Bounds on Hilbert Functions

Bounds on Hilbert Functions

ORNELLA GRECO

Licentiate Thesis Stockholm, Sweden 2013

TRITA-MAT-13-MA-03 ISSN 1401-2278

ISBN 978-91-7501-901-7

KTH Institutionen för Matematik 100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentia- texamen i matematik måndagen den 11 november 2013 kl 10.00 i sal 3418, Kungl Tekniska högskolan, Lindstedtsvägen 25, Stockholm.

Ornella Greco, 2013c

Tryck: Universitetsservice US AB

iii

Abstract

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectors of reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring.

iv

Sammanfattning

Denna avhandling består av två artiklar, båda relaterade till Hil- bert funktioner och h-vektorer. I den första artikel hanterar vi h- vektorer för reducerade noll-dimensionella scheman i det projektiva planet, och, i synnerhet, med problemet att finna möjliga h-vektorer för en union av två mängder av punkter med givna h-vektorer. I den andra artikeln, generaliserar vi Greens Hyperplansrestriktionssats till fallet av moduler över polynomringar.

Contents

Contents v

Part I: Introduction and summary

1 Introduction 1

1.1 Hilbert functions and h-vectors . . . . 2 1.2 Lexicographic Ideals and Modules . . . 3 1.3 Bounds on the Hilbert function . . . 8 1.4 The h-vector of a reduced zero-dimensional scheme inP² . . . 12

2 Summary of results 17

2.1 Paper A . . . 17 2.2 Paper B . . . 18

References 21

Part II: Scientific papers

Paper A

The h-vector of the union of two sets of points in the projective plane.

Paper B

Green’s Hyperplane Restriction Theorem: an extension to modules.

v

Acknowledgements

First of all, I would like to thank my supervisor, Mats Boij, for the support, for his scientific advises and also for giving me the chance to visit him at MSRI.

I also would like to express my gratitude to Ralf Fröberg, for being present even before the start of my PhD studies, and for giving me always very appreciated suggestions and comments.

A deep Sepas also to my Iranian support, Sadna and Afshin! And big Grazie to my Italian cheer, Alessandro and Martina!

I would like to thank also all the people met at MSRI, in particular Emanuele, Antonello, Emma, Ali, Jack and Jonathan, they all contributed to make me live an amazing period in Berkeley.

Last but not least, I must thank Ivan, for being everything to me, and fi- nally my parents, Saro and Annamaria, for being such good listeners and supporters.

vii

Chapter 1

Introduction

This thesis is constituted of two papers. The topic of both papers has been the study of an important invariant in commutative algebra: the Hilbert function, and, in relation with it, the h-vectors of rings and modules.

In the first paper (see [G-M-S2]), we study the Hilbert functions of sets of points in P²: namely we consider the coordinate ring associated with a set of points, and then calculate the first difference of the Hilbert function of this ring. In this way, we get a vector of finite length, called the h-vector of the set of points, which encodes information about the configuration of the set of points.

Then, we partially answer to the following question: given the h-vectors of two sets of points in the projective plane (there can be many configurations of points with the same h-vector), what are all possible h-vector for the union of two sets of points associated with the given h-vectors? We give two kinds of bounds on the h-vectors, and we also provide an algorithm that calculates many possible h-vectors, but we are not able to determine if they are all.

In the second paper, we study Hilbert functions in a more general set- ting. Namely, we generalize to the case of modules the Hyperplane Re- striction Theorem proved by M. Green in [Gr], which gives a bound on the Hilbert function of a general linear section of a symmetric algebra in terms of Macaulay’s binomial representations and lexicographic ideals.

1

2 CHAPTER 1. INTRODUCTION

In the next sections of this chapter, we are going to summarize the different concepts that are needed for the understanding of the two papers.

1.1 Hilbert functions and h-vectors

Now we are going to give an overview on the basic concepts of Hilbert functions, Hilbert series, and h-vectors. First of all, let us provide the setting in which we are going to work.

Let us consider a field k. A standard graded k-algebra S = ⊕_i≥0Si is a finitely generated k-algebra with all generators in degree 1. The main exam- ple of such an algebra is the polynomial ring, k[x₁, . . . , x_n], with the standard grading, or a quotient of the polynomial ring, k[x₁, . . . , xn]/I, where I is a standard graded ideal of k[x₁, . . . , xn].

Let M be a graded finitely generated S-module, and denote by M_i the i-th graded component.

Definition 1.1.1. The Hilbert function of M is a numerical function defined by H(M, d) = dim_k(M_d). The Hilbert series of M is defined as H_M(t) = P

i∈ZH(M, i)tⁱ.

Here we give a survey on some properties of Hilbert functions and Hilbert series. There are more complete surveys on this topic: see, for example, Chapter 4 in [B-H], or Chapter 6 in [H-H].

Theorem 1.1.2. [H-H, Theorem 6.1.3] Let d be the Krull dimension of M , then:

• there exists a Laurent polynomial Q_M ∈ Z[t, t⁻¹] such that H_M(t) =

QM

(1−t)^d;

• there exists a polynomial P_M ∈Q[x] of degree d−1 such that H(M, i) = PM(i) for all i > deg(Q_M) − d (i.e. the Hilbert function is of polyno- mial type).

From the previous theorem, we derive the following definitions.

