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 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 disjoint even cycles. In this case, a fomula 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

Regularity and algebraic properties of certain lattice ideals

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. Then, we give some applications. For finite fields, we give a formula for the regularity of the vanishing ideal of a degenerate torus in terms of the Frobenius number of a semigroup. We construct vanishing ideals, over finite fields, with prescribed regularity and degree of a certain type. Let X be a subset of a projective space over a field K. It is shown that the vanishing ideal of X is a lattice ideal of dimension 1 if and only if X is a finite subgroup of a projective torus. For finite fields, it is shown that X is a subgroup of a projective torus if and only if X is parameterized by monomials. We express the regularity of the vanishing ideal over a bipartie graph in terms of the regularities of the vanishing ideals of the blocks of the graph.Comment: Bull. Braz. Math. Soc. (N.S.), to appea

New examples of Calabi-Yau threefolds and genus zero surfaces

We classify the subgroups of the automorphism group of the product of 4 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 non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=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.Comment: 18 pages; v2: simplified some arguments in the last section, final version to appear on Communications in Contemporary Mathematic

Unprojection and deformations of tertiary Burniat surfaces

We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an \'etale (Z/2)^3-cover T of a surface with p_g=0 and K^2=3 and assume that it may be embedded in a Fano 3-fold V. We construct V by using the theory of parallel unprojection. Since V is an Enriques--Fano 3-fold, considering its Fano cover yields the simple description of the universal covers above

Ventajas de la tintura por pulverización mediante la aplicación de pigmentos cromotrópicos.

Los pigmentos cromotrópicos o termocrómicos dan un buen resultado en la obtención de efectos especiales en el diseño textill si bien su elevado coste impide un uso más amplio. La aplicación mediante pulverización implica una significativa reducción de costes porque se necesita una cantidad inferior a la utilizada en los procesos de estampado tradicionales y también porque la no necesidad de preparar plantillas-tamiz hace que el tiempo requerido sea menor, adaptándose al propio tiempo, a las nuevas técnicas de gestión integrada en la cadena de abastecimiento, especialmente en lo que se refiere a respuesta rápida.Chromotropic or thermochromic pigments are a good way to obtain special effects in textile design, but its price is a impediment for a wider divulgation. The application by pulverization allo ws a significant reduction of costs because we need a smaller quantity than traditional printing processes, but also because as we don't need to prepare screens the time needed is reduced and is adapted to the new techniques of integrated management on the supplying chain of, namely the quick response.Les pigments chromotropiques ou thermotropiques donnent de bons résultats pour obtenir des effets spéciaux d'impression textile, mais leur coût élevé empeche de les utiliser plus sovent. L'application par pulvérisation implique une réduction significative des coûts. Il faut en effet des quantités moindres que les quantités utilisées avec des prix traditionnels d'impression et il n'est pas nécessaire de préparer des patrons-tamis, d'où un gain de temps et une meilleure adaptation aux nouvelles techniques de gestion intégrée dans la chaîne d'approvisionnement, notamment en ce qui concerne la réponse rapide.Peer Reviewe

Codes over a weighted torus

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