# Moduli of Distributions via Singular Schemes

Maurício Corrêa  Departamento de Matemática, Instituto de Ciências Exatas,
UFMG, Av. Antônio Carlos 6627, 30123-970 Belo Horizonte MG, Brazil
<EMAIL_ADDRESS>,  Marcos Jardim  IMECC - UNICAMP
Departamento de Matemática
Rua Sérgio Buarque de Holanda, 651
13083-970 Campinas-SP, Brazil<EMAIL_ADDRESS>and  Alan Muniz
<EMAIL_ADDRESS>
(Date: October 2020)

###### Abstract.

Let $X$ be a smooth projective variety with
$\operatorname{Pic}(X)\simeq\mathbb{Z}$. We show that the map that sends a
codimension one distribution on $X$ to its singular scheme is a morphism from
the moduli space of distributions into a Hilbert scheme. We describe its
fibers and, when $X={\mathbb{P}^{n}}$, compute them via syzygies. As an
application, we describe the moduli spaces of degree 1 distributions on
${\mathbb{P}^{3}}$. We also give the minimal graded free resolution for the
ideal of the singular scheme of a generic distribution on ${\mathbb{P}^{3}}$.

###### Key words and phrases:

Distributions, Hilbert scheme, Singular scheme, Syzygy

###### 2020 Mathematics Subject Classification:

Primary 58A17, 14D20, 14J60; Secondary 14D22, 14F06, 13D02

###### Contents

  1. 1 Introduction
  2. 2 Families of Distributions
  3. 3 Moduli of Distributions and Hilbert Schemes
  4. 4 Syzygies and Distributions on $\mathbb{P}^{n}$
  5. 5 Distributions of Degree 1
    1. 5.1 Twisted Cubic
    2. 5.2 Line and Two Points
    3. 5.3 Conic and One Point
  6. A Tensor-Hom Relation for Sheaves

##  1\. Introduction

Let $X$ be a smooth projective variety over an algebraically closed field
$\kappa$ of characteristic 0. A _codimension $r$ distribution_ ${\mathscr{F}}$
on $X$ is given by an exact sequence

$\mathscr{F}:\ 0\longrightarrow
T_{\mathscr{F}}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}TX\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}N_{{\mathscr{F}}}\longrightarrow
0,$

where $T_{\mathscr{F}}$ is a reflexive sheaf of rank $m:=\dim(X)-r$, and
$N_{{\mathscr{F}}}$ is a torsion free sheaf, respectively called the _tangent_
and the _normal_ sheaves of $\mathscr{F}$. Two distributions $\mathscr{F}$ and
${\mathscr{F}}^{\prime}$ are said to be isomorphic if there exists an
isomorphism $\beta\colon T_{\mathscr{F}}\rightarrow T_{\mathscr{F}^{\prime}}$
such that $\phi^{\prime}\circ\beta=\phi$. Moreover ${\mathscr{F}}$ is called a
_foliation_ if $T_{\mathscr{F}}\subset TX$ is integrable in Frobenius’ sense
i.e. it is invariant under the Lie bracket,
$[T_{\mathscr{F}},T_{\mathscr{F}}]\subset T_{\mathscr{F}}$.

The theory of distributions and foliations has its origins in the studies of
nineteenth-century mathematicians such as Grassmann, Jacobi, Clebsch, Cartan
and Frobenius. They were motivated by the fundamental work due to Pfaff, who
proposed a geometric approach to the study of differential equations, see [17,
Chapter III]. The qualitative study of foliations induced by polynomial
differential equations was investigated by Poincaré, Darboux, Painlevé. In
modern terminology, this corresponds to the study of codimension one
holomorphic foliations on complex projective spaces. The classification and
description of spaces of codimension one foliations on projective spaces,
which are of induced by polynomial differential 1-forms, have been initiated
by Jouanolou [22] (degrees 0 and 1) and Cerveau and Lins Neto [9] (degree 2).
This is still a very active area of research.

A systematic study of (not necessarily integrable) distributions was more
recently initiated in [4]. The authors showed that there exists a quasi-
projective moduli scheme $\mathcal{D}^{P}(X)$ that parametrizes isomorphism
classes of distributions on $X$ for which the Hilbert polynomial of
$T_{\mathscr{F}}$ is equal to $P$; a more detailed description of
$\mathcal{D}^{P}(X)$ is given in Section 2 below, where we also address some
technical issues. In addition, they proved that there exists a the forgetful
morphism

$\varpi\colon\mathcal{D}^{P}\longrightarrow
M^{P},\quad\varpi([{\mathscr{F}}]):=[T_{\mathscr{F}}],$

where $M^{P}$ denotes the moduli space of stable reflexive sheaves with the
same Hilbert polynomial $P$. This technique was successfully applied to the
description of certain moduli spaces of distributions, see [4, Section 11].

However, this approach is limited by the assumption that the tangent sheaf is
stable. Such hypothesis fails, for instance, when $T_{\mathscr{F}}$ splits a
sum of line bundles. Furthermore, even then $T_{\mathscr{F}}$ is stable, the
geometry of the moduli space $M^{P}$ may not be well understood.

The aim of this work is to establish the existence of a morphism from the
moduli space of the distributions $\mathcal{D}^{P}$ to a Hilbert scheme

(1)
$\Sigma\colon\mathcal{D}^{P}\longrightarrow\operatorname{Hilb}^{Q},\quad\Sigma([{\mathscr{F}}]):=[\operatorname{Sing}({\mathscr{F}})],$

where the polynomial $Q$ is determined by $P$, see Theorem 3.2 below.

The advantage of the morphism $\mathcal{D}^{P}(X)\rightarrow{\rm Hilb}^{Q}(X)$
vis-à-vis the morphism $\varpi\colon\mathcal{D}^{P}\rightarrow M^{P}$ is that
the former always exists, while the latter requires the stability of the
tangent sheaves.

Two important problems in the theory of holomorphic distributions are related
to properties of the morphism $\Sigma$:

  1. (a)

determination of the distributions by their singular schemes
$\longleftrightarrow$ the injectivity of the map $\Sigma$;

  2. (b)

construction of distributions with prescribed singular schemes
$\longleftrightarrow$ surjectivity of the map $\Sigma$.

Problem (a) was considered by Gomez-Mont and Kempf [18] for foliation by
curves in projective spaces, with reduced, zero dimensional singular schemes.
Campillo and Olivares showed in [7] that when the hypothesis that the singular
scheme is reduced may be removed. Corrêa and Araujo [1] have provided
sufficient conditions for distributions of arbitrary dimension to be uniquely
determined by their singular schemes.

On the other hand, Problem (b) was first considered by Campillo and Olivares
in [6] in order to understand under which conditions a subscheme of
${\mathbb{P}^{2}}$ can be the singular subscheme of a foliation; they provided
a Cayley–Bacharach type theorem for codimension one foliations on
${\mathbb{P}^{2}}$. The same problem for the case of codimension one
foliations in ${\mathbb{P}^{3}}$ also has been proposed by Cerveau in [8].

Our first main result, proved in Sections 3 and 4, provides a description of
the space of codimension one distributions with fixed singular scheme as well
as an obstruction to the existence of distributions with prescribed singular
scheme.

######  Main Theorem 1.

Let $X$ be a smooth projective scheme over an algebraically closed ground
field $\kappa$ of characteristic 0 with ${\rm Pic}(X)=\mathbb{Z}$. Let
$\mathcal{D}^{P}$ be the moduli scheme parametrizing isomorphism classes of
codimension one distributions on $X$ with tangent sheaves having Hilbert
polynomial equal to $P$. For any ${\mathscr{F}}\in\mathcal{D}^{P}(X)$, the
fiber of the morphism $\Sigma$, parameterizing distributions with singular
scheme equal to $Z\in\operatorname{Hilb}^{Q}$ is a Zariski open subset of
$\mathbb{P}\left(H^{0}(\Omega^{1}_{X}(c)\otimes\mathcal{I}_{Z})\right)$ for
some integer $c$.

When $X={\mathbb{P}^{n}}$ and $Z$ is not contained in a hypersurface of degree
less than or equal to $d$, then
$H^{0}(\Omega^{1}_{{\mathbb{P}^{n}}}(d+2)\otimes\mathcal{I}_{Z})$ is
isomorphic to the space of linear first syzygies of the homogeneous ideal
associated with $Z$.

As a first application of Main Theorem 1, we consider generic codimension one
distributions on ${\mathbb{P}^{n}}$, that is, those distributions with
0-dimensional singular schemes. It was proved in [1, Theorem 1.4] that if
$d\geq n$, then a generic distribution degree $d$ on ${\mathbb{P}^{n}}$ with
zero dimensional singularities is determined by its singular scheme; however,
a careful reader will notice that their argument works in greater generality;
in fact, it holds for every $d$ and $n$, except for $d=1$ and $n=2,3$. Degree
1 distributions (foliations) on $\mathbb{P}^{2}$ are not determined by their
singular schemes, as one can easily define non isomorphic foliations with
three nondegenerate singularities in general position, see [5, p. 882]. For
$n=3$ we will use our syzygetic approach to describe distributions with a
given 0-dimensional singular scheme.

######  Main Theorem 2.

Let ${\mathscr{F}}$ be a degree $d\geq 1$ distribution on $\mathbb{P}^{3}$. If
$Z={\rm Sing}({\mathscr{F}})$ is zero-dimensional then the homogeneous ideal
$I(Z)$ has minimal graded free resolution given by

$0\longrightarrow S(-3d-2)\longrightarrow S(-2d-1)^{4}\oplus
S(-d-2)\longrightarrow S(-d-1)^{4}\oplus S(-2d)\longrightarrow
I(Z)\longrightarrow 0$

where $S=\kappa[x_{0},\dots,x_{3}]$. Moreover, the space of distributions
singular at $Z$ has dimension $4$ if $d=1$ and, if $d\geq 2$ then
${\mathscr{F}}$ is the unique degree $d$ distribution singular at $Z$.

In particular, we obtain the following corollary.

######  Corollary 1.

Let ${\mathscr{F}}$ be a degree $d$ distribution on $\mathbb{P}^{n}$ with
isolated singularities. If $(d,n)\notin\\{(1,2),(1,3)\\}$ then ${\mathscr{F}}$
is determined by its singular scheme.

Main Theorem 1 can also be used to give a full description of all the
irreducible components of the moduli space of isomorphism classes of
codimension one distributions of degree 1 on ${\mathbb{P}^{3}}$. A
classification of such distributions was given in terms of the possible Chern
classes of the tangent sheaf $T_{\mathscr{F}}$. In fact, it was shown that
$\operatorname{Sing}({\mathscr{F}})$ is never planar, and $T_{\mathscr{F}}$
either splits as a sum of line bundles or is $\mu$-stable; the relevant
information is summarized in Table 1.

$c_{2}(T_{\mathscr{F}})$ | $c_{3}(T_{\mathscr{F}})$ | $T_{\mathscr{F}}$ | $\operatorname{Sing}({\mathscr{F}})$
---|---|---|---
3 | 5 | stable | 5 points
2 | 2 | stable | line and 2 points
1 | 1 | stable | conic and 1 point
0 | 0 | split | twisted cubic
Table 1. Classification of codimension one distributions of degree 1 on
${\mathbb{P}^{3}}$, according to [4, Section 8]; the last column describes the
singular scheme of a generic distribution whose tangent sheaf has the
corresponding Chern classes.

Let $\mathcal{D}(a,b)$ denote the moduli space of degree 1 distributions
$\mathscr{F}$ such that
$(c_{2}(T_{\mathscr{F}}),c_{3}(T_{\mathscr{F}}))=(a,b)$. We prove:

######  Main Theorem 3.

For each possible pair $(a,b)$, the moduli space $\mathcal{D}(a,b)$ is an
irreducible, smooth quasi-projective variety. In addition:

  * •

the image of the morphism $\Sigma\colon\mathcal{D}(0,0)\rightarrow{\rm
Hilb}^{3t+1}({\mathbb{P}^{3}})$ is an open subset of the aCM component of
${\rm Hilb}^{3t+1}({\mathbb{P}^{3}})$, and its fibres are open subsets of
${\mathbb{P}^{1}}$; $\dim\mathcal{D}(0,0)=13$.