Definition 1.1.3. The polynomial P_M is called the Hilbert polynomial of M . Let us write Q_M =^Pb_i=ah_itⁱ for a, b ∈Z, then the vector (h^a, h_a+1, . . . , h_b) is the h-vector of M .

1.2. LEXICOGRAPHIC IDEALS AND MODULES 3

The theorem below relates Hilbert functions and graded Betti numbers.

Theorem 1.1.4. [B-H, Lemma 4.1.13] Let M be a finitely generated S- module of finite projective dimension, and let us give a graded free resolution of M :

0 −→ ⊕_jS(−j)^βpj −→ · · · −→ ⊕_jS(−j)^β0j −→ M −→ 0,

then H_M(t) = S_M(t)H_S(t), where S_M(t) =^P_i,j(−1)ⁱβijt^j, and, in particu- lar, if S = k[x₁, . . . , xn], H_M(t) = _(1−t)^SM(t)n.

1.2 Lexicographic Ideals and Modules

In this section, we will introduce the concepts of monomial orders, initial ideals and lexicographic ideals. Then we will extend these concepts to free S-modules. Let S denote the polynomial ring k[x₁, . . . , x_n].

Definition 1.2.1. A monomial order on S (or equivalently onNⁿ) is a total order, <, on the set of monomials in S, Mon(S), such that:

• if u, v ∈ Mon(S) and u < v, then uw < vw for all w ∈ Mon(S).

• < is a well ordering onNⁿ.

Well known examples of monomial orders, where we have x₁ > x₂ > · · · >

xn, are:

• the pure lexicographic order (lex), in which x^a <_lex x^b iff the first left non zero component in a − b is negative;

• the graded lexicographic order (deglex), defined by

x^a <deglex x^b ⇔ |a| < |b| or |a| = |b| and a <_lexb.

• the reverse lexicographic order (revlex), where it is defined x^a <revlex

x^b iff either |a| < |b| or |a| = |b| and last right non zero component in a − b is positive.

Example 1.2.2. InN³ we have that a = (1, 0, 0) >_lex (0, 3, 2) = b, since (0, 3, 2) − (1, 0, 0) = (−1, 3, 2), but since |a| < |b|, we have that a <_deglex b.

Moreover, c = (1, 2, 2) > b in all three monomial orders, since they have same total degree, and c − b = (1, −1, 0), since the first non zero entry is positive, and the last non zero entry is negative.

4 CHAPTER 1. INTRODUCTION

Let us fix a monomial order in S, <, and let u be a monomial in degree d.

Then, we give the following notations:

L_u= {v ∈ Mon(S_d)| v < u}, R_u = {v ∈ Mon(S_d)| v ≥ u}.

Definition 1.2.3. A monomial ideal I in S is called lexicographic ideal if, for every degree d, I_d is a lex-segment, i.e. a saturated chain (with respect to the chosen monomial order) of monomials starting from the largest, i.e.

if there is a monomial u ∈ S_d such that I_d= R_u.

Given a monomial order <, we can order the terms of a polynomial f ∈ S in a unique way, and in particular we can consider the leading (or initial) term of f , denoted by in_<(f ).

Definition 1.2.4. Given an ideal I in S the initial ideal of I with respect to

< is the ideal generated by the initial terms (with respect to the monomial order <) of the polynomials in I. We denote this ideal by in_<(I).

The initial ideals have nice properties, in particular we have many results that compare the invariants of graded ideals and their initial ideals.

Proposition 1.2.5. [H-H] Let I be a graded ideal of S, and let < be a monomial order on S. Then S/I and S/in_<(I) have the same Hilbert func- tion.

Theorem 1.2.6. [H-H, Theorem 3.3.4] Let I be a graded ideal of S and <

a monomial order on S. Then the following properties hold:

1. dim(S/I) = dim(S/in_<(I));

2. projdim(S/I) ≤ projdim(S/in_<(I));

3. reg(S/I) ≤ reg(S/in_<(I));

4. depth(S/I) ≥ depth(S/in_<(I));

Generic Initial ideals

We now define the concept of generic initial ideal. Let k be an infinite field, and again let us denote by S the polynomial ring, k[x₁, . . . , xn].

Consider the general linear group, GL(n), i.e. the group of all invertible n×n matrices with entries in k. This group can be equipped with the Zariski

1.2. LEXICOGRAPHIC IDEALS AND MODULES 5

topology, inherited from k^n×n; moreover, GL(n) is Zariski open: this implies that a subset of GL(n) is open if and only if it is a Zariski open set in k^n×n. Let us consider the action of GL(n) on the polynomial ring S: a matrix (a_ij) ∈ GL(n) sends x_j to ^Pn_i=1aijxi; this action can be then extended to any f ∈ S.

We have the following result.

Theorem 1.2.7. [H-H, Theorem 4.1.2] Let I be a graded ideal in S, and

< a monomial order on S, then there exists a non-empty Zariski open set U ⊆ GL(n), such that ∀ g, h ∈ U , in_<(gI) = in_<(hI).

Definition 1.2.8. The ideal in_<(gI), where g ∈ U , is called generic initial ideal of I, with respect to <, and denoted by gin_<(I) (or simply gin(I), when the monomial order is clear).

