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 configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say $p_1,...,p_r$ and $p'_1,...,p'_r$ have the same {\it configuration type} if for all choices of nonnegative integers $m_i$, $Z=m_1p_1+...+m_rp_r$ and $Z'=m_1p'_1+...+m_rp'_r$ have the same Hilbert function.) Assuming either that $7 \le r\le 8$ (see recent work of Guardo and the second author for the cases $r\le 6$) or that the points $p_i$ lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients $m_i$ determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of $Z=m_1p_1+...+m_rp_r$. We demonstrate our results by explicitly listing all Hilbert functions for schemes of $r\le 8$ double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.
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…
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…
The purpose of this note is to study initial sequences of zero-dimensional subschemes of Hirzebruch surfaces and classify subschemes whose initial sequence has the minimal possible growth.
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.
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…
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.
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…
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.
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 $n>8$.
In this paper, we study the Waldschmidt constant of a generalized fat point subscheme $Z=m_1p_1+\cdots+m_rp_r$ of $\mathbb{P}^2$, where $p_1,\cdots,p_r$ are essentially distinct points on $\mathbb{P}^2$, satisfying the proximity inequalities. Furthermore, we prove its lower semi-continuity for $r\le 8$. Using this property, we also calculate the Waldschmidt constants of the fat point subschemes $Z=p_1+\cdots+p_5$ giving weak del Pezzo surfaces of degree 4.
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…
Motivated by questions in interpolation theory and on linear systems of rational varieties, one is interested in upper bounds for the Castelnuovo-Mumford regularity of arbitrary subschemes of fat points. An optimal upper bound, named after Segre, was conjectured by Trung and, independently, by Fattabi and Lorenzini. It is shown that this conjecture is true. Furthermore, an alternate regularity bound is established that improves the Segre bound in some cases. Among the arguments is a new partition result for matroids.
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 order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it additional general points, the least degrees in which the ideals of the subschemes are nonzero. This point of view gives rise to the following conjecture: if C is any reduced irreducible curve on a surface X obtained by blowing up P^2 at any finite number of generic points, then C^2 > g-2, where g is the arithmetic genus of C. This conjecture is shown to be …
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 is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at $n\le8$ essentially distinct points of the projective plane. Each type gives rise to a surface $X$ obtained by blowing up the points. We classify those types such that $n=6$ and $-K_X$ is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configura…
In this paper we study the tangent space to the Hilbert scheme $\mathrm{Hilb}^d \mathbf{P}^3$, motivated by Haiman's work on $\mathrm{Hilb}^d \mathbf{P}^2$ and by a long-standing conjecture of Briançon and Iarrobino on the most singular point in $\mathrm{Hilb}^d \mathbf{P}^n$. For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Brianço…
Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a K3 surface M, the Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperkaehler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This implies that a generic deformation of the Hilbert scheme of K3 has no complex subvarieties.
For a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^n$ over a perfect field $K$, we first embed the homogeneous coordinate ring $R$ into its truncated integral closure $\widetilde{R}$. Then we use the corresponding map from the module of Kähler differentials $Ω^1_{R/K}$ to $Ω^1_{\widetilde{R}/K}$ to find a formula for the Hilbert polynomial ${\rm HP}(Ω^1_{R/K})$ and a sharp bound for the regularity index ${\rm ri}(Ω^1_{R/K})$. Additionally, we extend this to formulas for the Hilbert polynomials ${\rm HP}(Ω^m_{R/K})$ and bounds for the regularity indices of the higher modules of Kähler diffe…
Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers d…
