Jorge Neves

Vanishing Ideals Over Graphs and Even Cycles

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit combinatorial description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise vertex disjoint even cycles. In this case, a formula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.

A CONSTRUCTION OF NUMERICAL CAMPEDELLI SURFACES WITH TORSION Z/6

We produce a family of numerical Campedelli surfaces with Z/6 torsion by constructing the canonical ring of the etale 6 to 1 cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.

New examples of Calabi-Yau threefolds and genus zero surfaces

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

Parallel Kustin-Miller unprojection with an application to Calabi-Yau geometry

Kustin--Miller unprojection constructs more complicated Gorenstein rings from simpler ones. Geometrically, it inverts certain projections, and appears in the constructions of explicit birational geometry. However, it is often desirable to perform not only one but a series of unprojections. The main aim of the present paper is to develop a theory, which we call parallel Kustin--Miller unprojection, that applies when all the unprojection ideals of a series of unprojections correspond to ideals already present in the initial ring. As an application of the theory, we explicitly construct 7 families of Calabi--Yau 3-folds of high codimensions.

Codes over a weighted torus

We define the notion of weighted projective Reed-Muller codes over a subset X ź P ( w 1 , ź , w s ) of a weighted projective space over a finite field. We focus on the case when X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of the numerical semigroup Q = { w 1 , ź , w s } ź N . We compute the index of regularity of the vanishing ideal of X in terms of the weights of the projective space and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.