We now discuss some properties of generic initial ideals, namely they are Borel-fixed, and, in certain characteristic of the field k, they are strongly stable.

Let us consider the Borel subgroup of GL(n), denoted by B(n), i.e. the group of all upper triangular invertible n × n matrices. From the definition of the action, a matrix in B(n) will send x₁to a multiple of itself, and x_nto the linear combination of all variables x_i’s with coefficients the last column of tha matrix.

Definition 1.2.9. An ideal I in S is Borel-fixed if gI = I for every matrix g ∈ B(n).

Definition 1.2.10. A monomial ideal I in S is strongly stable if for every monomial u ∈ I and for j such that x_j divides u, we have that x_i(u/x_j) ∈ I,

∀ i < j.

Theorem 1.2.11. [H-H, Theorem 4.2.1] Let I be a graded ideal in S and

< a monomial order on S, then gin_<(I) is a Borel-fixed ideal.

The following theorem relates Borel-fixed ideals with strongly stable ideals.

Theorem 1.2.12. [H-H, Proposition 4.2.4]

• Let I be a graded ideal in S, if I is Borel-fixed, then it is a monomial ideal;

• If I is strongly stable, then it is Borel-fixed;

6 CHAPTER 1. INTRODUCTION

• Let I be a Borel-fixed ideal, and let a be the biggest exponent appearing among the generators of I, then, if char(k) = 0 or char(k) > a, I is strongly stable.

Definition 1.2.13. We say that a property P holds for a generic linear form ` if there is a non-empty Zariski open set U ⊆ S<sub>1</sub> such that P holds for all ` ∈ U .

In case we choose the monomial order < to be the reverse lexicographic order, we have the following result:

Proposition 1.2.14. [Gr2, Corollary 2.15] Let h be a generic linear form and I a graded ideal in F , then gin_<(I_h) = gin_<(I)_xn.

Lexicographic Modules and Generic Initial Modules Let us extend these definitions to free S-modules.

Let F be a free finitely generated S-module and let us fix an homogeneous basis {e₁, e2, . . . , er} and let deg(e_i) = f_i, where, without loss of generality, we may assume that f₁ ≤ f₂ ≤ · · · ≤ f_r.

We now define the monomial modules and we induce a lexicographic order on F in such a way that the concept of lexicographic module may be defined.

Definition 1.2.15. A monomial in F is an element of the form me_i where m ∈ Mon(S).

A submodule M ⊆ F is monomial if it is generated by monomials, in this case it can be written as I₁e1⊕ I₂e2 ⊕ · · · ⊕ I_rer, where I_i is a monomial ideal for i = 1, 2, . . . , r.

Let us now extend the concept of monomial order to modules, in particular let us define the graded lexicographic order and the reverse lexicographic order on F .

Definition 1.2.16. Given two monomials in F , me_i and ne_j, we say that mei>deglexnej if either i = j and m >_deglexn in R or i < j. In particular, we have that e₁ > e₂ > · · · > er.

Definition 1.2.17. A monomial submodule L is a lexicographic module if, for every degree d, L_d is spanned by the largest, with respect to the lexicographic order, H(L, d) monomials.

1.2. LEXICOGRAPHIC IDEALS AND MODULES 7

Definition 1.2.18. The reverse lexicographic order on F is defined by choosing an order on the basis of F , say e₁ > · · · > er and by setting me_i >_revlexne_j iff either deg(me_i) > deg(ne_j) or the degrees are the same and m >_revlexn in S or m = n and i < j.

Remark 1.2.19. The graded lexicographic order is a "position over term"

monomial order, on the contrary the reverse lexicographic order is "term over position" monomial order.

Definition 1.2.20. The initial module of a submodule M with respect to a chosen monomial order on F , denoted by in_<(M ), is the submodule of F generated by the set {in<(m)| m ∈ M }, i.e. by all the leading terms of elements in M .

In case we choose the reverse lexicographic order, we have some nice results.

Namely:

Proposition 1.2.21. [Ei, Proposition 15.12] Suppose that F is a free S- module with basis {e₁, . . . , e_r} and reverse lexicographic order. Let M be a graded submodule. Then:

• in(M + x_nF ) = in(M ) + x_nF ;

• (in(M ) :_F x_n) = in(M :_F x_n).

Let us recall the concept of generic initial module. In his thesis, see [Pa], K. Pardue defined this concept in a more general setting, here we will restrict to the case is needed in the second paper.

Let us consider the group GL(F ) of the S-module automorphisms of F . An element φ in GL(F ) is a homogeneous automorphism and can be represented by a matrix (t_ij), with t_ij ∈ S_fi−fj, where φ(e_i) = ^Pr_j=1tijej. Moreover, the general linear group GL(n) acts on F k-linearly; and we have also an action of GL(n) on GL(F ), given by a · φ = aφa⁻¹, for all a ∈ GL(n) and φ ∈ GL(F ).

So, let us denote by G the semidirect product G = GL(n)o GL(F ).

As done for the ideals, let us consider the Borel subgroup of G, B, which is obtained throught the semidirect product of B(n), the subgroup of GL(n) of upper triangular invertible matrices, and B(F ), the subgroup of all au- tomorphisms in GL(F ) represented by lower triangular matrices (they send each e_l to an S-linear combination of e₁, . . . , el).

