Castelnuovo Regularity Research Articles

Graded mapping cone theorem, multisecants and syzygies

Let X be a reduced closed subscheme in P n . As a slight generalization of property N p due to Green–Lazarsfeld, one says that X satisfies property N 2 , p scheme-theoretically if there is an ideal I generating the ideal sheaf I X / P n such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. Eisenbud et al. (2005) [8], Vermeire (2001) [20]). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N 2 , p (cf. Choi, Kwak, and Park (2008) [6], Eisenbud et al. (2005) [8], Kwak and Park (2005) [15]). In this case, the Castelnuovo regularity and normality can be obtained by the blowing-up method as reg ( X ) ⩽ e + 1 where e is the codimension of a smooth variety X (cf. Bertram, Ein, and Lazarsfeld (2003) [3]). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry (cf. Beheshti and Eisenbud (2010) [2], Kwak (1998) [14], Kwak and Park (2005) [15], Lazarsfeld (1987) [16]. We first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the non-complete embedding of X. For applications, we give an optimal bound on the length of zero-dimensional intersections of X and a linear space L in terms of graded Betti numbers. We also deduce several theorems about the relationship between X and its projections with respect to the geometry and syzygies for a projective scheme X satisfying property N 2 , p scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers of the projection for the case of N d , p , d ⩾ 2 , but also geometric structures for projections according to moving the center.

Supporting degrees of multi-graded local cohomology modules

For a finitely generated graded module M over a positively-graded commutative Noetherian ring R, the second author established in 1999 some restrictions, which can be formulated in terms of the Castelnuovo regularity of M or the so-called a ∗ -invariant of M, on the supporting degrees of a graded-indecomposable graded-injective direct summand, with associated prime ideal containing the irrelevant ideal of R, of any term in the minimal graded-injective resolution of M. Earlier, in 1995, T. Marley had established connections between finitely graded local cohomology modules of M and local behaviour of M across Proj ( R ) . The purpose of this paper is to present some multi-graded analogues of the above-mentioned work.

The bipartite Brill-Gordan locus and angular momentum

Given integers n,d,e with $1 \leqslant e < \frac{d}{2},$ let $X \subseteq {\Bbb P}^{\binom{d+n}{d}-1}$ denote the locus of degree d hypersurfaces in ${\Bbb P}^n$ which are supported on two hyperplanes with multiplicities d-e and e. Thus X is the Brill-Gordan locus associated to the partition (d-e,e). The main result of this paper is an exact determination of the Castelnuovo regularity of the ideal of X. Moreover, we show that X is r-normal for $r \geqslant 3.$ In the case of binary forms (i.e., for n = 1) we give an invariant-theoretic description of the ideal generators and, furthermore, exhibit a set of two covariants which define this locus set-theoretically. In addition to the standard cohomological tools in algebraic geometry, the proof crucially relies on the nonvanishing of certain 3j-symbols from the quantum theory of angular momentum.

Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space P n correspond to points of P N , where N = ( n + d d ) − 1 . Now assume d = 2 e is even, and let X ( n , d ) ⊆ P N denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X ( n , d ) , and show that this variety is r-normal for r ⩾ 2 . The latter result is representation-theoretic, and says that a certain GL n + 1 -equivariant morphism S r ( S 2 e ( C n + 1 ) ) → S 2 ( S r e ( C n + 1 ) ) is surjective for r ⩾ 2 ; a statement which is reminiscent of the Foulkes–Howe conjecture. For its proof, we reduce the statement to the case n = 1 , and then show that certain transvectants of binary forms are nonzero. The latter part uses explicit calculations with Feynman diagrams and hypergeometric series. For ternary quartics and binary d-ics, we give explicit generators for the defining ideal of X ( n , d ) expressed in the language of classical invariant theory.

The depth of the associated graded ring of ideals with any reduction number

Let R be a local Cohen–Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen–Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r+ℓ powers of I, where r denotes the Castelnuovo regularity of G and ℓ denotes the analytic spread of I.

Remarks on the Defining Equations of Smooth Threefolds in P 5

In this paper, we prove the vanishing of the first cohomology groups of ideal sheaves twisted by small integers for smooth threefolds in P5 defined over the complex number field C by using Le Potier vanishing theorem and Castelnuovo regularity for vector bundles, which contributes Peskine–Van de Ven conjecture for 2-normality.

A generalization of Castelnuovo regularity to Grassmann varieties

Using the Beilinson–Kapranov spectral sequence, we define Castelnuovo regularity for a coherent sheaf on a Grassmann variety. We show that many formal properties of regularity over projective spaces continue to hold in this generality and derive concrete bounds on the degrees of defining equations for subvarieties of the Grassmannian.

