Emanuela De Negri

M-Sequences, Graph Ideals, and Ladder Ideals of Linear Type☆

In this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow-up algebras. Our main tools are Grobner bases and Sagbi bases deformations and the notion of M-sequence of monomials. w x Let R s K X be a polynomial ring over a field K equipped with a monomial order t . Let I be an ideal of R generated by polynomials f , . . . , f . Consider the presentation 1 s w x w x w x c : R T s R T , . . . , T a R I s R f t , . . . , f t Ž . 1 s i s

Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions

Abstract In this paper we study some problems concerning bigraded ideals. By introducing the concept of bigeneric initial ideal, we answer an open question about diagonal subalgebras and we give a necessary condition for a function to be the bigraded Hilbert function of a bigraded algebra. Moreover, we give an upper bound for the regularity of a bistable ideal in terms of the degrees of its generators.

Pfaffian ideals of ladders

Abstract In this paper we study rings defined by ideals of pfaffians of a ladder. We prove that they are Cohen-Macaulay normal domains and we determine those which are Gorenstein. Further we compute the a -invariant in terms of the shape of the ladder.

On the rate of points in projective spaces

The rate of a standard gradedK-algebraR is a measure of the growth of the shifts in a minimal free resolution ofK as anR-module. It is known that rate(R)=1 if and only ifR is Koszul and that rate(R) ≥m(I)−1 wherem(I) denotes the highest degree of a generator of the defining idealI ofR. We show that the rate of the coordinate ring of certain sets of pointsX of the projective space Pn is equal tom(I)−1. This extends a theorem of Kempf. We study also the rate of algebras defined by a space of forms of some fixed degreed and of small codimension.

Divisor class group and canonical class of rings defined by ideals of pfaffians

In this paper we study the rings defined by ideals of pfaffians of a skew symmetric matrix of indeterminates. We analyze the case in which the pfaffians are not necessarily of fixed size. We prove that such rings are Cohen-Macaulay normal domains and we compute the divisor class group and the canonical class. It allows us to determine which of our rings are Gorenstein.

Multigraded Generic Initial Ideals of Determinantal Ideals

Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In previous work we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.

ON THE COORDINATE RING OF PAIRS OF ALTERNATING MATRICES WITH PRODUCT ZERO

Given an integer n ≥ 2, let X = (xi j ) and Y = (yi j ) be two generic alternating n × n matrices over a commutative ring k. Denote by k[X, Y ] the polynomial ring with indeterminates the entries of X and Y . Moreover denote by I1(XY ) the ideal generated by the entries of the product of X and Y . The ring k[X, Y ]/I1(XY ) is the coordinate ring of the variety of pairs (U, V ) of alternating n × n matrices with entries in k, such that U V = 0. In this note we give a k-basis of that coordinate ring, under the assumption that n! is a unit in k. We use some ideas of De Concini, Procesi and Strickland (1–4), ideas which have been also applied in (5–7), papers beneficial to us.