8 CHAPTER 1. INTRODUCTION

In Pardue’s thesis, there is the generalization to modules of many results on generic initial ideals. In particular, we have the following result.

Proposition 1.2.22. [Pa, Chapter 1] Let M ⊂ F be a graded module, and let < be a monomial order on F , then there exists a Zariski open set U ⊆ G such that in_<(φM ) is constant for every φ ∈ U . Moreover, in_<(φM) is fixed by the action of group B.

Definition 1.2.23. The monomial submodule in_<(φM ), φ ∈ U , is called the generic initial module of M , and denoted by gin_<(M).

Theorem 1.2.24. [Pa, Proposition 5, Chapter 1] A submodule M ⊆ F is fixed by B if and only if:

• M is a monomial submodule, i.e. can be written as I₁e₁⊕ · · · ⊕ I_re_r;

• the ideals I_j are Borel-fixed;

• for every i < j, (x₁, . . . , xn)^fj−fiIj ⊆ I_i.

1.3 Bounds on the Hilbert function

Let us now recall some results on extremal properties of Hilbert functions, and bounds on Hilbert functions. One of the most important result in this area is Macaulay’s theorem (see [Ma, B-H, K-R]), which characterizes the possible Hilbert functions of homogeneous k-algebras. Before announcing this theorem, we need the concept of Macaulay’s representations of integers.

Definition 1.3.1. Given a, d ∈N, the d-th Macaulay representation of a is the only way of writing a in the following way:

a_d d

!

+ a_d−1 d − 1

!

+ · · · + a₁ 1

! ,

where a_d> a_d−1> · · · > a₁ ≥ 0.

Observation 1.3.2. If a = ^a_d^d
+ ^a_d−1^d−1
+ · · · + ^a₁¹
and b = ^b_d^d
+ ^b_d−1^d−1
+

· · · + ^b₁¹
, a ≥ b if and only if (a_d, . . . , a₁) ≥_lex(b_d, . . . , b₁).

If c < d, then _d^c
= 0.

1.3. BOUNDS ON THE HILBERT FUNCTION 9

Moreover, we introduce three numerical functions, defined by means of the d-th Macaulay representation of a = ^a_d^d
+ ^a_d−1^d−1
+ · · · + ^a₁¹
, namely:

a^hdi = a_d+ 1 d + 1

!

+ a_d−1+ 1 d

!

+ · · · + a₁+ 1 2

! ,

a_hdi = a_d− 1 d

!

+ a_d−1− 1 d − 1

!

+ · · · + a₁− 1 1

! ,

a^(d) = a_d d + 1

!

+ a_d−1 d

!

+ · · · + a₁ 2

! .

The growth of the lexicographic ideals is related to the first function, namely:

Proposition 1.3.3 (Macaulay). [B-H, Proposition 4.2.9] Let I be a homo- geneous ideal in S, and d ∈N.

Then

H(S/I, d + 1) ≤ H(S/I, d)^hdi,

and equality holds if I_d is a lex-segment space and I_d+1= S₁· I_d.

The following result gives us the characterization of the Hilbert function.

Theorem 1.3.4 (Macaulay). [B-H, Theorem 4.2.10] Let k be a field, and h :N → N. The following conditions are equivalent:

1. there is a homogeneous k-algebra S with Hilbert function H(S, d) = h(d);

2. there is a homogeneous k-algebra with monomial relations S with Hilbert function H(S, d) = h(d);

3. h(0) = 1, h(d + 1) ≤ h(d)^hdi for all d ≥ 1.

Another classical result in this topic is the Hyperplane Restriction The- orem, proved by Green in [Gr], which gives a bound for the codimension of the generic linear restriction of a vector space generated in a certain degree by the codimension of a lex-segment space with same degree and dimension.

This result was also used by Green to give a new, less technical, proof of Macaulay’s Theorem.

10 CHAPTER 1. INTRODUCTION

Let ` ∈ S<sub>1</sub> be a generic linear form, and I a homogeneous ideal, let us denote by (S/I)<sub>`</sub> the reduction to this hyperplane, i.e. the ring S/[I + (`)].

Similarly, if V ⊆ S_d a lex-segment space, we denote by V<sub>`</sub> the image of V in S/(`).

Proposition 1.3.5. [K-R, Proposition 5.5.23] Let k be an infinite field, d ∈N, V ⊆ S^d a lex-segment space. Then:

1. For a generic linear form ` ∈ S<sub>1</sub>, there is a homogeneous linear change of coordinates φ : S → S such that φ(V ) = V and φ(`) = x_n;

2. For a generic linear form ` ∈ S₁, we have

codim_k(V_`) = codim_k(V_xn) = codim_k(V )_hdi.

Theorem 1.3.6 (Green). [B-H, Theorem 4.2.12] Let I be a homogeneous ideal in S, d ∈N, then

H((S/I)`, d) ≤ H(S/I, d)hdi,

where ` is a generic linear form. Equality holds if I_dis a lex-segment space.

The analogous result of Macaulay’s Theorem for the exterior algebra was proved by Kruskal and Katona (see [Ka, Kr]), and it provides a characteri- zation of f -vectors of simplicial complexes.

Let k be a field, W an n-dimensional k-vector space with basis {e₁, . . . , e_n}.

