Rafael H. Villarreal

Cohen-macaulay graphs

For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.

Rees algebras of edge ideals

Let G be a graph and let I be its edge ideal. We express the presentation ideal of R(I), the Rees algebra of I, in terms of the syzygies of I and the presentation ideal of the special fiber of R(I). A description of the elementary integral vectors of the kernel of the incidence matrix of G is given and then used to study the special fiber of R(I) via Grobner bases.

Edge ideals: algebraic and combinatorial properties

Boij–Söderberg theory describes the Betti diagrams of graded modules over the polynomial ring, up to multiplication by a rational number. Analog Eisenbud–Schreyer theory describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas.

Associated primes of powers of edge ideals

Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of Ii and

Blowup Algebras of Square-Free Monomial Ideals and Some Links to Combinatorial Optimization Problems

{\overline{I^i}}

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

stabilize for large i. We show that their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.

Relative volumes and minors in monomial subrings

Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.

Affine cartesian codes

Let I = (x v 1 , . . . , x v q ) be a square-free monomial ideal of a polynomial ring K[x1, . . . , xn] over an arbitrary field K and let A be the incidence matrix with column vectors v1, . . . , vq. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedra and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and