Cohen-Macaulay Subscheme Research Articles

Derived categories of projectivizations and flops

We prove a generalization of Orlov's projectivization formula for the derived category Dcohb(P(E)), where E does not need to be a vector bundle; Instead, E is a coherent sheaf which locally admits two-step resolutions. As a special case, this also gives Orlov's generalized universal hyperplane section formula. As applications, (i) we obtain a blowup formula for blowup along codimension two Cohen-Macaulay subschemes, (ii) we obtain new “flop-flop=twist” results for a large class of flops obtained by crepant resolutions of degeneracy loci. As another consequence, this gives a perverse schober on C. (iii) we give applications of above results to symmetric powers of curves and Θ-flops, following Toda [79].

Glicci ideals

AbstractA central problem in liaison theory is to decide whether every arithmetically Cohen–Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can indeed be achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+ 1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen–Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.

Star configurations in [formula omitted

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 define arithmetically Cohen–Macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals. As an application, we compute the resurgence for the ideal of the codimension n−1 star configuration in Pn in the monomial case (i.e., when the number of hyperplanes is n+1).

Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces

A Pfaff field on \mathbb{P}^n_k is a map \eta \colon \Omega^s_{\mathbb{P}^n_k} \to \mathcal{L} from the sheaf of differential s -forms to an invertible sheaf. The interesting ones are those arising from a Pfaff system, as they give rise to a distribution away from their singular locus. A subscheme X \subseteq \mathbb{P}^n_k is said to be invariant under \eta if \eta induces a Pfaff field \Omega^s_X\to\mathcal{L} |_X . We give bounds for the Castelnuovo–Mumford regularity of invariant complete intersection subschemes (more generally, arithmetically Cohen–Macaulay subschemes) of dimension s , depending on how singular these schemes are, thus bounding the degrees of the hypersurfaces that cut them out.

Compactifications of rational maps, and the implicit equations of their images

In this paper, we give different compactifications for the domain and the codomain of an affine rational map f which parameterizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify A n − 1 into an ( n − 1 ) -dimensional projective arithmetically Cohen–Macaulay subscheme of some P N . One particular interesting compactification of A n − 1 is the toric variety associated to the Newton polytope of the polynomials defining f . We consider two different compactifications for the codomain of f : P n and ( P 1 ) n . In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established by Laurent Busé and Jean-Pierre Jouanolou (2003) [12], Laurent Busé et al. (2009) [9], Laurent Busé and Marc Dohm (2007) [11], Nicolás Botbol et al. (2009) [5] and Nicolás Botbol (2009) [4].

Codimension 3 Arithmetically Gorenstein Subschemes of projective N-space

Nous étudions le problème de savoir si tous les sous-schémas arithmétiquement de Cohen-Macaulay de ℙ N sont “glicci” dans le cas de plus petite dimension, c’est-à-dire le cas de sous-schémas de dimension zéro de ℙ 3 . Nous prouvons qu’il n’y a pas de liaisons ni de biliaisons de Gorenstein descendantes d’un ensemble d’au moins 56 points généraux de ℙ 3 . Pour démontrer ce théorème, nous établissons plusieurs résultats concernant les sous-schémas arithmétiquement de Gorenstein de ℙ 3 .

Gorenstein liaison and ACM sheaves

We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of ℙn is in the Gorenstein liaison class of a complete intersection and we are able to characterize the Gorenstein liaison classes of curves on a nonsingular quadric threefold in ℙ4.

Gorenstein liaison of 0-dimensional schemes

The theory of Gorenstein liaison has been developed during the last 3 years to generalize liaison theory of codimension 2 schemes to schemes of codimension ≥ 3 in a projective space. One of the main open questions in Gorenstein liaison theory is whether any arithmetically Cohen-Macaulay subscheme of ℙn is in the Gorenstein liaison class of a complete intersection. In this paper we prove that any set of general points lying on a rational normal scroll surface is in the Gorenstein liaison class of a complete intersection.

Gorenstein liaison of divisors on standard determinantal schemes and on rational normal scrolls

Let C⊂ P n be an arithmetically Cohen–Macaulay subscheme. In terms of Gorenstein liaison it is natural to ask whether C is in the Gorenstein liaison class of a complete intersection. In this paper, we study the Gorenstein liaison classes of arithmetically Cohen–Macaulay divisors on standard determinantal schemes and on rational normal scrolls. As main results, we obtain that if C is an arithmetically Cohen–Macaulay divisor on a “general” arithmetically Cohen–Macaulay surface in P 4 or on a rational normal scroll surface S⊂ P n , then C is glicci (i.e. it belongs to the Gorenstein liaison class of a complete intersection).

Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetically Cohen-Macaulay. The module version of this result implies splitting criteria for reflexive sheaves.

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.

Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function

We give geometric constructions of families of graded Gorenstein Artin algebras, some of which span a component of the space Gor(T ) parametrizing Gorenstein Artin algebras with a given Hilbert function T . This gives a lot of examples where Gor(T ) is reducible. We also show that the Hilbert function of a codimension four Gorenstein Artin algebra can have an arbitrarily long constant part without having the weak Lefschetz property. In R = k[x;y] and in R = k[x;y;z], the parameter space Gor(T ) of graded Gorenstein Artin quotients of a given Hilbert function T is irreducible for all T . The codimension two case is implied by F. S. Macaulay’s result that Gorenstein height two ideals ink[x;y] are complete intersections [8], and the fact that the family of graded complete intersection ideals of given generator degrees is irreducible. The codimension three case was proved S. Diesel [4]. A. Iarrobino and V. Kanev [7] have recently shown that in codimension ve and higher, Gor(T ) can be reducible. In this paper, we will give a large class of examples where Gor(T ) is reducible, in codimension four and higher. A. Iarrobino and V. Kanev also proved that there is a close relation between graded Gorenstein Artin algebras of codimension three and nite length Cohen-Macaulay sub-schemes of P 2 . They show that whenever the Hilbert function T is equal to s for at least three degrees, there is a bration Gor(T )! H(T ) Hilb s (P 2 ) which takes the form f to the initial part of the ideal annR(f). We prove that there can be no such results in higher codimensions, by showing that for r 4, there are codimension r Gorenstein Artin algebras with arbitrary long constant part in the Hilbert function, whose initial ideal does not dene a sub-scheme of length s in P r 1 . The

Degrees of generators of ideals defining curves in projective space

For an arithmetically Cohen–Macaulay subscheme of projective space, there is a well-known bound for the highest degree of a minimal generator for the defining ideal of the subscheme, in terms of the Hilbert function. We prove a natural extension of this bound for arbitrary locally Cohen–Macaulay subschemes. We then specialize to curves in P 3, and show that the curves whose defining ideals have generators of maximal degree satisfy an interesting cobomological property. The even liaison classes possessing such curves are characterized, and we show that within an even liaison class, all curves with the property satisfy a strong Lazarsfeld–Rao structure theorem. This allows us to give relatively complete conditions for when a liaison class contains curves whose ideals have maximal degree generators, and where within the liaison class they occur.