Let E = ⊕_i∧ⁱW be the exterior algebra. A monomial in E is an element e_G= e_j1 ∧ · · · ∧ e_ji, where G = {j₁ < · · · < j_i} ⊆ [n].

In a natural way we can define the lexicographic order over the monomials in the exterior algebra: e_G<lexe_L iff x^G<lexx^L in S.

Let V = {v₁, . . . , v_n} be a finite set. A simplicial complex ∆ on V is a collection of subsets of V such that {v_i} ∈ ∆ for every i, and if G ∈ ∆ and F ⊆ G, then F ∈ ∆. The elements of ∆ are called faces of ∆. For a given face F ∈ ∆, dim(F ) = |F | − 1.

Given a simplicial complex ∆, for all i = −1, 0, . . . , dim(∆), let f_i be the number of i-dimensional faces of ∆. Note that ∅ ∈ ∆, so f−1 = 1 and f₀= |V |.

Here we state a theorem which characterizes the possible f -vectors for simplicial complexes.

1.3. BOUNDS ON THE HILBERT FUNCTION 11

Theorem 1.3.7 (Kruskal-Katona). A vector (f−1, f0, . . . , fd−1) ∈ Z^d+1 is the f -vector of some (d − 1)-dimensional simplicial complex if and only if

0 < f_i+1≤ f_i⁽ⁱ⁺¹⁾, ∀ 0 ≤ i ≤ d − 2.

Given a simplicial complex ∆ over V = {v₁, . . . , vn}, we can associate a monomial ideal in the exterior algebra E, in the following way:

e_i1 ∧ · · · ∧ e_it ∈ I_∆ ⇔ {v_i1, . . . , v_it} /∈ ∆.

The algebra E/I_∆ is called Stanley-Reisner algebra, and it is denoted by k{∆}. Notice that

f_d(∆) = H(k{∆}, d),

this lets us interpret the Kruskal-Katona Theorem as a Macaulay bound on the growth of the Hilbert functions for exterior algebra.

Macaulay’s Theorem and Green’s Hyperplane Restriction Theorem have been extended to modules by many authors, in particular by Hulett in [Hu]

and Gasharov in [Ga]. Here we recall their results.

Theorem 1.3.8. [Hu, Theorem 1] Let H be the Hilbert function of a graded quotient F/M , where M is a homogeneous submodule of F . Then there is a lexicographic module L ⊆ F such that H(F/L, d) = H(d) for all d, and βi,j(F/L) ≥ β_i,j(F/M ).

As a consequence, we get:

Theorem 1.3.9. [Hu, Corollary 6] Let M be a graded submodule of F , then

∃ N ≤ t such that we have the unique expression for H(F/M, d) =

N −1

X

i=1

n + d − fi− 1 d − f_i

!

+ a0

d − f_N

!

+ · · · + as

d − f_N− s

! ,

where _d−f^a0

N

+ · · · + _d−f^as

N−s

< ^n+d−f_d−f^N−1

N

, and it is the Macaulay repre- sentation of a lex-segment in degree d − f_N in the N -th component of F . Moreover,

H(F/M, d+1) ≤

N −1

X

i=1

n + d − f_i d − f_i+ 1

!

+ a₀+ 1 d − f_N + 1

!

+· · ·+ a_s+ 1 d − f_N − s + 1

! , where the bound is achieved by the lexicographic submodule of the previous theorem.

12 CHAPTER 1. INTRODUCTION

In the following result, due to Gasharov, he describes a bound on the growth of Hilbert function of modules, and on the generic hyperplane sec- tion. Let again F = ⊕^t_i=1Se_i, and deg(e_i) = f_i, with f_i ≥ f_i+1.

Theorem 1.3.10. [Ga, Theorem 4.2] Let N ⊆ F be a graded submodule, then ∀ p ≥ 0 and ∀ d ≥ p + f₁+ 1 we have that:

1. H(F/N, d + 1) ≤ H(F/N, d)^hd−f1−pi;

2. H(F_{`</sub>/N<sub>`}, d) ≤ H(F/N, d)_hd−f1−pi, where ` is a generic linear form.

1.4 The h-vector of a reduced zero-dimensional scheme in P

The study of sets of points in the projective plane constitute a really im- portant topic in algebra. There are several surveys dealing with it (see [Ei2, Gr2]). This section will give a summary of some results and definitions useful to understand the results contained in the first paper.

Let k be an algebraically closed field, P² = P²(k) be the projective plane over k and S = k[x₀, x1, x2] its homogeneous coordinate ring. In this section, we will refer to a finite set of distinct points in the projective plane as a reduced zero-dimensional scheme.

If X is a zero-dimensional scheme in P²(k), we denote by H_X(i) = dim_k((S/I_X)_i) its Hilbert function and by h = (h₀, . . . , ht) its h-vector where h₀ = 1 and h_i= 4H_X(i) = H_X(i) − H_X(i − 1) ∀ i > 0.

In case X is reduced, the number of points in X, also called degree of X, is given by h₀+ · · · + h_t.

It is important to point out that to a given h-vector may correspond sev- eral configurations of points, i.e. several different reduced zero-dimensional schemes.

Remark 1.4.1. To a given h-vector, (h₀, h₁, . . . , h_t), we can assign a diagram by drawing columns of h_i boxes for all i. For example if h = (1, 2, 3, 2, 2, 1), we will draw the following diagram.

