Multigraded regularity of complete intersections

Journal of Pure and Applied Algebra, 2001

To an arbitrary ideal I in a local ring (A; m) one can associate a multiplicity j(I; A) that generalizes the classical Hilbert-Samuel multiplicity of an m-primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen-Macaulay in A and satisÿes a suitable Artin-Nagata condition then our main result states that j(I; M ) is given by the length of I=(x1; : : : ; x d-1 ) + x d I where d:=dim A and x1; : : : ; x d are su ciently generic elements of I . This generalizes the classical length formula for m-primary ideals in Cohen-Macaulay rings. Applying this to an hypersurface H in the a ne space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle H c+1 with multiplicity e( jac H; C ; OH;C ), where jac H is the Jacobian ideal generated by the partial derivatives of a deÿning equation of H .

Journal of Pure and Applied Algebra, 1985

Geometriae Dedicata, 2019

Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d = (d 1 ,. .. , ds). It is known that, if t := s i=1 d i +k k − (k + 1)(m − k) 0 and Π s i=1 d i > 2, for Y a general complete intersection as above, then F k (Y) has dimension −t. In this paper we consider the case t > 0. Then the locus W d,k of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general [Y ] ∈ W d,k the scheme F k (Y) is zero-dimensional of length one. This implies that W d,k is rational.

Mathematische Zeitschrift, 2013

We address the conjecture of [Durfee1978], bounding the singularity genus pg by a multiple of the Milnor number µ for an n-dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely (n+1)!·pg ≤ µ, fails whenever the codimension r is greater than one. Moreover, we proposed a new inequality Cn,r · pg ≤ µ, and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a 'combinatorial inequality', that might have an independent interest.

Proceedings of the American Mathematical Society, 1976

The concept of a generalized complete intersection (GCI) of affine schemes is introduced. The proofs of the following theorems are then

2004

Compositio Mathematica - COMPOS MATH, 2002

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 arith...

Journal of Algebra, 2018

Given an ideal I = (f1, . . . , fr) in C[x1, . . . , xn] generated by forms of degree d, and an integer k > 1, how large can the ideal I k be, i.e., how small can the Hilbert function of C[x1, . . . , xn]/I k be? If r ≤ n the smallest Hilbert function is achieved by any complete intersection, but for r > n, the question is in general very hard to answer. We study the problem for r = n + 1, where the result is known for k = 1. We also study a closely related problem, the Weak Lefschetz property, for S/I k , where I is the ideal generated by the d'th powers of the variables.

Transactions of the American Mathematical Society

We compute the intersection number between two cycles A and B of complementary dimensions in the Hilbert scheme H parameterizing subschemes of given finite length n of a smooth projective surface S. The (n + 1)-cycle A corresponds 40 the set of finite closed subschemes the support of which has cardinality 1. The (n - 1)-cycle B consists of the closed subschemes the support of which is one given point of the surface. Since B is contained in A, indirect methods are needed. The intersection number is A.B = (-1)(n-1)n, answering a question by H. Nakajima.