The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A, m) in terms of local cohomology modules of Rees algebras ...

This paper considers the following conjecture: If R is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal J generated by a system of parameters, the Chern coefficient e1(J) < 0 is equivalent to R being non Cohen-Macaulay. The conjecture is established if R is a homomorphic image of a Gorenstein ring, and for all universally cate...

In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen-Macaulay curves in P. We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen-Macaulay curves Ct and Ct−r in P 3 defined by the maximal minors of a t× (t+ 1), resp. (t− r)× (t − r + 1), matrix with linear entries...

A canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over C is obtained by taking ultraproducts of absolute integral closures in positive characteristic. Among the applications are a new tight closure characterization of rational singularities in characteristic zero, and a necessary condition for Q-Gorenstein logterminal singularities. In particular, it ...

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring (R, m). For a Noetherian (good) filtration A of m-primary ideals, the positivity and bounds for e1(A) are well-studied if R is Cohen-Macaulay, or more broadly, if R is a Buchsbaum ring or mild generalizations thereof. For arbitrary geomet...

We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (Q-)factorial, or (Q-)Gorenstein singularities. We also give...

In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it.

In this expository paper we survey results proved during the last fifty years that relate Hilbert coefficients e0(I) and e1(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using...

The Hilbert functions of sets of distinct points in Pn have been characterized. We show that if we restrict to sets of distinct of points in Pn1 ×· · ·×Pnk that are also arithmetically Cohen-Macaulay (ACM for short), then there is a natural generalization of this result. We begin by determining the possible values for the invariants K-dimR/IX and depthR/IX, where R/IX is the coordinate ring ass...

This paper explores the interplay between the Frobenius functor and Serre’s vanishing conjecture over a Noetherian local ring R of prime characteristic p. We show that the Frobenius functor induces a diagonalizable map on certain Q-vector spaces, which are tensor products of Q with quotients of Grothendieck groups. This allows us to decompose an element (representing a bounded complex of finite...