1.4. THE H-VECTOR OF A REDUCED ZERO-DIMENSIONAL

SCHEME INP² 13

We now define three measures of the h-vectors.

Let X be a reduced zero-dimensional scheme and let h be its h-vector, then the length of h is τ (h) + 1, where:

τ (h) = max{i | h_i 6= 0} = min{i | H_X(i) = H_X(i + 1)}.

We define b(h) = max{h_i| i = 0, 1, . . . , τ (X)} as the height of the h-vector h, which is also equal to the least degree of a generator of IX, the vanishing ideal of X.

Sometimes we will use τ (X) and b(X) instead of τ (h) and b(h).

We define also η(h, d) := ^{Pτ (X)}_i=0 min{h_i, d}. If d is greater or equal to the height of h, then η(h, d) is just equal to the degree of the scheme.

The following theorem due to E. D. Davis (see [Da]) provides a descrip- tion of the structure of the h-vector of a reduced zero-dimensional scheme, and gives also information about the configuration of the sets of points as- sociated to an h-vector.

Theorem 1.4.2 (Davis). The h-vector (h₀, . . . , h_{τ (X)}) of a reduced zero- dimensional scheme X ⊂P² satisfies the following conditions:

1. h_d= d + 1, for d = 0, 1, . . . , b − 1, and h_b ≤ b;

2. h_d+1≤ h_d, for d ≥ b − 1;

3. If h_d= h_d+1= e for some d ≥ b − 1, the generators of I_X of degree at most d + 1 have a common factor of degree e. This leads to a partition of X into X₁∪ X₂, where X₁ lies on a curve of degree e and X₂ has the h-vector given by (h_e− e, h_e+1− e, . . . , h_d−1− e) .

Where b is the least degree of a generator of I_X.

In the case described in part (3) of the theorem, we say that the h-vector of X has a flat of height e.

It is possible to introduce a natural partial ordering in the set of the Hilbert functions of reduced zero-dimensional scheme in the projective plane

H := {4H_X| X ⊆P²is a finite set of points}.

Namely, letH¹= (H_X(i))_i∈NandH²= (H_Y(i))_i∈Nbe two Hilbert functions, we will say thatH² is more generic than H¹, and we will write H¹ ≤_g H²,

14 CHAPTER 1. INTRODUCTION

if H_X(i) ≤ H_Y(i), ∀ i ∈ N. We say also in this situation that H¹ is more special thanH2.

This partial order induce also a partial order (with the same notation, ≤_g) on the h-vectors. Indeed, if (h₀, h1, . . . , hs) and (h⁰₀, h⁰₁, . . . , h⁰_t) are the h- vectors of two finite sets of points inP². Then:

(h₀, h₁, . . . , h_s) ≤_g (h⁰₀, h⁰₁, . . . , h⁰_t) :⇐⇒

j

X

i=0

h_i≤

j

X

i=0

h⁰_i, ∀ j = 0, . . . , s

Going to the diagram of an h-vector, this means that, by moving one box from a row to an upper row in such a way that the result is admissible (i.e.

satisfies the conditions in Theorem 1.4.2), we get a more generic h-vector.

In the first paper, we provided two algorithms that build configurations of points associated to an h-vector, and that calculate a range of possible h-vectors for the union. In both of them, we have used heavily the theory of 2-type vectors, pseudo type vectors and linear configurations, that we are now going to introduce. This theory is fully described in [G-M-S1]

Definition 1.4.3. A 2-type vector is a vector (d₁, d₂, . . . , d_t), where 0 <

d1 < d2< · · · < dt .

To any h-vector, associated to a reduced zero-dimensional subscheme ofP², corresponds only one 2-type vectors. The following theorem explains this correspondence.

Theorem 1.4.4. [G-M-S1, Theorem 2.4, Theorem 2.5] Let S₂ denote the collection of Hilbert functions of all reduced zero-dimensional schemes in P². Then, there is a 1-1 correspondence between S₂ and the set of 2-type vectors. Moreover, let T = (d₁, d₂, . . . , dt) be a 2-type vector , and H_i the Hilbert function of d_i collinear points. Then, T corresponds to the Hilbert function defined by H(j) = H_t(j) + · · · + H₁(j − (t − 1)).

Once we have the definition of 2-type vector we can define the concept of linear configuration.

Definition 1.4.5. Let T = (d₁, . . . , dt) be a 2-type vector. Let L₁, . . . , Lt

be t distinct lines in P² and Xⁱ a set of d_i distinct points on L_i, for all i = 1, . . . , t. Moreover, we suppose that, for i 6= j, L_i does not contain any point ofX^j. Then,X = ∪^ti=1X^j is called a linear configuration of type T .

1.4. THE H-VECTOR OF A REDUCED ZERO-DIMENSIONAL

SCHEME INP² 15

The following result shows that the Hilbert function associated to a linear configuration of a given type T depends only on the type, and not on the choice of the lines and of the points on them.

Theorem 1.4.6. [G-M-S1, Theorem 2.8] LetX be a linear configuration of type T , and let H be the Hilbert function associated to T . Then the Hilbert function ofX, HX is H.

Remark 1.4.7. Given a reduced zero-dimensional scheme which is also a linear configuration, its type is nothing else than a partition of the degree of the scheme constituted by strictly increasing positive integers.