Generic projections, the equations defining projective varieties and Castelnuovo regularity

For a reduced, irreducible projective variety X of degree d and codimension e in \(\mathbb{P}^N\) the Castelnuovo-Mumford regularity \({\rm reg} X\) is defined as the least k such that X is k-regular, i.e., \(H^i(\mathbb{P}^N,\mathcal{I}_X(k-i))= 0\) for \(i\ge1\), where \(\mathcal {I}_X\subset\Cal O_{\mathbb{P}^N}\) is the sheaf of ideals of X. There is a long standing conjecture about k-regularity (see [5]): \({\rm reg}{X}\le d-e+1\). Here we show that \({\rm reg}{X}\le(d-e+1)+10\) for any smooth fivefold and \({\rm reg}{X}\le(d-e+1)+20\) 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 \(\rho_a\) (see Theorem 4.1).

The Castelnuovo regularity of the Rees algebra and the associated graded ring

The Castelnuovo regularity of the Rees algebra and the associated graded ring

Computing minimal finite free resolutions

In this paper we address the basic problem of computing minimal finite free resolutions of homogeneous submodules of graded free modules over polynomial rings. We develop a strategy, which keeps the resolution minimal at every step. Among the relevant benefits is a marked saving of time, as the first reported experiments in CoCoA show. The algorithm has been optimized using a variety of techniques, such as minimizing the number of critical pairs and employing an “ad hoc” Hilbert-driven strategy. The algorithm can also take advantage of various a priori pieces of information, such as the knowledge of the Castelnuovo regularity.

Castelnuovo Regularity and Graded Rings Associated to an Ideal

Castelnuovo Regularity and Graded Rings Associated to an Ideal

Castelnuovo regularity and graded rings associated to an ideal

We compare the Castelnuovo regularity defined with respect to different homogeneous ideals in a graded ring and use the result we obtain to prove a generalized Goto-Shimoda theorem for ideals of positive height in a Cohen-Macaulay local ring.

Bounds for the cohomology and the Castelnuovo regularity of certain surfaces

Let X ⊆ Pr be a reduced, irreducible and non-degenerate projective variety over an algebraically closed field K of characteristic 0. Let reg(x) be the Castelnuovo-Mumford regularity of the sheaf of ideals associated to X.Then it is an open problem—due to D. Eisenbud (see e.g. [E-Go])—whether(0.1) reg(X) ≤ deg(x) - codim (x) + 1,where deg(x) denotes the degree of X and codim(x) denotes the codimension of X. In many cases, this inequality has been proven to hold true.

Bounds for Castelnuovo's regularity and the genus of projective varieties

Bounds for Castelnuovo's regularity and the genus of projective varieties

Castelnuovo's regularity and cohomological properties of sets of points in ℙ n

Castelnuovo's regularity and cohomological properties of sets of points in ℙ n

On bounds for Castelnuovo’s index of regularity

For coherent sheaves on a projective space Castelnuovo's index of regularity was first defined by D. Mumford [11], who attributes the idea to G. Castelnuovo. In fact, he used it to show that certain twists of a coherent sheaf are generated by its g lo b a l sec tio n s . In a m o re algebraic setting Castelnuovo's regularity was defined by D. Eisenbud and S. Goto [2] and A. Ooishi [12], see (2.2). I t c o m e s out that Castelnuovo's index of regularity gives an upper bound for the maximal degree of the syzygies in a m inim al free resolution, see (2.8) fo r th e precise statem ent. This fact is very im portant for the complexity o f a program for a numerical computation of syzygies. As shown by examples of D . Bayer and M. Stillm an, see [1], Castelnuovo's regularity can be o f exponential growth with respect to the K rull dim ension. While these examples have rather wild singularities, one might hope to get better bounds in the case of tame singularities. This point of view is pursued further in the papers [2], [10], [12], [16], [17], [18]. O n e o f o u r m a in results, see (3.5), supplies an upper bound for reg R, Castelnuovo's regularity, o f a graded k-algebra R th a t is Cohen-Macaulay or locally Cohen-Macaulay and unm ixed . I n a geometric context, w e get a satisfactory bound in the case of a Cohen-Macaulay projective v a rie ty . In particular, with (3.5) a) we solve a problem posed in [10] in the affirmative. The same result was shown independently a n d b y a different argum ent b y J . Stuckrad and W. Vogel in [18]. In the case of a Cohen-Macaulay ring R it follows, see [2], that

CASTELNUOVO‐Regularität und HILBERTreihen

AbstractLet A be an equidimensional locally Cohen‐Macaulay graded k‐algebra. Following [16] we have upper bounds for Castelnuovo's regularity of A (see [16], main theorem und theorem 1.). The aim of this paper is to characterize the equalities of these bounds and Castelnuovo's regularity by using the theory of Hilbert series. For this reason the inequalities of [16] are new proved by applying our methods. Moreover, we improve and extend main results and investigations of [11], [7], [18], [14], [16].For monomial ideals the conjecture of D. BAYER and M. STILLMAN [4] is proved.

Castelnuovo's regularity and multiplicity

Castelnuovo's regularity and multiplicity

Castelnuovo's regularity of graded rings and modules

Castelnuovo's regularity of graded rings and modules