  * •

the morphism $\Sigma\colon\mathcal{D}(2,2)\rightarrow{\rm
Hilb}^{t+3}({\mathbb{P}^{3}})$ is dominant, and its fibres are open subsets of
${\mathbb{P}^{5}}$; $\dim\mathcal{D}(0,0)=15$.

  * •

the image of the morphism $\Sigma\colon\mathcal{D}(1,1)\rightarrow{\rm
Hilb}^{2t+2}({\mathbb{P}^{3}})$ is an open subset of the non pure component of
${\rm Hilb}^{2t+2}({\mathbb{P}^{3}})$, and its fibres are open subsets of
${\mathbb{P}^{1}}$; $\dim\mathcal{D}(0,0)=14$.

Further detailed information about the moduli spaces, the image of the
morphism $\Sigma$ as well as its fibres can be found in the Theorems 5.1, 5.2,
5.3 below. The proof of smoothness uses technical lemmata established in the
Appendix A and which might be of independent interest.

We point out that the fact that $\mathcal{D}(1,1)$ is an irreducible, smooth
quasi-projective variety of dimension $14$ was already established in [4,
Proposition 11.2] by analyzing the corresponding forgetful morphism. Let us
also mention, for the sake of completeness, that $\mathcal{D}(3,5)$, being the
moduli space of generic distributions of degree 1, is an open subset of
${\mathbb{P}^{19}}$.

Finally, we need to mention that the computations in this article were done,
in part, with the help of the computer algebra system Macaulay2 [19]. Besides
speeding up the work, it was crucial to building up intuition to achieve some
of our results.

### Acknowledgments

MC is supported by CNPq grant numbers 202374/2018-1, 302075/2015- 1,
400821/2016-8. MJ is supported by the CNPQ grant number 302889/2018-3 and the
FAPESP Thematic Project 2018/21391-1. We thank Renato Vidal Martins for his
participation in the initial stages of this project and many important
discussions. We also acknowledge the financial support from Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under the Finance Code
001.

##  2\. Families of Distributions

Let $\kappa$ be an algebraically closed field and let $X$ be a smooth
projective variety over $\kappa$. A _codimension $r$ distribution_
${\mathscr{F}}$ on $X$ is given by an exact sequence

(2) $\mathscr{F}:\ 0\longrightarrow
T_{\mathscr{F}}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}TX\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}N_{{\mathscr{F}}}\longrightarrow
0,$

where $T_{\mathscr{F}}$ is a coherent sheaf of rank $m:=\dim(X)-r$, and
$N_{{\mathscr{F}}}$ is a torsion free sheaf. The sheaves $T_{\mathscr{F}}$ and
$N_{{\mathscr{F}}}$ are called the _tangent_ and the _normal_ sheaves of
$\mathscr{F}$, respectively. Note that $T_{\mathscr{F}}$ must be reflexive
[21, Proposition 1.1].

Two distributions $\mathscr{F}$ and ${\mathscr{F}}^{\prime}$ are said to be
isomorphic if there exists an isomorphism $\beta\colon
T_{\mathscr{F}}\rightarrow T_{\mathscr{F}^{\prime}}$ such that
$\phi^{\prime}\circ\beta=\phi$. By taking exterior power
$\bigwedge^{m}\phi\colon\det(T_{\mathscr{F}})\rightarrow\bigwedge^{m}TX$,
every codimension $r$ distribution induces a section of
$H^{0}(X,\bigwedge^{m}TX\otimes\det(T_{\mathscr{F}})^{\vee})$. If
$\mathscr{F}$ and ${\mathscr{F}}^{\prime}$ are isomorphic distributions,
$\bigwedge^{m}\phi^{\prime}\circ\det(\beta)=\bigwedge^{m}\phi$; but
$\det(\beta)=\lambda$ for some constant $\lambda\in\kappa^{*}$ since
$\det(T_{\mathscr{F}})\simeq\det(T_{{\mathscr{F}}^{\prime}})$ i.e.
$\bigwedge^{m}\phi^{\prime}=\lambda\cdot\bigwedge^{m}\phi$. Therefore, every
isomorphism class of codimension $r$ distribution on $X$ induces an element in
the projective space

$\mathbb{P}\\!\left(H^{0}\\!\left(\bigwedge^{m}TX\otimes\det(T_{\mathscr{F}})^{\vee}\right)\right)=\mathbb{P}\\!\left(H^{0}\\!\left(\Omega^{r}_{X}\otimes\det(TX)\otimes\det(T_{\mathscr{F}})^{\vee}\right)\right).$

The _singular scheme_ of $\mathscr{F}$ is defined as follows. Consider the the
dual morphism $\phi^{\vee}\colon\Omega^{1}_{X}\rightarrow
T_{\mathscr{F}}^{\vee}$. Taking the maximal exterior power and a twist we
obtain

$\Omega^{m}_{X}\otimes\det(T_{\mathscr{F}})\longrightarrow\mathcal{O}_{X};$

the image of such morphism is the ideal sheaf $\mathcal{I}_{Z}$ of a subscheme
$\operatorname{Sing}({\mathscr{F}}):=Z\subset X$, called the singular scheme
of $\mathscr{F}$.

Next, we examine the notion of a family of distributions parameterized by a
scheme $S$ of finite type over $\kappa$. In fact, the authors of [28] and [4]
have proposed two slightly different notions for a _family of distributions
parameterized by $S$_, namely:

######  Definition 2.1.

According to [28], a family of distributions parameterized by $S$ is given by
an $S$-scheme $\mathcal{X}\longrightarrow S$ and a short exact exact sequence

$0\longrightarrow\mathbf{F}\longrightarrow
T\mathcal{X}_{S}\longrightarrow\mathbf{N}\longrightarrow 0,$

where $T\mathcal{X}_{S}$ stands for the relative tangent sheaf. Moreover, the
family is called flat over $S$ if so is $\mathbf{N}$.

######  Definition 2.2.

Let $X$ be a smooth projective variety and let $TX_{S}$ denote the relative
tangent sheaf of $X\times_{\kappa}S\longrightarrow S$. According to [4], a
family of distributions on $X$ parameterized by $S$ is given by a subsheaf
$\boldsymbol{\phi}\colon\mathbf{F}\hookrightarrow TX_{S}$ such that for each
$s\in S$ the restriction
$\boldsymbol{\phi}_{s}\colon\mathbf{F}|_{s}\rightarrow TX_{s}$ is injective
and $\operatorname{coker}(\phi_{s})$ is torsion free.

Throughout this work will stick with the later definition but, in fact, we can
reconcile both proposals. In order to achieve that we must make some technical
considerations. Recall the following consequence [16, Exposé IV, p.81] of the
Local Flatness Criterion, see also [21, p.35].

######  Lemma 2.3.

Let $(A,\mathfrak{m})\longrightarrow(B,\mathfrak{n})$ be a morphism of
Noetherian local rings $(\mathfrak{m}B\subset\mathfrak{n})$ and let $u\colon
M^{\prime}\longrightarrow M$ be a morphism of finitely generated $B$-modules.
Suppose that $M$ is flat over $A$. Then the following are equivalent:

  1. (1)

$u$ is injective and $\operatorname{coker}(u)$ is flat over $A$;

  2. (2)

$u\otimes 1\colon M^{\prime}\otimes_{A}A/\mathfrak{m}\longrightarrow
M\otimes_{A}A/\mathfrak{m}$ is injective.

Suppose that $S$ and $X$ are noetherian and $X$ is flat over $S$. As any
coherent sheaf $\mathcal{N}$ on $X$ is the quotient of a locally free sheaf,

$0\longrightarrow\ker(\phi)\longrightarrow
E\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\mathcal{N}\longrightarrow
0.$

Then Lemma 2.3 tells us that $\mathcal{N}$ is flat over $S$ if and only if the
sequence above base changes correctly i.e. for every $s\in S$ we have

$\ker(\phi)|_{s}\simeq\ker(\phi_{s}).$

Note that we have only used that $X$ is flat over $S$ to imply that $E$ being
locally free (hence flat over $X$) is also flat over $S$.

######  Proposition 2.4.

A family of distributions according to Definition 2.2 is equivalent to a flat
family in Definition 2.1 such that $\mathcal{X}=X\times_{\kappa}S$.

###### Proof.

Since $X$ is trivially flat over $\operatorname{Spec}(\kappa)$ we know that
$X\times S$ is flat over $S$ since flatness has base change property. Also
note that both are noetherian since they are of finite type over $\kappa$. Now
$TX_{S}$ is a locally free sheaf on $X\times S$ hence flat over $X\times S$
and, by transitivity of flatness [20, Proposition 9.1A], $TX_{S}$ is flat over
$S$. Then we apply Lemma 2.3 to the morphism
$\boldsymbol{\phi}\colon\mathbf{F}\longrightarrow TX_{S}$ to show that
$\operatorname{{coker}}(\boldsymbol{\phi})$ is flat over $S$ if and only if
$\boldsymbol{\phi}_{s}\colon\mathbf{F}|_{s}\longrightarrow TX_{s}$ is
injective for every $s\in S$. ∎

######  Remark 2.5.

Note that since $TX_{S}$ is flat over $S$,
$\operatorname{{coker}}(\boldsymbol{\phi})$ being flat implies that
$\mathbf{F}$ is flat over $S$ as well. This follows from the long exact
sequence of $\operatorname{Tor}$ sheaves.

##  3\. Moduli of Distributions and Hilbert Schemes

We now recall the definition of the moduli space of distributions, which is
based on the notion of family of distributions explained in the previous
section. Let $X$ be a smooth projective variety with ${\rm
Pic}(X)\simeq\mathbb{Z}$ and denote by $\mathcal{O}_{X}(1)$ the ample
generator. We will write $\mathcal{O}_{X}(k)$, $k\in\mathbb{Z}$, for its
tensor powers.

Let $\mathfrak{Sch}_{/\kappa}$ denote the category of schemes of finite type
over $\kappa$, and $\mathfrak{Sets}$ be the category of sets. Fix a polynomial
$P\in\mathbb{Q}[t]$, and consider the functor

${\mathcal{D}}{\it ist}^{P}_{X}\colon\mathfrak{Sch}_{/\kappa}^{\rm
op}\longrightarrow\mathfrak{Sets},\quad{\mathcal{D}}{\it
ist}^{P}_{X}(S):=\\{(\mathbf{F},\boldsymbol{\phi})\\}/\sim,$

where $\\{(\mathbf{F},\boldsymbol{\phi})\\}$ is the set of families of
distributions on $X$, in the sense of Definition 2.2, and we say that
$(\mathbf{F},\boldsymbol{\phi})\sim(\mathbf{F}^{\prime},\boldsymbol{\phi}^{\prime})$
if there exists and isomorphism
$\boldsymbol{\beta}\colon\mathbf{F}\rightarrow\mathbf{F}^{\prime}$ such that
$\boldsymbol{\phi}=\boldsymbol{\phi}^{\prime}\circ\boldsymbol{\beta}$.

One can show that the functor ${\mathcal{D}}{\it ist}^{P}_{X}$ is represented
by a quasi-projective scheme $\mathcal{D}^{P}(X)$, that is, there exists an
isomorphism of functors ${\mathcal{D}}{\it
ist}^{P}_{X}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}(\makebox[6.45831pt]{{$\cdot$}},\mathcal{D}^{P}(X))$
[4, Proposition 2.4]. Furthermore, the scheme $\mathcal{D}^{P}(X)$ is an open
subset of the quot scheme ${\rm Quot}^{P_{TX}-P}(TX)$. This last observation
implies that the Zariski tangent space of $\mathcal{D}^{P}(X)$ at the point
$[F,\phi]$ is $\operatorname{Hom}(F,\operatorname{{coker}}\phi)$ and leads to
the following smoothness criterion as an immediate consequence of [21,
Proposition 2.2.8]

######  Proposition 3.1.

Let $[\mathscr{F}]\in\mathcal{D}^{P}(X)$ be the isomorphism class of a
distribution on $X$. If
$\operatorname{Ext}^{1}(T_{\mathscr{F}},N_{\mathscr{F}})=0$, then
$\mathcal{D}^{P}(X)$ is smooth at $[\mathscr{F}]$ and $\dim
T_{[\mathscr{F}]}\mathcal{D}^{P}(X)=\dim\operatorname{Hom}(T_{\mathscr{F}},N_{\mathscr{F}})$.