A similar definition of linear configurations can be given for partitions of the degree of the scheme constituted by non-decreasing positive integers.

Definition 1.4.8. A pseudo type vector is a sequence of positive integers T = (d₁, . . . , d_t), where d_i≤ d_i+1 ∀ i, and if d_i−1= d_i, then d_i< d_i+1. A pseudo linear configuration of type T is a set of pointsX = ^St

i=1Xi, where Xi is a set of d_i distinct points on a line L_i. The lines L₁, . . . , L_t are all different, and none of the points ofXⁱ lies on L_j for i 6= j.

Also in this case, an O-sequence can be associated to a pseudo type vec- tor (see [G-M-S1]), but in general the Hilbert function of a pseudo linear configuration of type T = (d₁, . . . , d_t) is not uniquely determined, unless it satisfies the following:

between any two zero entries of ∆T there is at least one entry > 1. (1.1) To a given h-vector, (h₀, . . . , h_t), one can associate a monomial ideal I such that the standard graded k-algebra S/I has the given h-vector.

1. From the h-vector (h₀, . . . , ht) we pass to its geometric representation by drawing h_i boxes for all i.

2. By Davis’s Theorem we have that h_i = i + 1 for i = 0, . . . , b − 1 and h_i ≥ h_i+1 ∀i ≥ b − 1. Denote by d_i the number of squares in the i-th row, for i = 1, . . . , b. Notice that the vector D = (d1, . . . , db) is a 2-type vector.

3. Let I be the ideal generated by x^db, x^db−1y, . . . , y^b. Then the k-algebra S/I has h-vector (h₀, h₁, . . . , h_t). Note also that the ideal I is a lexi- cographic ideal.

16 CHAPTER 1. INTRODUCTION

To the ideal I = (x^db, x^db−1y, . . . , y^b) we can assign a set of points inP², whose defining ideal has the same h-vector. This set of points is a linear configuration of type D = (d₁, . . . , d_b). In order to get it we first choose two sets of distinct elements in k, {α₁, . . . , αd_b} and {β₁, . . . , βb}, and then replace every generator xⁱy^j, of I by (x − α₁z) . . . (x − αiz)(y − β₁z) . . . (y − β_jz).

In the special case where α₁ = 0, . . . , α_db = d_b− 1 and β₁ = 0, . . . , β_b= b − 1, we get the defining ideal of the following set of points:

db points with coordinates (i : 0 : 1), i = 0, . . . , d_b− 1;

d_b−1points with coordinates (i : 1 : 1), i = 0, . . . , d_b−1− 1;

...

d₁ points with coordinates (i : b − 1 : 1), i = 0, . . . , d₁− 1.

We will call this the standard linear configuration of type D.

Finally, we need the definition of sum of two partitions.

Remark 1.4.9. Given a partition of a number n, i.e. (c₁, . . . , ct), where ci ≤ c_i+1 and c₁+ · · · + c_t = n, then clearly by adding zero entries we will still get a partition of n. If the first partition is associated to a scheme, so are the ones with the zeros.

Definition 1.4.10. Let c = (c₁, . . . , c_t) and d = (d₁, . . . , d_v) be two par- titions of n respectively m. Assume in addition that at least one of those partitions is either a 2-type or a pseudo type vector whose first difference satisfies the Condition (1.1). We say that a partition of n + m is the sum of c and d, if it is obtained by ordering the sequence

{c_i+ d_j}i=1,...,t;j=1,...,v

(where each c_i and d_j appear exactly once in the sums) in a non-decreasing way.

Chapter 2

Summary of results

2.1 Paper A

The h-vector of the union of two sets of points in the projective plane.

The first paper deals with the problem of finding all the possible h-vectors of the union of two sets of points with given h-vectors h and h⁰.

In order to exclude an h-vector from the set of the possible ones, we found some bounds on the main units of measure, which are b, τ and η.

Namely, we proved the following inequalities:

Theorem 2.1.1. Given two sets of points X and Y in P², with give h- vectors respectively h and h⁰ we have that

max{τ (X), τ (Y )} ≤ τ (X ∪ Y ) ≤ τ (X) + τ (Y ) + 1.

Theorem 2.1.2. Given two sets of points X and Y in P², we have the following bounds for the height of the resulting h-vector:

max{b(X), b(Y )} ≤ b(X ∪ Y ) ≤ min{b(X) + b(Y ), b(G)}

where G consists of deg(X ∪ Y ) generic points.

Theorem 2.1.3. Let h be the h-vector of X with flat of height r, h⁰ the h-vector of Y with flat of height s and h⁰⁰ the h-vector of X ∪ Y . Then we have

η(h⁰⁰, r + s) ≥ η(h, r) + η(h⁰, s).

17

18 CHAPTER 2. SUMMARY OF RESULTS

Theorem 2.1.4. Given two h-vectors h and h⁰ with a flats of degree d and d⁰ respectively. For the union we can then exclude h-vectors h⁰⁰ which have a flat of degree d⁰⁰≥ d, d⁰ with

max



η(h, d⁰⁰) + η(h⁰, d⁰⁰+ d⁰) − η(h⁰, d⁰) +d⁰+ 2 2



− 2, η(h⁰, d⁰⁰)+

η(h, d⁰⁰+ d) − η(h, d) +d + 2 2



− 2



< η(h⁰⁰, d⁰⁰) < η(h, d) + η(h⁰, d⁰).

