Searcharxiv⌕ Search

arXiv subjects

Matteo Varbaro

Publications and source records attributed to Matteo Varbaro.

53 records · Page 3Linked to original sources
arXiv2013

Componentwise regularity (I)

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gröbner bases; the proof of our criterion relies on a strengthening of a lifting lemma of Buchsbaum and Eisenbud. This criterion helps us to show a stronger version of Green's crystallization theorem in a quite general setting, according to the componentwise regularity of the initial object. Finally we show a necessary condition, given a submodule $M$ of a free one over the polynomial ring and a weight such that $in(M)$ is componentwise linear, for the existence of an $i$ such that $β_i(M)=β_i(in(M))$.

math.AC↗
arXiv2013

Partitions of single exterior type

We characterize the irreducible representations of the general linear group GL(V) that have multiplicity 1 in the direct sum of all Schur modules of a given exterior power of V. These have come up in connection with the relations of the lower order minors of a generic matrix. We show that the minimal relations conjectured by Bruns, Conca and Varbaro are exactly those coming from partitions of single exterior type.

math.RT↗
arXiv2013

Relations between the minors of a generic matrix

It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.

math.AC↗
arXiv2013

Maximal minors and linear powers

An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.

math.AC↗
arXiv2012

On a conjecture by Kalai

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that $h$-vectors of flag Cohen-Macaulay simplicial complexes are $h$-vectors of Cohen-Macaulay balanced simplicial complexes.

math.AC↗
arXiv2012

h-vectors of matroid complexes

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.

math.AC↗
arXiv2012

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.

math.AC↗
arXiv2011

Groebner deformations, connectedness and cohomological dimension

This paper is an outcome of the author's master thesis written under the supervision of Aldo Conca. We prove some results relating connectedness properties with (local) cohomological dimension. As an interesting corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected.

math.AC↗
arXiv2011

Unmixed Graphs that are Domains

Given an arbitrary graph G, we study its basic covers algebra, which is the symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the case when G is bipartite, we characterize in a combinatorial fashion the situations when: 1) the basic covers algebra is a domain, and 2) it is a domain and in addition (the edge ideal of) G is unmixed. It turns out that the last result gives a complete characterization of those graphs for which any symbolic power of the edge ideal is generated by monomials of the same degree.

math.AC↗
arXiv2011

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.

math.AC↗
arXiv2011

On the h-vectors of Cohen-Macaulay Flag Complexes

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

math.AC↗
arXiv2011

Cohen-Macaulayness of generically complete intersection monomial ideals

In this paper we discuss the problem of characterizing the Cohen-Macaulay property of certain families of monomial ideals with fixed radical. More precisely, we consider generically complete intersection monomial ideals whose radical corresponds to special classes of simplicial complexes.

math.AC↗
arXiv2011

Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain cohomology modules vanish, a classical problem raised by Grothendieck. In the second chapter we will use local cohomology to study the connectedness behavior during a Groebner deformation and the arithmetical rank of certain varieties. In the third chapter, we will investigate the relations between the minors of a fixed size of a generic matrix by using tools from the representation theory of the general linear group (the results of this chapter will appear in a joint paper with Bruns and Conca). In the last chapter we will use combinatorial methods to study the Cohen-Macaulay property of the symbolic powers of Stanley-Reisner ideals. In the thesis are included five appendixes with some basic needed facts and a preliminary chapter introducing to local cohomology.

math.AC↗
arXiv2010

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 characteristic-zero-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 holds true also in characteristic-zero up to dimension three.

math.AG↗
arXiv2010

Koszulness, Krull Dimension and Other Properties of Graph-Related Algebras

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.

math.AC↗
arXiv2009

Dimension, depth and zero-divisors of the algebra of basic $k$-covers of a graph

We study the basic $k$-covers of a bipartite graph $G$; the algebra $\AG$ they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when $\AG$ is a domain in terms of the combinatorics of $G$; if follows from a result of Hochster that when $\AG$ is a domain, it is also Cohen-Macaulay. We then study the dimension of $\AG$ by introducing a geometric invariant of bipartite graphs, the "graphical dimension". We show that the graphical dimension of $G$ is not larger than $\dim(\AG)$, and equality holds in many cases (e.g. when $G$ is a tree, or a cycle). Finally, we discuss applications of this theory to the arithmetical rank.

math.AC↗