Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ?³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ℙ³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

A family of cohen-macaulay modules over singularities of type xt + y3

Let K be a field with char K ≠ 3 and it two positive integers such that 1 ≤i <t/2,t ≠ 3i. The classification problem for maximal Cohen-Macaulay modules over K[[X,Y]]/(X^t+Y³ ) is complicated if t≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modules N, such that N/YN is a direct sum of copies of K[[X]]/(X ⁱ)K[[X]]/(X^t-i ).     

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     

Arithmetically Buchsbaum divisors on varieties of minimal degree

In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.

     

Seminormal Varieties,Torsionfree Sheaves,and Picard groups

We give a construction of torsionfree sheaves on a seminormal variety Y using torsionfree sheaves on the normalization X and the non-normal locus W. We use it to find a relation between Picard groups of X, Y, and W. We apply it to determine the Picard groups of the generalized Jacobian, the compactified Jacobian and some subschemes associated to the moduli spaces of torsionfree sheaves of rank 2 and odd degree on a nodal curve.     

Generic projections, the equations defining projective varieties and Castelnuovo regularity

For a reduced, irreducible projective variety X of degree d and codimension e in the Castelnuovo-Mumford regularity is defined as the least k such that X is k-regular, i.e., for , where is the sheaf of ideals of X. There is a long standing conjecture about k-regularity (see [5]): . Here we show that for any smooth fivefold and for any smooth sixfold by extending methods used in [10]. Furthermore, we give a bound for the regularity of a reduced, connected and equidimensional locally Cohen-Macaulay curve or surface in terms of degree d, codimension e and an arithmetic genus (see Theorem 4.1). Received November 12, 1998; in final form May 4, 1999     

The seminormality property of circular complexes

In this paper we prove that the ring R[X, Y]/(X. Y, Y.X) is seminormal, whereR is a Cohen-Macaulay normal domain andX, Y are matrices of indeterminates.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

Reduced class groups grafting relative invariants

Let (X,T) be a regular stable conical action of an algebraic torus on an affine normal conical variety X defined over an algebraically closed field of characteristic zero. We define a certain subgroup of Cl(X//T) and characterize its finiteness in terms of a finite T-equivariant Galois descent of X. Consequently we show that the action (X,T) is equidimensional if and only if there exists a T-equivariant finite Galois covering such that is cofree. Moreover the order of is controlled by a certain subgroup of Cl(X). The present result extends thoroughly the equivalence of equidimensionality and cofreeness of (X,T) for a factorial X. The purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus actions as above. The generalization of R.P. Stanley?s criterion for freeness of modules of relative invariants plays an important role in showing key assertions.     

Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.     

The canonical sheaf of Du Bois singularities

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π_∗ωX^′(G)?ωX for a log resolution π:X^′→X, where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.     

Extensions of the Multiplicity conjecture

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded -algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-space, and propose an upper bound for finitely generated graded torsion modules. We establish this bound for torsion modules whose codimension is at most two.

     

Metric Projections and Polyhedral Spaces

We prove that if X is a Banach space and Y is a proximinal subspace of finite codimension in X such that the finite dimensional annihilator of Y is polyhedral, then the metric projection from X onto Y is lower Hausdorff semi continuous. In particular this implies that if X and Y are as above, with the unit sphere of the annihilator space of Y contained in the set of quasi-polyhedral points of X ^*, then the metric projection onto Y is Hausdorff metric continuous. Partially supported under project DST/INT/US-NSF/RPO/141/2003.     

The lazarsfeld-rao property on an arithmetically gorenstein variety

We show how to use all the machinery of liaison techniques for the study of two-codimensional subschemes of a smooth arithmetically Gorenstein subscheme ofP ⁿ.     

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety Y is flexible if its special group SAut(Y) of automorphisms (generated by the elements of one-dimensional unipotent subgroups of Aut(Y)) acts transitively on Y, and an algebraic variety is stably flexible if its product with some affine space is flexible. An irreducible algebraic variety X is locally stably flexible if it is a union of a finite number of Zariski open sets each of which is stably flexible. The main result of this paper states that the blowup of a locally stably flexible variety along a smooth algebraic subvariety (not necessarily equidimensional or connected) is subelliptic, and, therefore, it is an Oka manifold.     

Isomorphisms of algebras of generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the algebras of generalized functions (in the sense of Colombeau) on X, resp. Y is given by composition with a unique generalized function from Y to X. We also characterize the multiplicative linear functionals from the Colombeau algebra on X to the ring of generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.     

Schlicht Envelopes of Holomorphy and Foliations by Lines

Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation ℱ of X with no compact leaves. We call a domain Y in X an interval domain with respect to ℱ if Y intersects every leaf of ℱ in a nonempty connected set. We show that if X is Stein and if ℱ satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness.      

Distributive extensions of modules

Let X be a submodule of a module M. The extension is said to be distributive if X ∩ (Y + Z) = X ∩ Y + X ∩ Z for any two submodules Y and Z of M. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) is a distributive extension; (2) for any submodule Y of the module M, no simple subfactor of the module X/(X∩Y ) is isomorphic to any simple subfactor of Y/(X∩Y) (3) for any two elements x ∈ X and m ∈ M, there does not exist a simple factor module of the cyclic module xA/(X ∩ mA) that is isomorphic to a simple factor module of the cyclic module mA/(X ∩ mA). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 141–150, 2006.