Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.     

A stratification of Hilbert schemes by initial ideals and applications

We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points. As application, we describe the singular locus of the component of Hilb^{4 z +1} _ℙ4 containing the ACM curves of degree 4. Received: 30 November 1998 / Revised version: 16 September 1999     

A cohomological characterization of Alexander schemes

Vistoli defined Alexander schemes in [19], which behave like smooth varieties from the viewpoint of intersection theory with Q-coefficients. In this paper, we will affirmatively answer Vistoli’s conjecture that Alexander property is Zariski local. The main tool is the abelian category of bivariant sheaves, and we will spend most of our time for proving basic properties of this category. We show that a scheme is Alexander if and only if all the first cohomology groups of bivariant sheaves vanish, which is an analogy of Serre’s theorem, which says that a scheme is affine if and only if all the first cohomology groups of quasi-coherent sheaves vanish. Serre’s theorem implies that the union of affine closed subschemes is again affine. Mimicking the proof line by line, we will prove that the union of Alexander open subschemes is again Alexander. Oblatum 1-XII-1997 & 14-XII-1998 / Published online: 10 May 1999     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Liaison addition and the structure of a Gorenstein liaison class

We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.     

The Gale transform and multi-graded determinantal schemes

Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.     

Clutching and gluing in tropical and logarithmic geometry

The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in the tropical world and show that the resulting maps naturally commute with the process of tropicalization.     

Linear systems on a class of anticanonical rational threefolds

Let X be the blow-up of the three dimensional complex projective space along r   points in very general position on a smooth elliptic quartic curve B⊂P³$B\subset {\mathbb{P}}^3$ and let L∈Pic(X)$L\in \mathrm{Pic}(X)$ be any line bundle. The aim of this paper is to provide an explicit algorithm for determining the dimension of H⁰(X,L)$H^0(X,L)$.     

Segre classes on smooth projective toric varieties

We provide a generalization of the algorithm of Eklund, Jost and Peterson for computing Segre classes of closed subschemes of projective $k$ -space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].     

Minimal subschemes of the group association schemes of Mathieu groups

Minimal subschemes of the group association schemes of Mathieu groupsM _n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM _n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of. The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.     

On the reducibility of the postulation Hilbert scheme

We give a condition in terms of the possible graded Betti numbers compatible with a given Hilbert functionH of 0-dimensional subschemes of ℙ ⁿ which implies the reducibility of the postulation Hilbert scheme and of its subscheme which parametrizes reduced subschemes with Hilbert functionH.     

Conics in the Grothendieck ring

The subring of the Grothendieck ring of k-varieties generated by smooth conics is described, giving many zero divisors. The proof uses only elementary projective geometry.     

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction. Received: 19 February 1998 / Revised version: 28 December 1998     

Laplace equations,Lefschetz properties and line arrangements

In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6]. The starting point was an example in [3]. Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3].     

Subschemes of Hamming association schemes H(n,q), q≥4

The subschemes of the Hamming schemes H(n, q) for q greater than or equal to 4 are studied. We prove that there are no nontrivial subschemes when q > 4 and there exists only one nontrivial subscheme when q = 4.     

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus. This is related to classical beautiful results of Castelnuovo and Enriques that we revise here in terms of adjunction theory.