Unfortunately, the vanishing condition
$\operatorname{Ext}^{1}(T_{\mathscr{F}},N_{\mathscr{F}})=0$ is not satisfied
by most of the codimension one distributions of degree 1, treated in Section 5
below. For these examples we compute the $\dim\mathcal{D}^{P}(X)$ and
$\dim\operatorname{Hom}(T_{\mathscr{F}},N_{\mathscr{F}})$ separately and show
that they match.

We are finally in position to prove our first main result.

######  Theorem 3.2.

Let $X$ be a smooth projective scheme over an algebraically closed ground
field $\kappa$. Assume further that ${\rm Pic}(X)\simeq\mathbb{Z}$. Let
$\mathcal{D}^{P}(X)$ be the scheme parameterizing codimension one
distributions (up to isomorphism) on $X$ with tangent sheaves having Hilbert
polynomial equal to $P$. Then there is a morphism

$\Sigma\colon\mathcal{D}^{P}(X)\longrightarrow{\rm Hilb}^{Q}(X),$

where $Q(t):=P_{X}(t)-P_{TX}(t+c)+P(t+c)$, for some $c\in\mathbb{Z}$, which
assigns each distribution to its singular scheme.

###### Proof.

We claim that there is a natural transformation between the functors
$\mathcal{D}ist^{P}_{X}$ and $\mathcal{H}ilb^{Q}_{X}$. In fact, for each
$S\in\mathfrak{Sch}_{\kappa}$, we have that an element of
$\mathcal{D}ist^{P}_{X}(S)$ is a morphism of sheaves

$F\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}TX_{S}$

with $F$ coherent on $X\times S$, such that for each $s\in S$, the sequence

$0\longrightarrow
F_{s}\stackrel{{\scriptstyle\phi_{s}}}{{\longrightarrow}}TX\longrightarrow{\rm
coker}\,\phi_{s}\longrightarrow 0$

is a distribution on $X$ and $F_{s}$ has Hilbert polynomial $P$. As $F_{s}$ is
of codimension $1$ and $X$ is smooth, we have that ${\rm
coker}\phi_{s}=\mathcal{I}_{Z_{s}}\otimes\mathcal{L}_{s}$, where $Z_{s}$ is
the singular scheme of $F_{s}$ and $\mathcal{L}_{s}$ is a line bundle on $X$.
Thus $\mathcal{L}_{s}\cong\mathcal{O}_{X}(-c_{s})$ since ${\rm
Pic}(X)=\mathbb{Z}$.

Now, as the Hilbert polynomial of $F_{s}$ does not depend on $s$, it follows
that $d_{s}=d$ is constant. So set $\mathcal{I}:={\rm coker}\,\phi\otimes
p_{X}^{*}(\mathcal{O}_{X}(c))$. As $\mathcal{I}^{\vee\vee}$ is locally free
owing to [23, Lemma 6.13], and as ${\rm Pic}(X)=\mathbb{Z}$, we have that
$(\mathcal{I}^{\vee\vee})_{s}\cong\mathcal{O}_{X}$ for every $s\in S$.
Therefore [26, Proposition p.85] shows that
$\mathcal{I}^{\vee\vee}=\pi^{\ast}\mathcal{N}$ for some line bundle
$\mathcal{N}$ on $S$.

On the other hand, tensoring with a line bundle from the base does not affect
the family [21, Definition 4.1.1], so we assume that
$\mathcal{I}^{\vee\vee}\cong\mathcal{O}_{X\times S}$. Thus the natural map
$\mathcal{I}\longrightarrow\mathcal{I}^{\vee\vee}$ yields an inclusion of
sheaves $\mathcal{I}\hookrightarrow\mathcal{O}_{X\times
S}\in\mathcal{H}ilb^{Q}_{X}(S)$. This, by its turn, yields a morphism
$\Sigma\colon\mathcal{D}^{P}(X)\longrightarrow{\rm Hilb}^{Q}(X)$ which, by
construction, assigns each distribution to its singular scheme.

∎

In the study of the morphism $\Sigma$ it is useful to understand its fibers.
Let $X$ be as in Theorem 3.2 and let ${\mathscr{F}}\in\mathcal{D}^{P}(X)$ with
$Z=\operatorname{Sing}({\mathscr{F}})$. Then any distribution on $X$ singular
at $Z$ is defined by a twisted $1$-form $\omega\in
H^{0}(\Omega^{1}_{X}(c)\otimes\mathcal{I}_{Z})$. In particular, the fiber of
$\Sigma$ over $Z$ lies inside

$\mathbb{P}(H^{0}(\Omega^{1}_{X}(c)\otimes\mathcal{I}_{Z})).$

A priori, the singular scheme of $\omega$ could be larger, even of codimension
one. However, the locus of $1$-forms vanishing only on $Z$ is a (nonempty)
Zariski open set, thanks to the flattening stratification [25, Chapter 8].

######  Proposition 3.3.

Let $X$ be a smooth projective scheme over an algebraically closed ground
field $\kappa$. Assume further that ${\rm Pic}(X)\simeq\mathbb{Z}$. Let
${\mathscr{F}}\in\mathcal{D}^{P}(X)$. Then the fiber

$\Sigma^{-1}\left(\Sigma({\mathscr{F}})\right)\subset\mathbb{P}\\!\left(H^{0}(\Omega^{1}_{X}(c)\otimes\mathcal{I}_{Z})\right)$

parameterizing distributions with singular scheme equal to $Z$ is Zariski
open.

###### Proof.

Let
$S=\mathbb{P}\left(H^{0}(X,\Omega^{1}_{X}(c)\otimes\mathcal{I}_{Z})\right)$
and consider $\omega_{S}\in
H^{0}\\!\left(X_{S},\Omega^{1}_{X}(c)\boxtimes\mathcal{O}_{S}(1)\right)$ the
“tautological” $1$-form. Then we have the family $\mathcal{Z}$ of subschemes
defined by

$TX(-c)\boxtimes\mathcal{O}_{S}(-1)\stackrel{{\scriptstyle\omega_{S}}}{{\longrightarrow}}\mathcal{O}_{X_{S}}\longrightarrow\mathcal{O}_{\mathcal{Z}}\longrightarrow
0$

Note that for each point $s\in S$, the fiber $\mathcal{Z}_{s}$ the singular
scheme of $\omega_{s}$, the restriction of $\omega_{S}$ to the fiber over $s$.
We may apply the flattening stratification to $\mathcal{O}_{\mathcal{Z}}$,
[25, Corollary, p.60].

It follows that the map $s\mapsto\chi(\mathcal{O}_{\mathcal{Z}_{s}}(t))$ is
locally constant. We denote its finite set of values, by $N$. Then each $f\in
N$ corresponds to a stratum $S_{f}$ whose underlying set $|S_{f}|$ is composed
by the closed points $s\in S$ such that
$\chi(\mathcal{O}_{\mathcal{Z}_{s}}(t))=f(t)$. Moreover the Zariski closure of
$|S_{f}|$ is contained in $\cup_{g\geq f}|S_{g}|$, where $f>g$ if and only if
$f(t)>g(t)$ for $t\gg 0$.

Let $p$ be the Hilbert polynomial of $Z$. Note $p\in N$ since $Z$ is the
singular scheme of ${\mathscr{F}}$. We have that $Z\subset\mathcal{Z}_{s}$ for
every $s\in S$ (closed) hence $p$ is minimal in $N$. Therefore $S_{p}\subset
S$ is a nonempty open subscheme. ∎

######  Remark 3.4.

Note that the tautological $1$-form restricted to $S_{p}$ induces the
universal family of the fiber:

$0\longrightarrow T_{\mathscr{F}}\longrightarrow
TX_{S_{p}}\stackrel{{\scriptstyle\omega_{S_{p}}}}{{\longrightarrow}}\mathcal{I}_{\mathcal{Z}}\otimes\mathcal{O}_{S_{p}}(1)\boxtimes\mathcal{O}_{X}(c)\longrightarrow
0$

##  4\. Syzygies and Distributions on $\mathbb{P}^{n}$

Recall that the degree of a codimension one distribution ${\mathscr{F}}$ on
${\mathbb{P}^{n}}$ is geometrically defined by the tangency between
${\mathscr{F}}$ and a general line $L$. If a global section of
$\Omega_{\mathbb{P}^{n}}^{1}(k)$ defines ${\mathscr{F}}$ then its restriction
to $L$ defines a degree $k-2$ divisor (on $L$), called the tangency divisor.
Thus the degree is defined by $d=k-2$, see [12, section 2.5]. Note that we
also have $d=2-c_{1}(T_{\mathscr{F}})$.

Given $Z$ the singular scheme of a degree $d$ distribution on
$\mathbb{P}^{n}$, the distributions singular at $Z$ are defined by twisted
$1$-forms vanishing on $Z$. We will now show a correspondence between these
forms and linear first syzygies of the homogeneous ideal $I(Z)$, provided that
$Z$ is not contained in a hypersurface of degree $\leq d$.

As a consequence of the classification obtained in [4, Section 8], every
codimension one distribution of degree 1 on ${\mathbb{P}^{3}}$ satisfies the
condition posed in the previous paragraph, since
$h^{0}(\mathcal{I}_{Z}(d))=h^{1}(T_{\mathscr{F}}(-2))=0$ for every possible
tangent sheaf $T_{\mathscr{F}}$. In higher degree, one can show that
codimension one distribution on ${\mathbb{P}^{n}}$ with zero-dimensional
singular scheme also satisfies this hypothesis.

######  Definition 4.1.

A codimension one distribution on ${\mathbb{P}^{n}}$ is said to be _generic_
if its singular scheme is a (not necessarily reduced) zero-dimensional
subscheme of ${\mathbb{P}^{n}}$.

The previous definition is justified by the fact that codimension one
distributions of degree $d$ with zero-dimensional singular scheme form an open
subset of $\mathbb{P}(H^{0}({\mathcal{O}}_{\mathbb{P}^{n}}(d+2)))$.

######  Lemma 4.2.

Let $Z$ be a zero-dimensional subscheme of $\mathbb{P}^{n}$. If $Z$ is the
singular scheme of a degree $d\geq 1$ distribution then $Z$ is not contained
in a hypersurface of degree $\leq d$.

###### Proof.

Let $\mathcal{I}_{Z}$ be the ideal sheaf of $Z$. The scheme $Z$ is not
contained in a hypersurface of degree at most $d$ if and only if
$H^{0}(\mathcal{I}_{Z}(d))=\\{0\\}$. We will prove the later.

By hypothesis, $\mathcal{I}_{Z}$ is the image of a morphism $\omega\colon{\rm
T}{\mathbb{P}^{n}}(-d-2)\longrightarrow\mathcal{O}_{\mathbb{P}^{n}}$ and the
associated Koszul complex is exact. Then we may use it to compute
$H^{0}(\mathcal{I}_{Z}(d))$. Twisting the Koszul resolution by
$\mathcal{O}_{\mathbb{P}^{n}}(d)$ and splitting it into short exact sequences
yields the following.