Later, we used the concept of pseudo type vector (or partition) to give an algorithm that produces all the pseudo type vectors associated to a given h-vector h, i.e., using this algorithm, we are able to find all monomial ideals I in two variables, x, y, with pure powers of the variables, such that the ring k[x, y]/I has h-vector h.

Afterwards, we gave the definition of sum of two partitions, in such a way that we could give a new algorithm, able to produce possible h-vectors of the union. We conjectured that this algorithm gives all of them.

In the last section, we provided a way to construct the unique minimum, with respect to the partial ordering defined on the set of h-vectors, h-vector for the union.

Proposition 2.1.5. For any two given h-vectors, we can always construct the unique minimum h-vector for the union among all the admissible ones.

This h-vector achieves the lower bound for the height and the upper bound for the length.

The paper gives only a partial solution to the question, since we were not able to prove that the range of possible h-vectors found through the second algorithm covers all the possible h-vectors of the union, so we were not able to give a complete characterization of the h-vectors of the union.

2.2 Paper B

Green’s Hyperplane Restriction Theorem: an extension to modules.

The second paper gives a new generalization of Green’s Hyperplane Re- striction Theorem to the case of modules over the polynomial ring.

The tools we have used to prove the main theorem are mainly inequalities

2.2. PAPER B 19

regarding Macaulay representation of integers, lexicographic modules and generic initial modules. Namely, we have first proved the following inequal- ity:

Proposition 2.2.1. Given a, b ∈N, a ≤ N¹ = ^n+d_d¹⁻¹

1

, b ≤ N₂ = ^n+d_d²⁻¹

2

and d₁ ≥ d₂, d₁, d₂ ∈N, then

a_hd1i+ b_hd2i≤

((a + b)_hd2i if a + b ≤ N₂, (a + b − N₂)_hd1i+ (N₂)_hd2i if a + b ≥ N₂. Then, we have extended the inequality to a more general case.

Finally, we have used the theory on generic initial modules to restrict to the monomial case and, combining the Green’s Hyperplane Restriction Theorem with these inequalities on Macaulay representations, we were able to prove the main theorem.

Definition 2.2.2. Let M be a submodule in F , and m ∈N. Set di = m − f_i ({d₁, d2, . . . , dr} is a non-increasing sequence) and N_i = ^n+d_dⁱ⁻¹

i

= dim_kSdi. Then, if^Pr_i=j+1Ni ≤ H(F/M, m) ≤^Pr_i=jNi, for some j, we define

H(F/M, m)_{m,r}= (H(F/M, m) −

r

X

i=j+1

N_i)_hdji+

r

X

i=j+1

N_ihdii. Theorem 2.2.3. Let F = Se₁⊕ · · · ⊕ Se_r where deg(e_i) = f_i for all i. Let M be a submodule in F , then

H((F/M )_{`</sub>, m) ≤ H((F/L)<sub>`}, m)

where ` is generic linear form, m ∈N, and L is a submodule that in degree m is generated by a lex-segment of length H(M, m).

Moreover,

H((F/L)_`, m) = H(F/M, m)_{m,r}.

In the past, a generalization of Hyperplane Restriction Theorem has been done by Gasharov (see [Ga]). The reason why we did another generalization was to find a bound of the restriction of a module to a generic hyperplane fully described by lexicographic modules.

Green’s Hyperplane Restriction Theorem has been applied in several papers (see for instance [A-S], [B-Z], [M-Z]) to get some results about level and Gorenstein algebras, with focus on the weak Lefschetz property. For future work, we intend to use Theorem 2.2.3 to produce similar results.

References

[A-S] J. Ahn and Y. S. Shin, Artinian level algebras of codimension 3, J.

Pure Appl. Algebra 216 (2012), no. 1, 95–107.

[B-Z] M. Boij and F. Zanello, Some algebraic consequences of Green’s Hy- perplane Restriction Theorems, J. Pure Appl. Algebra 214 (2010), no.

7, 1263–1270.

[B-H] W. Bruns and J. Herzog, Cohen- Macaulay Rings, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1998.

[Da] E.D. Davis, Complete Intersections of Codimension 2 in P^r: The Bezout-Jacobi-Segre Theorem Revisited., Rend. Sem. Mat.Univers. Po- litecn. Torino, 43, 4 (1985), 333-353.

[Ei] D. Eisenbud, Commutative algebra. With a view toward algebraic geom- etry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.

[Ei2] D. Eisenbud, The geometry of syzygies. A second course in commuta- tive algebra and algebraic geometry. Graduate Texts in Mathematics, 229. Springer-Verlag, New York, 2005.

[Ga] V. Gasharov, Extremal properties of Hilbert functions, Illinois Journal of mathematics 41, 1997.

[G-M-S1] A.V. Geramita, J. Migliore and L. Sabourin, On the first infinites- imal neighborhood of a linear configuration of points in P².

J. Algebra 298 (2006), no. 2, 563-611.

[G-M-S2] O. Greco, M. Mateev and C. Söger, The h-vector of the union of two sets of points in the projective plane. Matematiche (Catania) 67 (2012), no. 1, 197–222.

21

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