Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases that n is even, conjectures for the resolution are also verified. Previously, by work of Evain (for the Hilbert function) and of Harbourne, Holay and Fitchett (for the resolution), these conjectures were known for infinitely many m for a given n > 9 only for n being a power of 4.
We study the Hilbert functions of fat points in P^1 x P^1. If Z is an arbitrary fat point subscheme of P^1 x P^1, then it can be shown that for every i and j the values of the Hilbert function H_Z(l,j) and H_Z(i,l) eventually become constant for l >> 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in P^1 x P^1. This enables us to compute all but a finite number values of H_Z without using the coordinates of points. We also characterize the ACM fat points schemes using our description of the eventual beha…
In this paper, we investigate the question of triviality of the rational Chow groups of complete intersections in projective spaces and obtain improved bounds for this triviality to hold. Along the way, we study the dimension and nonemptiness of some Hilbert schemes of fat $r$-planes contained in a complete intersection $Y$, generalizing well-known results on the Fano varieties of $r$-planes contained in $Y$.
We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat poi…
Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending results obtained in a previous paper using the second approach. In addition the second approach is used to obtain a complete determination of the Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points are smooth points of an irreducible plane cubic curve. Additional results are obtained using the first approach for $n\geq 9$ double points…
Given a fat point scheme $\mathbb{W}=m_1P_1+\cdots+m_sP_s$ in the projective $n$-space $\mathbb{P}^n$ over a field $K$ of characteristic zero, the modules of Kähler differential $k$-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of $\mathbb{W}$ when $k\in\{1,\dots, n+1\}$. In this paper we determine the value of its Hilbert polynomial explicitly for the case $k=n+1$, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme $\mathbb{Y} = (m_1-1)P_1 + \cdots + (m_s-1)P_s$. For $n…
The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P^2 are determined whenever Z is supported at any 6 or fewer distinct points. All results hold over an algebraically closed field k of arbitrary characteristic.
The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed fields of any characteristic. The algorithm, which works by giving a formula in certain cases and a reduction to these cases otherwise, does not involve Gröbner bases, and so is very fast, even for very large multiplicities. Partial information is also obtained in certain cases with $n>8$.
Given an ideal $I$, the containment problem is concerned about finding the values $m$ and $n$ such that the $m$-th symbolic power of $I$ is contained in its $n$-th ordinary power. In this paper we consider this problem focusing on two classes of ideals of fat points. In particular we study the ideal of $n$ points on a line in $\mathbb{P}^N$ and the ideal of three nonlinear points in $\mathbb{P}^N$ for $N\geq2$.
We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defi…
This paper is a natural sequel to [22] in that it tackles problems of the same nature. Here one aims at the ideal theoretic and homological properties of a class of ideals of general plane fat points whose second symbolic powers hold virtual multiplicities of proper homaloidal types. For this purpose one carries a detailed examination of their linear systems at the initial degree, a good deal of the results depending on the method of applying the classical arithmetic quadratic transformations of Hudson--Nagata. A subsidiary guide to understand these ideals through their initial linear systems …
One considers plane Cremona maps with proper base points and the {\em base ideal} generated by the linear system of forms defining the map. The object of this work is the interweave between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. One reveals conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here one emphasizes also the role of one single…
We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in P^n. Our main theorem is that these ideals are componentwise linear. This result yields a number of corollaries, including the multiplicity conjecture of Herzog, Huneke, and Srinivasan in this case. On the computational side, using an iterated mapping cone process, we compute formulas for the graded Betti numbers of ideals associated to two fat points in P^n, verifying a conjecture of Fatabbi, and at most n+1 general double points in P^n.
In this paper we find an algorithm which computes the Hilbert function of schemes $Z$ of "fat points" in $\PP3$ whose support lies on a rational normal cubic curve $C$. The algorithm shows that the maximality of the Hilbert function in degree $t$ is related to the existence of fixed curves (either $C$ itself or one of its secant lines) for the linear system of surfaces of degree $t$ containing $Z$.
In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms of certain local data and the, so called, power structure over the corresponding ring. Here we give an analogue of these formulae for equivariant (with respect to an action of a finite group on a smooth variety) Hilbert schemes of zero-dimensional subschemes and compute some local generating series for an action of the cyclic group on a smooth surface.
Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space P^d. Assume that the given points lie in a hyperplane P^{d-1} in P^d, and that the ground field k is algebraically closed of characteristic 0. We give an explicit minimal graded free resolution of I(m,d) in k[P^d] in terms of the minimal graded free resolutions of the ideals I(j,d-1) in k[P^{d-1}] with j < m+1. As a corollary, we give the following formula for the Poincare polynomial …
A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the projective plane. We study this problem for $r$ points in the plane over an algebraically closed field $k$ of arbitrary characteristic in case either $r \le 8$ or the points lie on a (possibly reducible) conic. In either case, it follows from work of the second author that there are only finitely many configuration types of points, where our notion of confi…
We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discu…
Let I be the ideal corresponding to a set of general points $p_1,...,p_n \in P^2$. There recently has been progress in showing that a naive lower bound for the Hilbert functions of symbolic powers $I^{(m)}$ is in fact attained when n>9. Here, for m sufficiently large, the minimal free graded resolution of $I^{(m)}$ is determined when n>9 is an even square, assuming only that this lower bound on the Hilbert function is attained. Under ostensibly stronger conditions (that are nonetheless expected always to hold), a similar result is shown to hold for odd squares, and for infinitely many m for ea…
We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.
