Apostolos Thoma

MINIMAL SYSTEMS OF BINOMIAL GENERATORS AND THE INDISPENSABLE COMPLEX OF A TORIC IDEAL

Let A = {a1, . . . , am} � Zn be a vector configuration and IA � K(x1, . . . , xm) its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of IA. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to A a simplicial complexind(A). We show that the vertices ofind(A) correspond to the indispensable monomials of the toric ideal IA, while one dimensional facets ofind(A) with minimal binomial A-degree correspond to the indispensable binomials of IA.

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.

On the universal Gröbner bases of toric ideals of graphs

The universal Grobner basis of an ideal is a Grobner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Grobner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.

On the set-theoretic complete intersection problem for monomial curves in An and Pn

Abstract In this paper we deal with the problem of the expression of monomial curves in the affine or projective n -dimensional space as set-theoretic complete intersections. We develop two techniques for finding monomial curves which are set-theoretic complete intersections. Using these two techniques we are able to generalize all previous known results and give infinitely many examples of monomial curves which are set-theoretic complete intersections in an affine or projective n -dimensional space, for any n .

On the binomial arithmetical rank

Abstract. The binomial arithmetical rank of a binomial ideal I is the smallest integer s for which there exist binomials f1,..., fs in I such that rad (I) = rad (f1,..., fs). We completely determine the binomial arithmetical rank for the ideals of monomial curves in

Toric sets and orbits on toric varieties

P_K^n

Minimal generating sets of lattice ideals

. In particular we prove that, if the characteristic of the field K is zero, then bar (I(C)) = n - 1 if C is complete intersection, otherwise bar (I(C)) = n. While it is known that if the characteristic of the field K is positive, then bar (I(C)) = n - 1 always.

MARKOV BASES AND GENERALIZED LAWRENCE LIFTINGS

Abstract Let D be an integer matrix. A toric set, namely the points in K n parametrized by the columns of D , and a toric variety are associated to D . The toric set is a subset of the toric variety. We describe the relation between the toric set and the toric variety, in terms of the orbits of the torus action on the toric variety. The toric set depends on the sign (+,−,0) pattern of the matrix D . Finally, we prove that any toric variety over an algebraically closed field can be expressed as a toric set, for an appropriate matrix.

Matchings in simplicial complexes, circuits and toric varieties

Let