Matteo Varbaro

Symbolic powers and matroids

We prove that all the symbolic powers of a Stanley-Reisner ideal are Cohen-Macaulay if and only if the associated simplicial complex is a matroid.

The possible extremal Betti numbers of a homogeneous ideal

We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field.

Cohomological and projective dimensions

In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a well-known theorem of Peskine and Szpiro. As a corollary, we give new examples of prime ideals that are not set-theoretically Cohen-Macaulay.

h-Vectors of Matroid Complexes

In this paper we partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley’s conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.

Multiplicities of classical varieties

The j-multiplicity plays an important role in the intersec- tion theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the e-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relation ship between the j-multiplicity of an ideal and the degree of its fiber cone. As a conse- quence, we are able to compute the j-multiplicity of all the ideals defin- ing rational normal scrolls. By using the standard monomial theory, we can also compute the j- and e-multiplicity of ideals defining determinan- tal varieties: The found quantities are integrals which, qu ite surprisingly, are central in random matrix theory.

Unmixed Graphs that are Domains

We extend a theorem of Villareal on bipartite graphs to the class of all graphs. On the way to this result, we study the basic covers algebra of an arbitrary graph G. We characterize with purely combinatorial methods the cases when 1) is a domain and 2) G is unmixed and is a domain.

Generic and special constructions of pure O-sequences

It is shown that the h-vectors of Stanley–Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of matroids, or more generally of Cohen–Macaulay complexes, (b) matroids whose dual is (rank + 2)-partite, and (c) matroids of Cohen–Macaulay type at most 5. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed.

On the arithmetical rank of certain segre embeddings

We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristiczero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine and Szpiro in positive characteristic also holds true in characteristic-zero up to dimension three.

Computing the Betti table of a monomial ideal: A reduction algorithm

Abstract In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform some numerical experiments. As a major byproduct, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.

F-Thresholds, Integral Closure and Convexity

The purpose of this note is to revisit the results of the paper of Henriques and Varbaro from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial f allows one to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.