$\displaystyle
0\longrightarrow\mathcal{O}_{\mathbb{P}^{n}}((d+1)(1-n))\longrightarrow$
$\displaystyle\bigwedge\limits^{n-1}{\rm
T}{\mathbb{P}^{n}}(-(n-1)(d+2)+d)\longrightarrow F_{n-2}\longrightarrow 0$
$\displaystyle\vdots$ $\displaystyle 0\longrightarrow F_{j}\longrightarrow$
$\displaystyle\bigwedge\limits^{j}{\rm
T}{\mathbb{P}^{n}}(-j(d+2)+d)\longrightarrow F_{j-1}\longrightarrow 0$
$\displaystyle\vdots$ $\displaystyle 0\longrightarrow F_{1}\longrightarrow$
$\displaystyle{\rm
T}{\mathbb{P}^{n}}(-2)\longrightarrow\mathcal{I}_{Z}(d)\longrightarrow 0$

Note that, for $1\leq j\leq n-1$ and $d\geq 1$,

$H^{j-1}\\!\left(\bigwedge^{j}{\rm
T}{\mathbb{P}^{n}}(-j(d+2)+d)\right)=\\{0\\}$

Then the long exact sequences of cohomology yields the following injections

$H^{0}(\mathcal{I}_{Z}(d))\hookrightarrow
H^{1}(F_{1})\hookrightarrow\dots\hookrightarrow
H^{n-2}(F_{n-2})\hookrightarrow
H^{n-1}(\mathcal{O}_{\mathbb{P}^{n}}((d+1)(1-n)))=\\{0\\}.$

This completes the proof. ∎

######  Remark 4.3.

The previous lemma has a simpler alternative proof in the case $n=3$. Let

${\mathscr{F}}~{}:~{}0\longrightarrow T_{\mathscr{F}}\longrightarrow{\rm
T}{\mathbb{P}^{3}}\longrightarrow\mathcal{I}_{Z}(d+2)\longrightarrow 0$

be a generic codimension one distribution of degree $d$ on ${\mathbb{P}^{3}}$;
set $Z:=\operatorname{Sing}({\mathscr{F}})$. Since $\dim Z=0$, we have that
${\mathcal{E}}{\it xt}^{1}(\mathcal{I}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})=0$.
Thus dualizing the exact sequence above, we obtain the exact sequence

$0\longrightarrow{\mathcal{O}}_{\mathbb{P}^{3}}(-d-2)\longrightarrow\Omega^{1}_{\mathbb{P}^{3}}\longrightarrow
T_{\mathscr{F}}(d-2)\longrightarrow 0$

where we used the identity $T_{\mathscr{F}}^{\vee}\simeq T_{\mathscr{F}}(d-2)$
valid for rank 2 reflexive sheaves. Using these two sequences we have that
$h^{0}(\mathcal{I}_{Z}(d))=h^{1}(T_{\mathscr{F}}(-2))=0$, as desired.

Furthermore, we can also check that if ${\mathscr{F}}$ codimension one
distribution on ${\mathbb{P}^{n}}$ whose tangent sheaf splits as a sum of line
bundles, then $\operatorname{Sing}({\mathscr{F}})$ is not contained in a
hypersurface of degree $d$. However, although examples of codimension one
distributions on ${\mathbb{P}^{n}}$ not satisfying this property are not known
to the authors, it seems quite hard to verify it for arbitrary distributions.

######  Proposition 4.4.

Let $Z\subset\mathbb{P}^{n}$ be a closed subscheme and let $d\geq 0$ be an
integer. Suppose that $Z$ is not contained in a hypersurface of degree less
than or equal to $d$. Then there exists a linear isomorphism between the
spaces of degree $d+2$ twisted $1$-forms singular at $Z$ and linear first
syzygies of the homogeneous ideal $I(Z)$.

###### Proof.

The space of degree $d+2$ twisted $1$-forms singular at $Z$ is
$H^{0}(\Omega^{1}_{\mathbb{P}^{n}}(d+2)\otimes\mathcal{I}_{Z})$, where
$\mathcal{I}_{Z}$ is the ideal sheaf of $Z$. Consider the saturated
homogeneous ideal is $I(Z)=\bigoplus_{j}H^{0}(\mathcal{I}_{Z}(j))$. By
hypothesis, $H^{0}(\mathcal{I}_{Z}(j))=\\{0\\}$ for $j\leq d$ hence the space
of linear first syzygies of $I(Z)$ is ${\rm Tor}^{S}_{1}(I(Z),\kappa)_{d+2}$,
where $S=\kappa[x_{0},\dots,x_{n}]$.

Consider the second exterior power of Euler sequence twisted by
$\mathcal{I}_{Z}(d+2)$:

$0\longrightarrow\Omega^{2}_{\mathbb{P}^{n}}(d+2)\otimes\mathcal{I}_{Z}\longrightarrow\mathcal{I}_{Z}(d)^{\oplus\binom{n+1}{2}}\longrightarrow\Omega^{1}_{\mathbb{P}^{n}}(d+2)\otimes\mathcal{I}_{Z}\longrightarrow
0$

which is exact since ${\rm
Tor}_{1}^{\mathcal{O}_{\mathbb{P}^{n}}}(\Omega^{1}_{\mathbb{P}^{n}}(d+2),\mathcal{I}_{Z})=\\{0\\}$.
From the long sequence of cohomology we have that

$H^{0}(\mathcal{I}_{Z}(d))^{\oplus\binom{n+1}{2}}\rightarrow
H^{0}(\Omega^{1}_{\mathbb{P}^{n}}(d+2)\otimes\mathcal{I}_{Z})\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}H^{1}(\Omega^{2}_{\mathbb{P}^{n}}(d+2)\otimes\mathcal{I}_{Z})\rightarrow
H^{1}(\mathcal{I}_{Z}(d))^{\oplus\binom{n+1}{2}}$

is exact. We have that $\phi$ is injective, by hypothesis, and from [14,
Theorem 5.8] the image of $\phi$ is precisely ${\rm
Tor}^{S}_{1}(I(Z),\kappa)_{d+2}$. Thus we have the desired isomorphism. ∎

Now we present this correspondence in an effective way. Fix
$Z\subset\mathbb{P}^{n}$ a closed subscheme with ideal sheaf
$\mathcal{I}_{Z}$. Suppose, as above, that $h^{0}(\mathcal{I}_{Z}(d))=0$. Also
fix $\\{G_{1},\dots,G_{r}\\}$ a (linear) basis of
$H^{0}(\mathcal{I}_{Z}(d+1))$.

First let

$\omega=F_{0}dx_{0}+\dots+F_{n}dx_{n}$

be a degree $d+2$ twisted $1$-form vanishing on $Z$ i.e. $F_{0},\dots,F_{n}\in
H^{0}(\mathcal{I}_{Z}(d+1))$. Then $F_{j}=\sum f_{ij}G_{j}$ for
$f_{ij}\in\kappa$. It follows that

$0=\iota_{R}\omega=\sum_{i=0}^{n}x_{i}F_{i}=\sum_{j=1}^{r}\left(\sum_{i=0}^{n}x_{i}f_{ij}\right)G_{j}$

hence $\left(\sum_{i=0}^{n}x_{i}f_{i1},\dots,\sum_{i=0}^{n}x_{i}f_{ir}\right)$
provides a first syzygy for any set of generators of $I(Z)$ containing
$\\{G_{1},\dots,G_{r}\\}$.

Conversely, fix a set of generators of $I(Z)$ of the form
$\\{G_{1},\dots,G_{r},H_{1}\dots,H_{k}\\}$. Note that $\deg H_{j}\geq d+2$ for
every $j$. Then a linear first syzygy for this set of generators can be
written as $(L_{1},\dots L_{r},0,\dots,0)$ for $L_{j}$ linear homogeneous
polynomials. Define

$\omega=\sum_{j=1}^{r}G_{j}dL_{j}.$

Note that $\iota_{R}\omega=\sum_{j=1}^{r}G_{j}L_{j}=0$ hence $\omega$ defines
a twisted $1$-form on $\mathbb{P}^{n}$ that is clearly singular at $Z$.

######  Remark 4.5.

The discussion above leads to an algorithm to compute the generic $1$-form
$\omega$ vanishing on a subscheme $Z$. It goes as follows: The imput is the
ideal $I(Z)$ and the output is the generic $1$-form $\omega$ singular at $Z$.

  * •

Take $G=(F_{0},\dots,F_{r})$ an ordered minimal generating set for $I(Z)$;

  * •

Compute $M$ the matrix of linear syzygies of $G$ and let $k+1$ be its number
of columns;

  * •

Define additional variables $(t_{0},\dots,t_{k})$;

  * •

Return

$\omega:=G\cdot dM\cdot(t_{0},\dots,t_{k})^{T},$

where $dM$ stands for the entrywise exterior derivative and the dots are
matrix products.

To conclude this section we take a closer look at generic codimension one
distributions. For $d\geq n$, such distributions are determined by its
singular scheme, see [1, Theorem 1.4]. However a careful reader will notice
that their argument works in more generality. Indeed it holds for every $d$
and $n$, except for $d=1$ and $n=2,3$.

On the other hand, we argue that generic codimension one distributions of
degree 1 on ${\mathbb{P}^{2}}$ or ${\mathbb{P}^{3}}$ are not determined by
their singular schemes. Indeed, for $n=2$ one can easily define non isomorphic
foliations with three nondegenerate singularities in general position, see [5,
p. 882]. For $n=3$ we will use our syzygetic approach to show that there
exists a $4$-dimensional family of distributions with a given 0-dimensional
singular scheme.

######  Theorem 4.6.

Let ${\mathscr{F}}$ be a degree $d\geq 1$ distribution on $\mathbb{P}^{3}$. If
$Z={\rm Sing}({\mathscr{F}})$ is zero-dimensional then the homogeneous ideal
$I(Z)$ has minimal graded free resolution given by

$0\longrightarrow S(-3d-2)\longrightarrow S(-2d-1)^{4}\oplus
S(-d-2)\longrightarrow S(-d-1)^{4}\oplus S(-2d)\longrightarrow
I(Z)\longrightarrow 0$

where $S=\kappa[x_{0},\dots,x_{3}]$. Moreover, the space of distributions
singular at $Z$ has dimension $4$ if $d=1$ and, if $d\geq 2$ then
${\mathscr{F}}$ is the unique degree $d$ distribution singular at $Z$.

###### Proof.

Let $\omega=\sum_{j=0}^{3}A_{j}dx_{j}$ be a $1$-form that defines
${\mathscr{F}}$. We claim that $I(Z)=(A_{0},A_{1},A_{2},A_{3},P)$ where $P$ is
the Pfaffian of the matrix defined by $d\omega$ i.e. $P={\rm Pf}(J-J^{T})$ for
$J$ the jacobian matrix of $(A_{0},A_{1},A_{2},A_{3})$. It follows that $\deg
P=2d$ since $\deg A_{j}=d+1$.

From [24, Corollary 4.9] we have that $Z$ is arithmetically Gorenstein. Then
it follows from [3, Section 4] that $I(Z)$ has a minimal resolution given by

$0\longrightarrow\bigwedge^{5}F\longrightarrow\bigwedge^{4}F\longrightarrow
F\longrightarrow I^{+}\longrightarrow 0$

where $F$ is a free $S$-module. Therefore $F=S(-d-1)^{4}\oplus S(-2d)$ and the
resolution becomes

$0\longrightarrow S(-6d-4+l)\stackrel{{\scriptstyle
g^{T}}}{{\longrightarrow}}S(-5d-3+l)^{4}\oplus
S(-4d-4+l)\stackrel{{\scriptstyle f}}{{\longrightarrow}}S(-d-1)^{4}\oplus
S(-2d)\stackrel{{\scriptstyle g}}{{\longrightarrow}}I^{+}\longrightarrow 0$

where $l$ accounts for the shift made so that every map has degree zero. Since
$(x_{0},\dots,x_{3},0)$ is a first syzygy we can compute $l=3d+2$ whence
follows the statement.

From the degrees in the resolution we can compute the degrees of the entries
of the map $f$ to see that for $d\geq 2$ the only linear syzygies are
multiples of $(x_{0},\dots,x_{3},0)$. In the case $d=1$ we have
$-2d-1=-d-2=-3$ and $-d-1=-2d=-2$, hence every entry of $f$ is linear. It
follows from Propositions 4.4 and 3.3 that the space of distributions singular
at $Z$ has (projective) dimension $4$ if $d=1$ and dimension $0$ (i.e.
${\mathscr{F}}$ is unique) if $d\geq 2$.

