J_{F,5}:m^{2})=J_{F,3},$

this is, using this second order information, we can recover $J_{F,3}\subset
S^{3}V$.

Now, using Mather-Yau’s theorem we can recover $F$ from $J_{F}$.

Let us explain the dependence on $\operatorname{GL}(V)$ in order to
differentiate as we have done:

$S^{3}H^{2,1}(X)\to H^{0}(-K_{X})^{*}$

is intrinsic to $X$, and $s_{Y}\in H^{0}(-K_{X})$ is intrinsic to $(X,Y)$,
hence $S^{3}H^{2,1}(X)\overset{C}{\to}\mathbb{C}$ (up to a constant) is
intrinsic to $(X,Y)$. It does not depend on how $X$ sits in $\mathbb{P}^{4}$
as a cubic. Likewise $H^{0}(N_{Y/X})\cong H^{0}(-K_{X})/\mathbb{C}s_{Y}$ is
intrinsic to $(X,Y)$, so

$H^{0}(N_{Y/X})\to\frac{S^{3}H^{2,1}(X)^{*}}{\mathbb{C}C}$

is intrinsic to $(X,Y)$.

Now

$\displaystyle<\frac{\partial c}{\partial\nu},P>=0\quad\forall\nu\in
H^{0}(N_{Y/X})\iff P\in\ker(S^{3}H^{2,1}(X)\to H^{0}(-K_{X})^{*})$

Now, choosing an embedding $X\subset\mathbb{P}^{4}$ as a cubic,

$\ker(S^{3}H^{2,1}(X)\to H^{0}(-K_{X})^{*})\cong J_{F,3}$

and we recover $J_{F,3}$ and hence $F$.

We now know $F$, up to a constant, hence $X$. We also know $(J_{F,5}:Q)\subset
S^{3}V$

######  Lemma 4.21.

We can recover $Q$ from $F$ and $(J_{F,5}:Q)$.

###### Proof.

We may assume that $Q$ is irreducible. Now if

$(J_{F,5}:Q)=(J_{F,5}:Q^{\prime})$

for $Q,Q^{\prime}$ linearly independent, then

$Q(J_{F,5}:Q)\subseteq(Q)\cap(Q^{\prime})=(QQ^{\prime})\subseteq S^{5}V$

But $(QQ^{\prime})$ has dimension $5=\dim V$ in $S^{5}V$, whereas $(Q)$ has
dimension $35=\dim S^{3}V$, and $(J_{F,5}:Q)$ has $\operatorname{codim}1$ in
$(Q)$ because $J_{F,5}$ has $\operatorname{codim}1$ in $S^{5}$. Hence
$\dim(J_{F,5}:Q)=34$.

This is a contradiction, so $Q^{\prime}=cQ$. ∎

We thus have determined $F$ and $Q$ in $S^{\ast}H^{2,1}(X)$.

∎

##  Appendix A Macaulay2 code

This is the code used in Macaulay2 for the proof of Theorem 4.7.

    KK=ZZ/31991
    R=KK[a,b,c,d,e]
    F=random(R^1, R^{-3})
    Q=ideal(random(R^1, R^{-2}))
    FJac = ideal(jacobian(F))
    I2 =quotient(FJac,Q)
    m=matrix{{a,b,c,d,e}}
    m2=symmetricPower(2,m)
    m3=symmetricPower(3,m)
    m4=symmetricPower(4,m)
    m5=symmetricPower(5,m)
    m6=symmetricPower(6,m)
    L = intersect(I2,ideal(m3))
    Lgen=generators(L)
    Ldiff= diff(transpose(m3),Lgen)
    Lker= kernel(transpose(Ldiff))
    C=m3*generators(Lker)
    V=diff(transpose(m2), generators(I2))
    W=generators(kernel(transpose(V)))
    U=ideal(m2*W)
    U5= generators(intersect(U, ideal(m5)))
    Y = diff(transpose(m5),U5)
    Quin= generators(kernel(transpose(Y)))
    Quintic= m5*Quin
    Z= intersect(I2,ideal(m6))

We explain below the code.

  1. (1)

        KK=ZZ/31991
    R=KK[a,b,c,d,e]

We work in $\mathbb{Z}/31991$ so coefficients do not get too long. Now we work
on $\mathbb{Z}/31991[a,b,c,d,e]$, polynomial ring in $5$ variables.

  2. (2)

        F=random(R^1, R^{-3})
    Q=ideal(random(R^1, R^{-2}))

We get $F,Q$ random polynomials of degree $3$ and $2$.

  3. (3)

        FJac = ideal(jacobian(F))

$FJac$ is the ideal $J_{F}$.

  4. (4)

        I2 =quotient(FJac,Q)

$I2$ is $(J_{F}:Q)$. It consists of $10$ graded generators.

  5. (5)

        m=matrix{{a,b,c,d,e}}
    m2=symmetricPower(2,m)
    m3=symmetricPower(3,m)
    m4=symmetricPower(4,m)
    m5=symmetricPower(5,m)
    m6=symmetricPower(6,m)

This sets up matrices where entries are the monomials of degree $2,\ldots,6$.

  6. (6)

        L = intersect(I2,ideal(m3))
    Lgen=generators(L)

This computes the elements of $(J_{F}:Q)$ of degree $\geq 3$. As it happens,
this is generated by cubics. There are $34$ of these in $S^{3}V$, which has
dimension $35$.

  7. (7)

        Ldiff= diff(transpose(m3),Lgen)
    Lker= kernel(transpose(Ldiff))
    C=m3*generators(Lker)

This is the main idea. We want to compute the unique (up to scalars) $C\in
S^{3}V^{*}$ that is orthogonal to the $34$ cubics found in the previous step.
If $G_{1},\ldots,G_{34}$ are these cubics, the first line computes the
$35\times 34$ matrix whose rows are

$\frac{\partial^{3}G_{1}}{\partial{z^{*}}^{I}},\ldots,\frac{\partial^{3}G_{34}}{\partial{z^{*}}^{I}}.$

The entries are integers. The next line computes the kernel of the transpose -
this kernel is of dimension one and is a vector of $35$ integers. There third
line of this group converts these $35$ integers back into a cubic - this is
$C$ (up to a constant.

  8. (8)

        V=diff(transpose(m2), generators(I2))
    W=generators(kernel(transpose(V)))
    U=ideal(m2*W)

This does something similar to the $10$ quadrics that generate $I_{2}$. The
$5$-dimensional subspace of $S^{2}V^{*}$ that is orthogonal to these $10$
quadrics is $\operatorname{span}(\frac{\partial C}{\partial
a},\ldots,\frac{\partial C}{\partial e})$, i.e. $J_{C,2}$.

  9. (9)

        U5= generators(intersect(U, ideal(m5)))
    Y = diff(transpose(m5),U5)
    Quin= generators(kernel(transpose(Y)))
    Quintic= m5*Quin

This computes the generators of $J_{C}\cap S^{5}V^{*}$. These have
$\operatorname{codim}1$ in $S^{*}V^{*}$. Then it computes the one element of
$S^{5}V$ that is orthogonal to these, this is the quintinc.

  10. (10)

        Z= intersect(I2,ideal(m6))

This computes the generators of $J_{C}\cap S^{6}V^{*}$. If $C$ is smooth,
$S^{6}V^{*}/J_{C}=0$. If $C$ is singular, $J_{C}\cap S^{7}V^{*}$ is contained
in the set of sextics vanishing on the singular locus of $C$. We check that
$Z=J_{C}\cap S^{6}V^{*}$ equals $S^{6}V^{*}$, so $C$ is smooth.

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