Marcel Morales

Set-theoretic complete intersections on binomials

Let V be an affine toric variety of codimension r over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, V is a set-theoretic complete intersection on binomials if and only if V is a complete intersection. Moreover, if F 1 ,..., F r are binomials such that I(V) = rad(F 1 ,...,F r ), then I(V) = (F 1 ,...,F r ). While in the positive characteristic p case, V is a set-theoretic complete intersection on binomials if and only if V is completely p-glued. These results improve and complete all known results on these topics.

Noetherian Symbolic Blow-Ups

The following question was raised by Cowsik [Cw]: if P is a prime ideal in a regular local ring (R, n?), is the symbolic Rees ring Rep) := @,120 PC”’ a noetherian ring, where PC”’ is . the n-symbolic power of P (i.e., P’“‘= P”R, n R). This question appears in works of Rees [Re] and Nagata

On the length of generalized fractions

Abstract Let M be a finitely generated module over a Noetherian local ring (R, m ) with dimM=d. Let (x1,…,xd) be a system of parameters of M and (n1,…,nd) a set of positive integers. Consider the length of generalized fraction 1/(x1n1,…,xdnd,1) as a function in n1,…,nd. Sharp and Hamieh [J. Pure Appl. Algebra 38 (1985) 323–336] asked whether this function is a polynomial for n1,…,nd large enough. In this paper, we will give counterexamples to this question. We also study conditions on the system of parameters x , in order to show that the length of the generalized fraction 1/(x1n1,…,xdnd,1) is not a polynomial for n1,…,nd large enough.

Fiber Cone of Codimension 2 Lattice Ideals

Let I ⊂ ℛ: = 𝒦[x 1, x 2, …, x r ] be a codimension two lattice ideal. In this article we study the arithmetic properties of the blow-up of the ideal I in ℛ. Let ℱ(I) = ⨁ n≥0 I n /𝔪 I n be the Fiber cone of I, we prove that In addition, if 𝒦 is infinite and I is radical, noncomplete intersection, then: ℱ(I) has dimension 3, is reduced, arithmetically Cohen–Macaulay, of minimal degree. Moreover, a presentation of ℱ(I) is effective from the minimal system of generators of I. An explicit minimal reduction of ℱ(I) is given. The blow-up ring, or Rees ring ℛ(I) = ⨁ n≥0 I n , is arithmetically Cohen–Macaulay and has a presentation by linear and quadratic forms. This article completes and extends to the general case of codimension 2 lattice ideals previous results for the simplicial toric case by Morales and Simis (1992), Gimenez et al. (1999), and Barile and Morales (1998).

Monomial ideals with 3-linear resolutions

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these operations. We apply the operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper sub-clutter of it has a linear resolution.

The regularity of edge ideals of graphs

Abstract In this paper, we introduce some reduction processes on graphs which preserve the regularity of related edge ideals. Using these, an alternative proof for the theorem of R. Froberg on the linearity of the resolutions of edge ideals is given.

Generalized F-Modules and the Associated Primes of Local Cohomology Modules

ABSTRACT We introduce a class of modules called generalized f-modules, which contains strictly all f-modules and generalized Cohen–Macaulay modules. The properties of multiplicity, local cohomology modules, localization, completion… of these modules are presented. A result concerning the finiteness of associated primes of local cohomology modules with respect to generalized f-modules is given. Some connections to the coordinate rings of algebraic varieties and Stanley-Reisner rings are considered.

The analytic spread of the ideal of a monomial curve in projective 3-space

Let k be an algebraically closed field and let C ⊂ An stand for a quasi homogeneous surface admitting a rational parametrization given by monomials. Consider the blowing-up A n of A n with center C and look at the fibre of the structural morphism

Castelnuovo–Mumford regularity of projective monomial varieties of codimension two

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