Now we prove the claim. Let $I=(A_{0},A_{1},A_{2},A_{3})$ be the ideal
generated by the coefficients of $\omega$. Recall that
$I(Z)=\bigoplus_{l\in\mathbb{Z}}H^{0}(\mathcal{I}_{Z}(l))$ is the saturation
of $I$. Since $\dim Z=0$, the ideal sheaf $\mathcal{I}_{Z}$ has a resolution
given by the Koszul complex induced by $\omega$:

$0\longrightarrow\bigwedge^{3}\left({\rm
T}{\mathbb{P}^{3}}(-d-2)\right)\longrightarrow\bigwedge^{2}\left({\rm
T}{\mathbb{P}^{3}}(-d-2)\right)\longrightarrow{\rm
T}{\mathbb{P}^{3}}(-d-2)\longrightarrow\mathcal{I}_{Z}\longrightarrow 0.$

We break it into shot exact sequences that, with the appropriate
identifications, are:

(3) $\displaystyle
0\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-3d-2)\longrightarrow\,$
$\displaystyle\Omega^{1}_{\mathbb{P}^{3}}(-2d)\longrightarrow
T_{\mathscr{F}}(-d-2)\longrightarrow 0,$ (4) $\displaystyle 0\longrightarrow
T_{\mathscr{F}}(-d-2)\longrightarrow\,$ $\displaystyle{\rm
T}{\mathbb{P}^{3}}(-d-2)\longrightarrow\mathcal{I}_{Z}\longrightarrow 0.$

From (4) and the Euler sequence we note that the image of $H^{0}({\rm
T}{\mathbb{P}^{3}}(-d-2+l))\rightarrow H^{0}(\mathcal{I}(l))$ is precisely
$I_{l}$. On the other hand (3) and Bott formulae imply

$h^{1}(T_{\mathscr{F}}(-d-2+l))=\begin{cases}0&l\neq 2d\\\ 1&l=2d\end{cases}.$

Therefore $I_{l}=I(Z)_{l}$ except for $l=2d$ where the later has one more
generator. We only need to produce this missing generator. Recall that $s\in
I(Z)$ if there exist $m\geq 1$ and a $4\times 4$ matrix $Q$ with coefficients
in $S$ such that

$s\begin{bmatrix}x_{0}^{m}\\\ \vdots\\\
x_{3}^{m}\end{bmatrix}=Q\begin{bmatrix}A_{0}\\\ \vdots\\\ A_{3}\end{bmatrix}.$

Let $B$ be the matrix defined by $d\omega$ so that the Euler relation
$\iota_{R}d\omega=(d+2)\omega$ translates to

$B\begin{bmatrix}x_{0}\\\ \vdots\\\
x_{3}\end{bmatrix}=(d+2)\begin{bmatrix}A_{0}\\\ \vdots\\\ A_{3}\end{bmatrix}.$

To show that $P=\operatorname{Pf}(B)\in I(Z)$ we set $m=1$ and $Q=(q_{ij})$
where $(-1)^{i+j}q_{ij}$ is the Pfaffian of the submatrix of $B$ obtained by
eliminating the rows $i,j$ and the columns $i,j$. ∎

From the previous discussion and the Theorem we derive the following
corollary.

######  Corollary 4.7.

Let ${\mathscr{F}}$ be a generic codimension one distribution of degree $d$ on
$\mathbb{P}^{n}$. If $(d,n)\notin\\{(1,2),(1,3)\\}$ then ${\mathscr{F}}$ is
determined by its singular scheme.

Let $\mathcal{D}^{gen}({\mathbb{P}^{n}})$ denote the moduli space of
isomorphism classes of generic distributions on ${\mathbb{P}^{n}}$, and set

$\phi(d,n):=\displaystyle\int_{\mathbb{P}^{n}}c_{n}(\Omega_{\mathbb{P}^{n}}^{1}(d+2))=\frac{(d+1)^{n+1}-(-1)^{n+1}}{d+2}.$

The morphism of Theorem 3.2 then becomes

$\Sigma\colon\mathcal{D}^{gen}({\mathbb{P}^{n}})\longrightarrow{\rm
Hilb}^{\phi(d,n)}(\mathbb{P}^{n})$

Corollary 4.7 implies that this morphism is injective when
$(d,n)\notin\\{(1,2),(1,3)\\}$. For these two exceptional cases, we have that
$\Sigma$ is dominant. Indeed, note that $\phi(1,2)=3$ and $\phi(1,3)=5$. In
both cases the reduced schemes in linear general position are all projectively
equivalent. Since they are dense in the respective Hilbert schemes, we just
need to show one example for each. Then consider the distribution given, in
homogeneous coordinates $(x_{0},\dots,x_{n})$, by

$\omega=\sum_{j=0}^{n}(x_{j+1}x_{j}-x_{j-1}^{2})dx_{j}$

where the indices are taken modulo $n+1$. One may check that the singular
scheme is reduced.

On the other hand, Lemma 4.2 implies that
$\Sigma\colon\mathcal{D}^{gen}({\mathbb{P}^{n}})\rightarrow{\rm
Hilb}^{\phi(d,n)}(\mathbb{P}^{n})$ is not surjective, for any $(d,n)$.
Precisely determining the image of $\Sigma$ seems very difficult. For
instance, let $Z$ be a zero-dimensional subscheme of $\mathbb{P}^{3}$ of
length $5$. In order to be the singular scheme of a degree one distribution,
$Z$ must be nonplanar, arithmetically Gorenstein and locally complete
intersection, see the proof of Theorem 4.6 or [29, p.119]. We wonder if these
conditions are also sufficient.

##  5\. Distributions of Degree 1

In this section, we apply Theorem 3.2 and Proposition 4.4 to provide a
description of the irreducible components of the moduli space of codimension
one distributions of degree $1$ on ${\mathbb{P}^{3}}$. In this framework, we
can explicitly describe the image of the morphism
$\mathcal{D}^{P}(\mathbb{P}^{3})\rightarrow{\rm Hilb}^{Q}(\mathbb{P}^{3})$ and
compute its fibers.

In order to proceed with our intentions, let us recall the classification of
codimension one distributions $\mathscr{F}$ of degree 1 on ${\mathbb{P}^{3}}$
obtained in [4, Section 7]. Such distributions are given by exact sequences of
the following form

(5) $0\longrightarrow T_{\mathscr{F}}\longrightarrow{\rm
T}{\mathbb{P}^{3}}\longrightarrow\mathcal{I}_{Z}(3)\longrightarrow 0$

and the classification was given in terms of the Chern classes of the tangent
sheaf $T_{\mathscr{F}}$. It was shown that the $T_{\mathscr{F}}$ either splits
as a sum of line bundles or is $\mu$-stable; the relevant information is
summarized in Table 1.

We already looked at the generic distributions (corresponding to the first
line) in the end of the previous section. We analyze each of the other three
cases in the three subsequent sections below.

###  5.1. Twisted Cubic

Let $\mathcal{D}(0,0)$ be the scheme parameterizing isomorphism classes of
codimension one distributions $\mathscr{F}$ of degree 1 on $\mathbb{P}^{3}$
such that $(c_{2}(T_{\mathscr{F}}),c_{3}(T_{\mathscr{F}}))=(0,0)$; according
to the classification outlined in Table 1, we have that tangent sheaf splits
as a sum of line bundles; in fact,
$T_{\mathscr{F}}={\mathcal{O}}_{\mathbb{P}^{3}}\oplus{\mathcal{O}}_{\mathbb{P}^{3}}(1)$.
It follows from [4, Corollary 8.9] that $C:={\rm Sing}(\mathscr{F})$ is
arithmetically Cohen-Macaulay curve (aCM, for short) of degree 3 and genus 0,
that is, $C$ is a, possibly degenerate, twisted cubic.

Let $H$ be the irreducible component of ${\rm Hilb}^{3t+1}(\mathbb{P}^{3})$
that contains twisted cubics (recall that ${\rm Hilb}^{3t+1}(\mathbb{P}^{3})$
has a second irreducible component whose generic point corresponds to a plane
cubic plus a point). There exists a map $f\colon H\longrightarrow{\rm
Grass}_{3}(H^{0}(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(2)))$ sending a
curve to the net of quadrics that defines it. The image of the space of
determinantal nets and $f$ is an isomorphism on the aCM open subset, see [15,
30]. On the other hand, one can easily list the possible irreducible
components of an aCM degree three curve. The nonreduced ones are supported on
lines and such multiple structures are described in [27]. It follows that aCM
degree three curves defined by a determinantal net of quadrics are
projectively equivalent to one of the following:

  1. (1)

$I=(xz-y^{2},xw-yz,yw-z^{2})$ a twisted cubic;

  2. (2)

$I=(xz-y^{2},xw,yw)$ a conic meeting a noncoplanar line;

  3. (3)

$I=(xw,xy,yz)$ three noncoplanar lines meeting twice;

  4. (4)

$I=(xy,xz,yz)$ three noncoplanar concurrent lines;

  5. (5)

$I=(x^{2},xz,yz)$ a planar double line meeting a line not in its plane;

  6. (6)

$I=(x^{2},xw,xz-yw)$ a double line of genus -1 meeting a line;

  7. (7)

$I=(x^{2}-yz,xz,z^{2})$ a triple line in a quadratic cone;

  8. (8)

$I=(x^{2},xy,y^{2})$ a thick line of degree three.

Next, we use the results in the previous section to describe
$\mathcal{D}(0,0)$ in terms of the morphism
$\Sigma\colon\mathcal{D}(0,0)\rightarrow H$ and compute its dimension. For
each ideal $I$ listed above, we compute a generic $1$-form vanishing at
$V(I)$, showing that the image of the of the morphism
$\Sigma\colon\mathcal{D}(0,0)\rightarrow H$ is precisely the open subset
$U\subset H$ consisting of non planar aCM curves apart from triple lines in
case (8). We then find that the space of linear first syzygies of the ideals
in the list above have, in all cases, dimension equal to 2, thus, according to
Proposition 3.3 and Theorem 4.4, the fibers of $\Sigma$ are open subsets of
$\mathbb{P}^{1}$. We infer that they are of the form $D_{+}(h)$ for some
homogeneous polynomial $h\in\mathbb{C}[t_{0},t_{1}]$. As a consequence of the
theorem on the dimension of the fibers, we conclude that $\mathcal{D}(0,0)$ is
irreducible and

$\dim\mathcal{D}(0,0)=\dim H+1=13.$

Moreover, one can apply Proposition 3.1 to check that $\mathcal{D}(0,0)$ is
smooth, since

$\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=H^{1}(\mathcal{I}_{Z}(3))\oplus
H^{1}(\mathcal{I}_{Z}(2))=0.$

Using the procedure outlined in Remark 4.5, we can find explicit expressions
for 1-forms vanishing at each of the possible ideals listed above. In
addition, since the tangent sheaf is
${\mathcal{O}}_{\mathbb{P}^{3}}\oplus{\mathcal{O}}_{\mathbb{P}^{3}}(1)$, there
exists a constant vector field that is tangent to each distribution. All of
this information (generic 1-form vanishing at $V(I)$, vector field and
polynomial $h$) is listed in Table 2.

Case | Generic $1$-form | Vector field | $h$
---|---|---|---
(1) |  $\omega=t_{1}(z^{2}-yw)dx+(t_{0}(z^{2}-yw)-t_{1}(yz-xw))dy+$ $\quad+(t_{1}(y^{2}-xz)-t_{0}(yz-xw))dz+t_{0}(y^{2}-xz)dw$ |  $t_{0}^{3}\partial_{x}+t_{0}^{2}t_{1}\partial_{y}+$ $\quad+t_{0}t_{1}^{2}\partial_{z}+t_{1}^{3}\partial_{w}$ | 1
(2) |  $\omega=-t_{0}ywdx+(t_{1}yw+t_{0}xw)dy+t_{1}xwdz$ $\quad-t_{1}(y^{2}-xz)dw$ |  $t_{1}^{2}\partial_{x}+t_{0}t_{1}\partial y+$ $\quad+t_{0}^{2}\partial z$ | $t_{1}$
(3) |  $\omega=t_{1}yzdx-t_{0}xwdy-t_{1}xydz+t_{0}xydw$ |  $t_{0}\partial_{z}+t_{1}\partial_{w}$ | $t_{0}t_{1}$
(4) |  $\omega=t_{0}yzdx+t_{1}xzdy-(t_{0}+t_{1})xydz$ |  $\partial_{w}$ | $t_{0}t_{1}(t_{0}+t_{1})$
(5) |  $\omega=(t_{0}yz+t_{1}xz)dx-t_{0}xzdy-t_{1}x^{2}dz$ |  $\partial_{w}$ | $t_{0}t_{1}$
(6) |  $\omega=(t_{0}xw+t_{1}(yw-xz))dx-t_{1}xwdy+t_{1}x^{2}dz+$ $\quad-t_{0}x^{2}dw$ |  $t_{0}\partial_{z}+t_{1}\partial_{w}$ | $t_{1}$
(7) |  $\omega=(t_{0}z^{2}+t_{1}xz)dx-t_{1}z^{2}dy-(t_{0}xz+t_{1}(x^{2}-yz))dz$ |  $\partial_{w}$ | $t_{1}$
Table 2. Classification of distributions with singular scheme in ${\rm
Hilb}^{3t+1}(\mathbb{P}^{3})$.

