Resolutions of ideals of fat points with support in a hyperplane
Giuliana Fatabbi, Brian Harbourne, Anna Lorenzini
备注：
10 pages; to appear in Proc. Amer. Math. Soc.; some expositional changes; added a reference to paper of Geramita, Migliore and Sabourin (math.AC/0411445)
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 P_{m,d} of I(m,d) in terms of the Poincare polynomials P_{j,d-1} of I(j,d-1): P_{m,d} = (1 + XT)(Σ_{0<j\le m} T^{m-j}(P_{j,d-1} - 1)) + 1 + XT^m.
We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).
We compute the multiplier ideals of hyperplane arrangements via the interpretation of these ideals in terms of spaces of arcs, due to Ein, Lazarsfeld and the author.
The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.
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.
In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the mu…
M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give an alternate proof using a log resolution, which is simpler and allows us to consider non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini--Procesi, we also simplify the result.
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 the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projec…
Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.
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.
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 introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's conjecture says that freeness of $\mathcal A$ is determined by the combinatorics of the intersection lattice of $\mathcal A$. We study the Cohen-Macaulayness of three other ideals, all unmixed, that are closely related to $\mathcal A$. Let $\overline J = \mathfrak q_1 \cap \dots \cap \mathfrak q_s$ be the intersection of height two primary com…
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…
Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\bf P}^N$ for all $r>0$, but in the last year various counterexam…
Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f:=l_1\cdots l_m\in R$. Let $J_f$ denote the ideal generated by the partial derivatives of $f$ and let $\mathbb{I}$ designate the ideal generated by the $(m-1)$-fold products of $l_1,\ldots, l_m$. This paper is centered on the relationship between the two ideals $J_f, \mathbb{I}\subset R$,…
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.
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.
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…
Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal of Z. In this paper, we introduce the concept of a very unexpected hypersurface. This is a stronger condition which takes into account an explanation for some hypersurfaces previously considered unexpected. We then develop a duality theory to relate the study of very unexpected hypersurfaces to the derivations of dual hyperplane arrangements. This allows us …
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 $P^3$.
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…
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.
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…
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.
We prove a conjecture of Teissier asserting that if $f$ has an isolated singularity at $P$ and $H$ is a smooth hypersurface through $P$, then $\widetildeα_P(f)\geq \widetildeα_P(f\vert_H)+\frac{1}{θ_P(f)+1}$, where $\widetildeα_P(f)$ and $\widetildeα_P(f\vert_H)$ are the minimal exponents at $P$ of $f$ and $f\vert_H$, respectively, and $θ_P(f)$ is an invariant obtained by comparing the integral closures of the powers of the Jacobian ideal of $f$ and of the ideal defining $P$. The proof builds on the approaches of Loeser and Elduque-Mustata. The new ingredients are a result concerning the behav…
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…
The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
For a scheme of fat points $Z$ defined by the saturated ideal $\mathcal{I}_Z$, the regularity index computes the Castelnuovo-Mumford regularity of the Cohen-Macaulay ring $R/\mathcal{I}_Z.$ For points in " general position" we improve the bound for the regularity index computed by Segre for $\mathbb {P}^2$ and generalised by Catalisano, Trung and Valla for $\mathbb {P}^n$. Moreover, we prove that the generalised Segre's bound conjectured by Fatabbi and Lorenzini holds for $n+3$ arbitrary points in $\mathbb {P}^n$. We propose a modification of Segre's conjecture for arbitrary points and we disc…
We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linea…
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…
We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog [2000] on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.
Let $ V $ a vector space of dimension $n$. A $V$ family $ \{H_1, \ldots, H_p \} $ of vectorial hyperplanes being distinct two by two defines an arrangement $ {\cal A}_p = {\cal A} ( H_1, \ldots ,H_p ) $ of $ V $. For $ i \in \{ 1, \ldots, p \} $, let $ l_i $ be a linear form on $V$ with $H_i$ as kernel. This arrangement is generic if the intersection of every sub-family of $n$ hyperplanes of the arranfement is reduced to zero. Let $A_V ({\bf C}) $, be the Weyl algebra of algebraic differential operators with coefficients in the symetric algebra denoted $S$ of the dual of $V$. Following J. Bern…
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}.$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (…
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated binomial ideal, which defines a toric subvariety in a product of projective lines. Connections to graphic arrangements and to Beilinson's spectral sequence are explored.
We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…
Given an arrangement of hyperplanes in $¶^n$, possibly with non-normal crossings, we give a vanishing lemma for the cohomology of the sheaf of $q$-forms with logarithmic poles along our arrangement. We give a basis for the ideal $\cal J$ of relations for the Orlik-Solomon's algebra. Under certain genericity conditions it was shown by H.~Esnault, V.~Schechtman and E.~Viehweg that the cohomology of a local system is given by the Aomoto complex. We generalize this result to a deformation of local systems obtained via a deformation of our arrangement. We calculate the Gauß-Manin connection for thi…
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…
The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of…
Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\bf P}^N$. In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.
Let $X$ be an integral projective variety of codimension two, degree $d$ and dimension $r$ and $Y$ be its general hyperplane section. The problem of lifting generators of minimal degree $σ$ from the homogeneous ideal of $Y$ to the homogeneous ideal of $X$ is studied. A conjecture is given in terms of $d$, $r$ and $σ$; it is proved in the cases $r=1,2,3$. A description is given of linear systems on smooth plane curves whose dimension is almost maximal.
We study the footprint function, with respect to a monomial order, of complete intersection graded ideals in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we give a formula for the footprint function and a sharp lower bound for the corresponding minimum distance function. This allows us to recover a formula for the minimum distance of an affine cartesian code and the fact that in this case the minimum distance and the footprint functions coincide. Then we present an extension of a result of Alon and Füredi, a…