We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and only if I^(3) = I^3. We use these configurations to answer a question of Huneke that asks whether I^(m) = I^m for all m if equality holds when m equals the big height of the ideal I.
Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) sets of points $X$ in $\mathbb{P}^1\times\mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\ge…
This is an appendix to the recent paper of Favacchio and Guardo. In these notes we describe explicitly a minimal bigraded free resolution and the bigraded Hilbert function of a set of 3 fat points whose support is an almost complete intersection (ACI) in $\mathbb{P}^1\times\mathbb{P}^1.$ This solve the interpolation problem for three points with an ACI support.
If $I=(f_1,\ldots,f_r)$ is an ideal in $S=k[x_1,\ldots,x_n]$, and $f_i$ are "general" elements of given degrees, there is a conjecture on the Hilbert series of $S/I$. We are considering the corresponding concepts in bigraded rings.
An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gröbner cells and their Betti strata recovering results of Ellingsrud and Strømme, Göttsche and Iarrobino.
This paper is a survey on major results on Hilbert functions of multigraded algebras and mixed multiplicities of ideals, including their applications to the computation of Milnor numbers of complex analytic hypersurfaces with isolated singularity, multiplicities of blowup algebras and mixed volumes of polytopes.
B. Harbourne and C. Huneke conjectured that for any ideal $I$ of fat points in $P^N$ its $r$-th symbolic power $I^{(r)}$ should be contained in $M^{(N-1)r}I^r$, where $M$ denotes the homogeneous maximal ideal in the ring of coordinates of $P^N$. We show that this conjecture holds for the ideal of any number of simple (not fat) points in general position in $P^3$ and for at most $N+1$ simple points in general position in $P^N$. As a corollary we give a positive answer to Chudnovsky Conjecture in the case of generic points in $P^3$.
By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this preprint determines the number of generators in each degree in a minimal homogeneous set of generators for the homogeneous ideal defining any fat point subscheme supported at points of any plane conic, smooth or not. Special cases for points on a cubic are also studied. All work is over an algebraically closed field of arbitrary characteristic.
In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m_1P_1 + ... + m_iP_i + .... + m_sP_s in P^n and Z' = m_1P_1 + ... + (m_i-1)P_i + .... + m_sP_s. We associate to P_i a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P_i of multiplicity m_i, and we denote it by deg_Z(P_i) = (d_1,...,d_v). We show that if one knows deg_Z(P_i) and the Hilbert function of Z, one will also know the Hilbert function of …
Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.
Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $\mathbb{P}^n$. In this context, we look inside properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals…
In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences.
This paper is concerned with determining the number of generators in each degree for minimal sets of homogeneous generators for saturated ideals defining fat point subschemes $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$ of $P^2$. For thin points (i.e., m_i=1 for all i), a solution is known, in terms of a maximal rank property. Although this property in general fails for fat points, we show it holds in an appropriate asymptotic sense. In the uniform (i.e., $m_1= ... =m_r$) case, we determine all failures of this maximal rank property for $r\le 9$, and we develop evidence for the con…
This paper surveys certain problems involving numerical characters for ideals I(Z) defining fat points subschemes $Z=m_1p_1+...+m_np_n$ for general points $p_i\in {\bf P}^2$. It also presents some new results, and includes a suite of MACAULAY 2 scripts for computing actual or conjectured values of (or bounds on) these characters. One such script, for example, is findres, which implements an algorithm due to Fitchett, Harbourne and Holay for computing the syzygy modules in a minimal free resolution of the ideal I(Z) for any such Z with n<=8; since findres does not rely on a Gröbner basis calcul…
Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.
Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if $J$ defines a complete intersection.
Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates, and holds for any field.
The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.
Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.
Sets of zero-dimensional ideals in the polynomial ring $k[x,y]$ that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of $(x,y)$-primary ideals in Gröbner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by…
Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points which approach a divisor. On the other hand, Nagata in connection with its counter example to the fourteenth problem of Hilbert determined the Hilbert function H(d) of the union of k^2 points of the same multiplicity m in the plane up to degree d=km. We introduce a new method to determine…
Following the approach in the book "Commutative Algebra", by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the generic initial extensor of a subset of a Grassmannian and then the double-generic initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric p…
The diagonal in a product of projective spaces is cut out by the ideal of 2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For projective lines we obtain a space of trees that is irreducible but singular. All ideals in our Hilbert scheme are radical. We also explore connections to affine buildings and Deligne schemes.
Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, in this note we present three algorithms to produce all strongly stable ideals with certain prescribed properties: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbe…
We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of Borel-fixed ideals and our previous work classifying Hilbert schemes with unique Borel-fixed points.
Let $F$ be a line bundle on the blow-up $X$ of $P^2$ at $r$ general points $p_1, ..., p_r$ and let $L$ be the pullback to $X$ of the line bundle coming from a line on $P^2$. Under reasonable hypotheses that are conjectured always to hold if the points $p_1, ..., p_r$ are sufficiently general, it is shown that the computation of the dimension of the cokernel of the natural map $μ_F:Γ(F)\otimesΓ(L)\toΓ(F\otimes L)$ reduces to the case that $F$ is ample. As an application, a complete determination of the dimension of the cokernel of $μ_F$ is obtained when $r\le 7$, thereby solving the Ideal Gener…
Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original questio…
Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert coefficients. In addition, we give some examples in this regards.
Let $X$ be a set of $K$-rational points in $P^1 \times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $Ω^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler different…
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the system of equations}. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and it relies on global estimations for the Hilbert function of homogeneous polynomial ideals. In this direction, we obta…