The case (8) does not figure in the table above because a $1$-form singular at
such scheme is singular in codimension one. Indeed, the generic $1$-form is
$\omega=(t_{0}y+t_{1}x)(ydx-xdy)$. We also note that the integrable
distributions, i.e. foliations, that arise in this case must be of linear
pullback type $\mathcal{LPB}(1)$. A direct inspection shows that these are
precisely the cases (4), (5) and (7). Our conclusions can therefore be
summarized in the following statement.

######  Theorem 5.1.

Let $\mathcal{D}(0,0)$ be the scheme parameterizing distributions (up to
isomorphism) on $\mathbb{P}^{3}$ of codimension and degree $1$ whose tangent
sheaf is locally free. Let also $H$ be the irreducible component of ${\rm
Hilb}^{3t+1}(\mathbb{P}^{3})$ containing twisted cubics. Then:

  * (i)

There is a surjective morphism $\Sigma\colon\mathcal{D}(0,0)\rightarrow U$,
the open set consisting of non planar aCM curves in $H$ which are not of type
(8), which assigns each $\mathscr{F}\in\mathcal{D}(0,0)$ to its singular
scheme, and such that each fiber $\Sigma^{-1}(C)$ is naturally isomorphic to
an open set of $\mathbb{P}^{1}$.

  * (ii)

$\mathcal{D}(0,0)$ is an irreducible, smooth quasi-projective variety of
dimension 13.

  * (iii)

There exists a map $v\colon\mathcal{D}(0,0)\rightarrow\mathbb{P}^{3}$ that
assigns to each $\mathscr{F}\in\mathcal{D}(0.0)$ a constant vector field
$v(\mathscr{F})\in\mathbb{P}^{3}$ which is tangent $\mathscr{F}$. Moreover,
for any $C\in U$, the closure of the image $\overline{v(\Sigma^{-1}(C))}$
describes a rational normal curve of degree $d$ in some
$\mathbb{P}^{d}\subset\mathbb{P}^{3}$, where:

    * (a)

$d=3$ if and only if $C$ is a twisted cubic.

    * (b)

$d=2$ if and only if $C$ is a conic meeting a noncoplanar line;

    * (c)

$d=1$ if and only if $C$ is three noncoplanar lines meeting twice;

    * (d)

$d=0$ if and only if $C$ is three noncoplanar concurrent lines;

  * (iv)

$\mathscr{F}$ is integrable if and only if ${\rm Sing}(\mathscr{F})$ falls in
cases (4), (5) or (7). In this case $\mathscr{F}$ is a linear pullback
foliation.

###  5.2. Line and Two Points

The Hilbert scheme ${\rm Hilb}^{t+3}(\mathbb{P}^{3})$ of configurations of a
line and two points is an irreducible variety of dimension $10$, see [11,
Example 4.5]. The possible configurations arise by simply adding two points to
a line $X$. The resulting scheme is always a flat limit of $X$ plus two
isolated points, [11, Theorem 1.4]. Since we are concerned with nonplanar
schemes $V(I)\in{\rm Hilb}^{t+3}(\mathbb{P}^{3})$, we may consider, up to
projective equivalence, the $I$ listed below:

  1. (1)

$I=(x,y)\cap(x,z,w)\cap(y,z,w)=(yw,xw,yz,xz,xy)$, a line and two points;

  2. (2)

$I=(x,y)\cap(x,y^{2},z)\cap(y,z,w)=(xw,yz,xz,y^{2},xy)$, a line and two points
but one embedded;

  3. (3)

$I=(x,y)\cap(x,y^{2},z)\cap(x^{2},y,w)=(xw,yz,x^{2},y^{2},xy)$, a line with
two simple embedded points;

  4. (4)

$I=(x,y)\cap(x^{2},z,w)=(yw,xw,yz,xz,x^{2})$, a line and a double point;

  5. (5)

$I=(x,y)\cdot(x,y,z)=(x^{2},xy,xz,y^{2},yz)$, a line with embedded double
point.

The corresponding ideal sheaves have all the same resolution:

(6) $0\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-4)^{\oplus
2}\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-3)^{\oplus
6}\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-2)^{\oplus
5}\longrightarrow\mathcal{I}_{Z}\longrightarrow 0$

In particular, one can check that $h^{i}(\mathcal{I}_{Z}(k))=0$ for $i>0$ and
$k>0$.

Now we analyze each of the cases raised above following the same procedure
outline in the previous subsection. Generic 1-forms vanishing at $V(I)$ for
the ideals in items (1) through (5) are obtained following the algorithm given
in Remark 4.5; however, every 1-form vanishing at a scheme $Z=V(I)$ for an
ideal $I$ of type (5) also vanishes along three concurrent lines meeting at
the embedded double point, implying that the corresponding morphism ${\rm
T}{\mathbb{P}^{3}}\rightarrow\mathcal{I}_{Z}(3)$ cannot be surjective; the
explicit expression is in display (7) below. It thus follows that the image of
the morphism $\Sigma\colon\mathcal{D}(1,1)\rightarrow{\rm
Hilb}^{t+3}(\mathbb{P}^{3})$ is the open subset $U\subset{\rm
Hilb}^{t+3}(\mathbb{P}^{3})$ consisting of nonplanar schemes without embedded
double point (that is, the complement of the families of schemes of type (5)
and the planar ones).

Furthermore, we also compute the dimension of the the fibres of $\Sigma$ using
Proposition 4.4; it turns out that the
$h^{0}(\Omega^{1}_{{\mathbb{P}^{3}}}\otimes\mathcal{I}_{Z}(3))=6$ for each
possible scheme $Z$. Finally, the open subsets giving the fibers of the
morphism $\Sigma$ are of the form $D_{+}(h)$, for some homogeneous polynomial
$h\in\mathbb{C}[t_{0},\dots,t_{5}]$. We list these polynomials in Table 3
below, together with the generic 1-form of each case.

Case | Generic $1$-form | $h$
---|---|---
(1) |  $\omega=(t_{2}yw+t_{4}yz)dx+(t_{3}xw+t_{5}xz)dy+$ $\quad\,+(t_{0}yw+t_{1}xw-(t_{4}+t_{5})xy)dz+$ $\quad\,-(t_{0}yz+t_{1}xz+(t_{2}+t_{3})xy)dw$ | $t_{0}t_{1}(t_{3}t_{4}-t_{2}t_{5})$
(2) |  $\omega=(t_{3}yz-t_{5}y^{2})dx+(t_{1}yz+t_{2}xw+t_{4}xz+t_{5}xy)dy+$ $\quad\,+(t_{0}xw-t_{1}y^{2}-(t_{3}+t_{4})xy)dz-(t_{0}xz+t_{2}xy)dw$ | $t_{0}t_{3}(t_{0}t_{5}-t_{2}t_{3})$
(3) |  $\omega=(t_{2}yz+t_{3}y^{2}+t_{4}xw+t_{5}xy)dx+$ $\quad\,+(t_{0}yz+t_{1}xw-t_{3}xy-t_{5}x^{2})dy+$ $\quad\,-(t_{0}y^{2}+t_{2}xy)dz-(t_{1}xy+t_{4}x^{2})dw$ | $t_{0}t_{4}(t_{1}t_{2}-t_{0}t_{4})$
(4) |  $\omega=(t_{0}yw+t_{3}yz+t_{4}xw+t_{5}xz)dx-(t_{0}xw+t_{3}xz)dy+$ $\quad\,+(t_{1}yw+t_{2}xw-t_{5}x^{2})dz-(t_{1}yz+t_{2}xz+t_{4}x^{2})dw$ | $t_{1}(t_{4}t_{3}-t_{0}t_{5})$
Table 3. Classification of distributions with singular scheme in ${\rm
Hilb}^{t+3}(\mathbb{P}^{3})$.

We remark that the generic $1$-form in case (5) is

(7)
$\omega=(t_{1}xz+t_{3}y^{2}+t_{4}xz+t_{5}xy)dx+(t_{0}yz+t_{2}xz)dy-(t_{0}y^{2}+(t_{1}+t_{2})xy+t_{4}x^{2})dz.$

Since it does not depend on $w$, it defines a family of linear pullback
foliations. Each general member is singular at three lines crossing at the
double point.

######  Theorem 5.2.

$\mathcal{D}(2,2)$, the scheme parameterizing isomorphism classes of
codimension one distributions $\mathscr{F}$ of degree 1 on $\mathbb{P}^{3}$
such that $(c_{2}(T_{\mathscr{F}}),c_{3}(T_{\mathscr{F}}))=(2,2)$, is an
irreducible smooth quasi-projective variety of dimension 15. The image of the
morphism $\Sigma\colon\mathcal{D}(2,2)\rightarrow{\rm
Hilb}^{t+3}(\mathbb{P}^{3})$ is the open subset the open subset consisting of
nonplanar schemes without an embedded double point, and its fibers are open
subsets of $\mathbb{P}^{5}$.

The second part of the previous theorem was proved above, and it implies, by
the theorem of the dimension of the fibres, that $\mathcal{D}(2,2)$ is an
irreducible quasi-projective variety of dimension 15. In order to show that
$\mathcal{D}(2,2)$ is smooth, we argue, using Lemmata A.1 and A.2, that $\dim
T_{[{\mathscr{F}}]}\mathcal{D}(2,2)=15$ for every
$\mathscr{F}\in\mathcal{D}(2,2)$. Indeed, note that $T_{\mathscr{F}}^{\vee}$
is a rank 2 stable reflexive sheaf with Chern classes
$(c_{1}(T_{\mathscr{F}}^{\vee}),c_{2}(T_{\mathscr{F}}^{\vee}),c_{3}(T_{\mathscr{F}}^{\vee}))=(-1,2,2)$
so we can see from [10, Table 2.6.1] that

$h^{1}(T_{\mathscr{F}}^{\vee}(k))=0\text{ for }k\neq 0,\quad
h^{2}(T_{\mathscr{F}}^{\vee}(k))=0\text{ for }k\geq-1\,\text{ and
}\,h^{3}(T_{\mathscr{F}}^{\vee}(k))=0\text{ for }k\geq-4.$

We can then apply the functor $\makebox[6.45831pt]{{$\cdot$}}\otimes
T_{\mathscr{F}}^{\vee}(3)$ to the resolution in display (6), keeping in mind
item (2) of Lemma A.1, to conclude that
$h^{i}(T_{\mathscr{F}}^{\vee}\otimes\mathcal{I}_{Z}(3))=0$ for $i>0$, thus
$\operatorname{Ext}^{i}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=0$ for $i>1$ and

$\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))\simeq
H^{0}({\mathcal{E}}{\it
xt}^{1}(T_{\mathscr{F}},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z}(3))\simeq
H^{0}({\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z}),$

with the last isomorphism coming from the sequence that defines the
distribution, namely (5). The expression in display (17) gives, in the case at
hand,

$\dim\operatorname{Hom}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=\chi(T_{\mathscr{F}},\mathcal{I}_{Z}(3))+h^{0}({\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z})=9+6=15,$

with the two quantities in the middle being computed using (16) and Lemma A.2.

To compute $\chi(T_{\mathscr{F}},\mathcal{I}_{Z}(3))$ we need to compute ${\rm
ch}(T_{\mathscr{F}})^{\vee}$ and ${\rm ch}(\mathcal{I}_{Z}(3))$. Since we have
$c_{1}(T_{\mathscr{F}})=1$, $c_{2}(T_{\mathscr{F}})=c_{3}(T_{\mathscr{F}})=2$,
$c_{1}(\mathcal{I}_{Z}(3))=3$, $c_{2}(\mathcal{I}_{Z}(3))=1$ and
$c_{3}(\mathcal{I}_{Z}(3))=-5$, it follows that ${\rm
ch}(T_{\mathscr{F}})^{\vee}=\left(2,-1,-\frac{3}{2},-\frac{1}{6}\right)$ and
${\rm ch}(\mathcal{I}_{Z}(3))=\left(1,3,\frac{7}{2},\frac{1}{2}\right)$.
Therefore $\chi(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=9$.

Finally we compute $h^{0}({\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z})={\rm
length}\,{\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z}$.
We know that $Z$ is composed by a line $L$ and two points $p_{1}$ and $p_{2}$.
Table 3 shows that either: $p_{1},p_{2}\not\in L$ are isolated points;
$p_{1}\notin L$ and $p_{2}\in L$ is a simple embedded point; $p_{1},p_{2}\in
L$ are simple embedded points; or $p_{1}=p_{2}\notin L$ is a double point. In
any case, Lemma A.2 yields $h^{0}({\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z})=6$.

###  5.3. Conic and One Point

Recall that the Hilbert Scheme ${\rm Hilb}^{2t+2}(\mathbb{P}^{3})$ consists of
two irreducible components: one component contains two skew lines, while the
other contains non pure schemes consisting of a conic plus a point; let $H$
denote this second component. One can show that $H$ is a smooth irreducible
variety of dimension $11$, see [11, Example 4.6(b)]. We will only consider the
configurations where the point (embedded or not) does not lie on the same
plane as the conic. Then, up to projective equivalence, they are listed below:

  1. (1)

$I=(xy-z^{2},w)\cap(x,y,z)=(zw,yw,xw,xy-z^{2})$, a conic and a point;

  2. (2)

$I=(xy,w)\cap(x,y,z)=(zw,yw,xw,xy)$, a pair of concurrent lines and a point;

  3. (3)

$I=(z^{2},w)\cap(x,y,z)=(zw,yw,xw,z^{2})$, a planar double line and a point;

  4. (4)

$I=(xy-z^{2},w)\cap(y,z,w^{2})=(w^{2},zw,yw,xy-z^{2})$, a conic and an
embedded point;

  5. (5)

$I=(xy,w)\cap(x,z,w^{2})=(w^{2},zw,xw,xy)$, a pair of concurrent lines and an
embedded point in the smooth locus;

  6. (6)

$I=(xy,w)\cap(x,y,w^{2})=(w^{2},yw,xw,xy)$, a pair of concurrent lines and an
embedded point in the singular locus;

  7. (7)

$I=(z^{2},w)\cap(y,z,w^{2})=(w^{2},zw,yw,z^{2})$, a planar double line and an
embedded point.

The corresponding ideal sheaves have all the same resolution:

(8)
$0\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-4)\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-3)^{\oplus
4}\longrightarrow\mathcal{O}_{\mathbb{P}^{3}}(-2)^{\oplus
4}\longrightarrow\mathcal{I}_{Z}\longrightarrow 0$

Again, one can check that $h^{i}(\mathcal{I}_{Z}(k))=0$ for $i>0$ and $k>0$.

Now we analyze each of the cases listed above following the procedure outlined
in the previous subsections. Generic 1-forms vanishing at $V(I)$ for the
ideals in items (1) through (7) are obtained using the algorithm described in
Remark 4.5; we then find that every 1-form vanishing at a scheme $Z=V(I)$ for
ideals $I$ of type (6) or (7) also vanishes along three concurrent lines,
meaning that the corresponding morphism ${\rm
T}{\mathbb{P}^{3}}\rightarrow\mathcal{I}_{Z}(3)$ cannot be surjective; the
explicit expressions are in display (9) below. It thus follows that the image
of the morphism $\Sigma\colon\mathcal{D}(2,2)\rightarrow H$ is the open subset
$U\subset H$ consisting of nonplanar schemes which are not of type (6) or (7).

Finally, one can check that the dimension of the space of linear syzygies of
the ideals of type (1) through (5) is equal to 4, thus the fibers of
$\Sigma\colon\mathcal{D}(2,2)\rightarrow H$ are open subsets of
${\mathbb{P}^{3}}$ the form $D_{+}(h)$ for some homogeneous polynomial
$h\in\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. We provide explicit expressions for
the generic $1$-form and corresponding polynomials $h$ in Table 4.

Case | Generic $1$-form | $h$
---|---|---
(1) |  $\omega=w(t_{1}z+t_{2}y)dx+w(t_{0}z+(t_{3}-t_{2})x)dy+$ $\quad\,-w(t_{0}y+t_{1}x+t_{3}z)dz+t_{3}(z^{2}-xy)dw$ | $t_{3}(t_{0}t_{1}+t_{2}^{2}-t_{2}t_{3})$
(2) |  $\omega=w(t_{1}z+t_{2}y)dx+w(t_{0}z+t_{3}x)dy+$ $\quad\,-w(t_{0}y+t_{1}x)dz-(t_{2}+t_{3})xydw$ | $t_{0}t_{1}(t_{2}+t_{3})$
(3) |  $\omega=w(t_{1}z+t_{2}y)dx+w(t_{0}z-t_{2}x)dy+$ $\quad\,-w(t_{0}y+t_{1}x-t_{3}z)dz-t_{3}z^{2}dw$ | $t_{2}t_{3}$
(4) |  $\omega=t_{3}wydx+w(t_{1}w+t_{2}z)dy+w(t_{0}w-t_{2}y-t_{3}z)dz+$ $\quad\,-(t_{0}zw+t_{1}yw+t_{3}(xy-z^{2}))dw$ | $t_{3}(t_{0}t_{2}+t_{1}t_{3})$
(5) |  $\omega=w(t_{1}w+t_{2}z)dx+t_{3}xwdy+$ $\quad\,+w(t_{0}w-t_{2}x)dz-(t_{0}zw+t_{1}xw+t_{3}xy)dw$ | $t_{0}t_{2}t_{3}$
Table 4. Classification of distributions with singular scheme in ${\rm
Hilb}^{2t+2}(\mathbb{P}^{3})$.

Note that cases (6) and (7) do not figure in the table. The generic $1$-forms
are, respectively:

(9) $\displaystyle\omega_{6}$
$\displaystyle=w(t_{1}w+t_{2}y)dx+w(t_{0}w+t_{3}x)dy-(t_{0}yw+t_{1}xw+(t_{2}+t_{3})xy)dw,$
$\displaystyle\omega_{7}$
$\displaystyle=w(t_{1}w+t_{2}z)dy+w(t_{0}w-t_{2}y+t_{3}z)dz-(t_{0}zw+t_{1}yw+t_{3}z^{2})dw.$

They depend on three variables, hence they define families of linear pullback
foliations. Each general member of either family is singular at three
concurrent lines, the third one passing through the embedded point.

It is worth pointing out the integrable distributions, i.e. foliations, in
this case. These are the rational foliations $\mathcal{R}(2,1)$. Since the
singular scheme of such foliations are composed of a smooth conic and a point,
we only need to compute the Frobenius integrability condition $\omega\wedge
d\omega=0$ in case (1). It follows that the integrable distributions are those
that satisfy $t_{0}=t_{1}=0$. In particular, we have that $\mathcal{R}(2,1)$
is smooth. In [13, Theorem 2.1] is proved that the irreducible component
$R(2,1)$ is generically smooth.

######  Theorem 5.3.

Let $\mathcal{D}(1,1)$ be the scheme parameterizing isomorphism classes of
codimension one distributions $\mathscr{F}$ of degree 1 on $\mathbb{P}^{3}$
such that $(c_{2}(T_{\mathscr{F}}),c_{3}(T_{\mathscr{F}}))=(1,1)$. Then the
image of the morphism $\Sigma\colon\mathcal{D}(1,1)\rightarrow H$ is the open
subset consisting of non planar schemes in $H$ which are neither of type (6)
nor of type (7) in the list above, and its fibers are open subsets of
$\mathbb{P}^{3}$. In addition:

  1. (1)

$\mathcal{D}(1,1)$ is an irreducible smooth quasi-projective variety of
dimension 14.

  2. (2)

$\mathcal{D}(1,1)$ contains $\mathcal{R}(2,1)$ as a codimension two subset;
the image of the restricted morphism $\Sigma\colon\mathcal{R}(2,1)\rightarrow
H$ is the open subset consisting of the schemes of type (1) in the list above,
and its fibres are open subsets of ${\mathbb{P}^{1}}$.

It was already shown in [4, Proposition 11.2] that $\mathcal{D}(1,1)$ is
smooth but here we give a different proof. Our strategy is to show that $\dim
T_{[{\mathscr{F}}]}\mathcal{D}(1,1)=\dim\mathcal{D}(1,1)$ for every
$\mathscr{F}\in\mathcal{D}(1,1)$. As it was pointed out in [4, proof of
Theorem 8.5], the tangent sheaf $T_{\mathscr{F}}$ admits the following
resolution

(10)
$0\longrightarrow{\mathcal{O}}_{\mathbb{P}^{3}}(-1)\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}{\mathcal{O}}_{\mathbb{P}^{3}}^{\oplus
3}\longrightarrow T_{\mathscr{F}}\longrightarrow 0.$

Applying the functor
$\operatorname{Hom}(\makebox[6.45831pt]{{$\cdot$}},\mathcal{I}_{Z}(3))$ to
this sequence we get

$0\longrightarrow\operatorname{Hom}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))\longrightarrow
H^{0}(\mathcal{I}_{Z}(3))^{\oplus
3}\stackrel{{\scriptstyle\tilde{\mu}}}{{\longrightarrow}}H^{0}(\mathcal{I}_{Z}(4))\longrightarrow\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))\longrightarrow
0,$

since $h^{1}(\mathcal{I}_{Z}(3))=0$. Therefore

$\dim\operatorname{Hom}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=3\,h^{0}(\mathcal{I}_{Z}(3))-h^{0}(\mathcal{I}_{Z}(4))+\dim\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3)).$

Since $h^{0}(\mathcal{I}_{Z}(3))=12$ and $h^{0}(\mathcal{I}_{Z}(4))=25$ we
only need to show that
$\dim\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=3$. We will
use Lemmata A.1 and A.2 to achieve that.

Note that $T_{\mathscr{F}}^{\vee}=T_{\mathscr{F}}(-1)$, since
$T_{\mathscr{F}}$ is a rank 2 reflexive sheaf with $c_{1}(T_{\mathscr{F}})=1$,
thus
$T_{\mathscr{F}}^{\vee}\otimes\mathcal{I}_{Z}(3)=T_{\mathscr{F}}\otimes\mathcal{I}_{Z}(2)$.
Tensoring the exact sequence (10) with $\mathcal{I}_{Z}(2)$ and passing to
cohomology, we get

$H^{i}(\mathcal{I}_{Z}(2))^{\oplus 3}\longrightarrow
H^{i}(T_{\mathscr{F}}\otimes\mathcal{I}_{Z}(2))\longrightarrow
H^{i+1}(\mathcal{I}_{Z}(1)),$

but we already observed right after display (8) that
$h^{i}(\mathcal{I}_{Z}(k))=0$ for $i>0$ and $k>0$, thus
$h^{i}(T_{\mathscr{F}}\otimes\mathcal{I}_{Z}(2))=0$ for $i>0$. Item (5) in
Lemma A.1 yields

$\dim\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=h^{0}({\mathcal{E}}{\it
xt}^{1}(T_{\mathscr{F}},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z}(3))=h^{0}({\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z})$

and this is just the length of ${\mathcal{E}}{\it
xt}^{3}(\mathcal{O}_{Z},{\mathcal{O}}_{\mathbb{P}^{3}})\otimes\mathcal{I}_{Z}$
which will be computed with Lemma A.2.

We know that $Z$ is composed by a conic $C$ and a point $p$. If $p\not\in C$
then $\mathcal{I}_{Z,p}=\mathfrak{m}_{p}$ the maximal ideal at $p$. If $p\in
C$ i.e. an embedded point for $Z$. From the table 4 we know that we only need
to treat the case where $C$ is reduced and $p$ is embedded in the regular
part. In either case, Lemma A.2 shows that the length is three hence
$\dim\operatorname{Ext}^{1}(T_{\mathscr{F}},\mathcal{I}_{Z}(3))=3$.

##  Appendix A Tensor-Hom Relation for Sheaves

If $M$ and $N$ are modules over a commutative ring $R$, then there is a
canonical morphism of $R$-modules

$\displaystyle M^{\vee}\otimes N$
$\displaystyle\longrightarrow\operatorname{Hom}_{R}(M,N)$ $\displaystyle
u\otimes y$ $\displaystyle\longmapsto(x\mapsto u(x)\otimes y)$

see [2, Chapter II §4.2]. Now, given sheaves $F$ and $G$ on a variety $X$, the
previous observation leads to the following morphism of sheaves

$\eta\colon F^{\vee}\otimes G\longrightarrow{\mathcal{H}}{\it om}(F,G)$

which we call _Tensor-Hom relation_. The following technical lemma is useful
in our proof of smoothness for $\mathcal{D}(2,2)$ and $\mathcal{D}(1,1)$.

######  Lemma A.1.

Let $X$ be a smooth projective threefold and let $F$ and $G$ be sheaves on
$X$. If $F$ is reflexive and $G$ is torsion free then:

  1. (1)

${\mathcal{E}}{\it xt}^{1}(F,G)\simeq{\mathcal{E}}{\it
xt}^{1}(F,\mathcal{O}_{X})\otimes G$;

  2. (2)

$\operatorname{Tor}_{1}(F,G)=0$;

  3. (3)

The morphism $\eta$ is surjective and the induced cohomology map
$H^{i}(F^{\vee}\otimes G)\rightarrow H^{i}({\mathcal{H}}{\it om}(F,G))$ is
surjective for $i=0$ and an isomorphism for $i\geq 1$;

  4. (4)

$\eta$ is an isomorphism if either
$\operatorname{Sing}(F)\cap\operatorname{Sing}(G)=\emptyset$ or $G$ is
reflexive;

  5. (5)

if $H^{i}(F^{\vee}\otimes G)=0$ for $i>0$, then
$\operatorname{Ext}^{1}(F,G)=H^{0}({\mathcal{E}}{\it
xt}^{1}(F,\mathcal{O}_{X})\otimes G)$ and $\operatorname{Ext}^{i}(F,G)=0$ for
$i>1$.

###### Proof.

Since $F$ is reflexive, it admits a resolution of the form

(11) $0\longrightarrow
L_{1}\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}L_{0}\longrightarrow
F\longrightarrow 0,$

where $L_{1}$ and $L_{0}$ are locally free sheaves. Applying the functor
$\makebox[6.45831pt]{{$\cdot$}}\otimes G$ to this sequence we get

$0\longrightarrow\operatorname{Tor}_{1}(F,G)\longrightarrow L_{1}\otimes
G\stackrel{{\scriptstyle\mu\otimes\mathbf{1}_{G}}}{{\longrightarrow}}L_{0}\otimes
G\longrightarrow F\otimes G\longrightarrow 0.$

Note that $L_{0}\otimes G$ is torsion free, while the support of
$\operatorname{Tor}_{1}(F,G)$ is contained in
$\operatorname{Sing}(F)\cap\operatorname{Sing}(G)$. Then
$\operatorname{Tor}_{1}(F,G)\subset L_{1}\otimes G$ is only possible if
$\operatorname{Tor}_{1}(F,G)=0$. This proves (2).

Dualizing the sequence in display (10), we obtain

(12) $\displaystyle 0\longrightarrow F^{\vee}\longrightarrow$ $\displaystyle
L_{0}^{\vee}\stackrel{{\scriptstyle\mu^{\vee}}}{{\longrightarrow}}K\longrightarrow
0,$ (13) $\displaystyle 0\longrightarrow K\longrightarrow$ $\displaystyle
L_{1}^{\vee}\longrightarrow{\mathcal{E}}{\it
xt}^{1}(F,\mathcal{O}_{X})\longrightarrow 0,$

where $K:=\operatorname{{im}}\mu^{\vee}$. On the one hand, we apply the
functor $\makebox[6.45831pt]{{$\cdot$}}\otimes G$ to the sequence in display
(12) and obtain

(14) $0\longrightarrow\operatorname{Tor}_{1}(K,G)\longrightarrow
F^{\vee}\otimes G\longrightarrow L_{0}^{\vee}\otimes
G\xrightarrow{\mu^{\vee}\otimes\mathbf{1}_{G}}K\otimes G\longrightarrow 0$

On the other hand, the apply the functor ${\mathcal{H}}{\it
om}(\makebox[6.45831pt]{{$\cdot$}},G)$ to the sequence in display (11) to
obtain

(15) $0\longrightarrow{\mathcal{H}}{\it om}(F,G)\longrightarrow
L_{0}^{\vee}\otimes
G\xrightarrow{\mu^{\vee}\otimes\mathbf{1}_{G}}{L_{1}^{\vee}\otimes
G}\longrightarrow{\mathcal{E}}{\it xt}^{1}(F,G)\longrightarrow 0.$

As tensor product preserves cokernels, we get that ${\mathcal{E}}{\it
xt}^{1}(F,G)\simeq{\mathcal{E}}{\it xt}^{1}(F,\mathcal{O}_{X})\otimes G$. This
proves (1).

The sequence (15) also shows that
$\ker\mu^{\vee}\otimes\mathbf{1}_{G}={\mathcal{H}}{\it om}(F,G)$ and the
sequence in display (14) induces

$0\longrightarrow\operatorname{Tor}_{1}(K,G)\longrightarrow F^{\vee}\otimes
G\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}{\mathcal{H}}{\it
om}(F,G)\longrightarrow 0.$

In the same fashion, we apply the functor
$\makebox[6.45831pt]{{$\cdot$}}\otimes G$ to the sequence in display (13) to
obtain

$\operatorname{Tor}_{2}({\mathcal{E}}{\it
xt}^{1}(F,\mathcal{O}_{X}),G)\simeq\operatorname{Tor}_{1}(K,G);$

if $\operatorname{Sing}(F)\cap\operatorname{Sing}(G)=\emptyset$, then
$\operatorname{Tor}_{2}({\mathcal{E}}{\it xt}^{1}(F,\mathcal{O}_{X}),G)=0$
hence $\eta$ is an isomorphism. Alternatively, item (2) also implies that
$\operatorname{Tor}_{1}(K,G)=0$ when $G$ is reflexive, since the sequence (12)
implies that $K$ is torsion free. This completes the proof of item (4).

Moreover, $\operatorname{Tor}_{1}(K,G)$ is a zero-dimensional sheaf whenever
non trivial, so the induced cohomology map

$H^{i}(F^{\vee}\otimes G)\longrightarrow H^{i}({\mathcal{H}}{\it om}(F,G))$

is surjective for $i=0$ and an isomorphism for $i\geq 1$. This completes the
proof of (3).

Finally, the item (5) is an immediate consequence of the spectral sequence for
local to global Ext’s. Indeed, if $h^{i}(F^{\vee}\otimes G)=0$ for $i>0$, then
$h^{i}({\mathcal{H}}{\it om}(F,G))=0$ in the same range by item (3). Since
$\dim{\mathcal{E}}{\it xt}^{j}(F,G)=0$ due to the fact that $F$ is reflexive,
the only nontrivial entries in the second page of the spectral sequence
$E_{2}^{ij}=H^{i}({\mathcal{E}}{\it xt}^{j}(F,G))$ occur for
$(i,j)=(0,0),(0,1)$, thus, in addition, all differential $d_{2}^{ij}$ must
vanish. It follows that $E_{2}^{ij}=H^{i}({\mathcal{E}}{\it xt}^{j}(F,G))$
already converges in the second page, therefore
$\operatorname{Ext}^{1}(F,G)=H^{0}({\mathcal{E}}{\it xt}^{1}(F,G))$ and
$\operatorname{Ext}^{k}(F,G)=0$ for $k=2,3$. ∎

Next, we recall the following version of the Hirzebruch–Riemann–Roch theorem,
compare with [21, Lemma 6.1.3]:

(16) $\chi(E,F):=\sum_{k}(-1)^{k}\dim\operatorname{Ext}^{k}(E,F)=\int_{X}{\rm
ch}(E)^{\vee}\cdot{\rm ch}(F)\cdot{\rm td}(X)$

If we know that $\operatorname{Ext}^{k}(E,F)=0$ for $k>1$ (situation described
in the Lemma A.1), then

(17) $\dim\operatorname{Hom}(E,F)=\chi(E,F)+\dim\operatorname{Ext}^{1}(E,F).$

The calculation of $\dim\operatorname{Ext}^{1}(E,F)$ in our examples will rely
on the following technical fact from commutative algebra.

######  Lemma A.2.

Let $\kappa$ be a field and let $R=\kappa[x,y,z]_{(x,y,z)}$ be the local ring
at the origin of $\mathbb{A}^{3}_{\kappa}$. Let $I\subset R$ be an ideal, then

$\dim_{\kappa}\operatorname{Ext}^{3}_{R}(R/I,R)\otimes_{R}I=\begin{cases}3,&\text{if
$V(I)$ is the closed point;}\\\ 3,&\text{if $V(I)$ is a smooth rational curve
with a simple embedded point;}\\\ 6,&\text{if $V(I)$ is a double
point.}\end{cases}$

###### Proof.

First assume that $I=(x,y,z)$ the maximal ideal. Therefore
$\operatorname{Ext}^{3}_{R}(R/I,R)=R/I$ and

$\operatorname{Ext}^{3}_{R}(R/I,R)\otimes_{R}I=\frac{(x,y,z)}{(x,y,z)^{2}}\simeq\kappa^{3}.$

Next assume that $V(I)$ is a smooth rational curve with a simple embedded
point. Up to automorphism of $R$ we can suppose that
$I=(x,y)\cap(x^{2},y,z)=(x^{2},xz,y)$ which gives the following resolution

$0\longrightarrow R\xrightarrow{\begin{pmatrix}-y\\\ -x\\\
z\end{pmatrix}}R^{3}\xrightarrow{\begin{pmatrix}z&0&y\\\ -x&y&0\\\
0&-xz&-x^{2}\end{pmatrix}}R^{3}\xrightarrow{\begin{pmatrix}x^{2}&xz&y\end{pmatrix}}R\longrightarrow
R/I\longrightarrow 0$

Therefore $\operatorname{Ext}^{3}_{R}(R/I,R)$ is determined by the leftmost
map in this sequence, indeed it is the cokernel of its dual i.e.
$\operatorname{Ext}^{3}_{R}(R/I,R)=R/(x,y,z)$. It follows that

$\operatorname{Ext}^{3}_{R}(R/I,R)\otimes_{R}I=\frac{(x^{2},xz,y)}{(x^{2},xz,y)\cdot(x,y,z)}=\frac{(x^{2},xz,y)}{(x^{3},x^{2}z,x^{2}z,xy,y^{2},yz)}\simeq\kappa^{3}.$

Now consider that $V(I)$ is a double point. We will follow the same path. Up
to automorphism, $I=(x,y,z)^{2}+(x,y)=(x,y,z^{2})$. Then we have the
resolution

$0\longrightarrow R\xrightarrow{\begin{pmatrix}x\\\ y\\\
z^{2}\end{pmatrix}}R^{3}\xrightarrow{\begin{pmatrix}0&z^{2}&-y\\\
-z^{2}&0&x\\\
y&-x&0\end{pmatrix}}R^{3}\xrightarrow{\begin{pmatrix}x&y&z^{2}\end{pmatrix}}R\longrightarrow
R/I\longrightarrow 0$

and it follows that $\operatorname{Ext}^{3}_{R}(R/I,R)=R/I$. Therefore

$\operatorname{Ext}^{3}_{R}(R/I,R)\otimes_{R}I=\frac{I}{I^{2}}=\frac{(x,y,z^{2})}{(x^{2},xy,xz^{2},y^{2},yz^{2},z^{4})}\simeq\kappa^{6}.$

We can give a basis taking the classes of $\\{x,y,z^{2},xz,yz,z^{3}\\}$. ∎

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