Searcharxiv⌕ Search

arXiv subjects

Matteo Varbaro

Publications and source records attributed to Matteo Varbaro.

At least 37 records · Page 2Linked to original sources
arXiv2019

Linear syzygies, hyperbolic Coxeter groups and regularity

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen--Macaulay.

math.AC↗
arXiv2018

Hankel determinantal rings have rational singularities

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen-Macaulay module.

math.AC↗
arXiv2018

Simplicial complexes of small codimension

We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for ${\rm CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the ${\rm CM}_t$ property is a topological invariant of a simplicial complex.

math.AC↗
arXiv2018

Connectivity of hyperplane sections of domains

During the conference held in 2017 in Minneapolis for his 60th birthday, Gennady Lyubeznik proposed the following problem: Find a complete local domain and an element in it having three minimal primes such that the sum of any two of them has height 2 and the sum of the three of them has height 4. In this note this beautiful problem will be discussed, and will be shown that the principle leading to the fact that such a ring cannot exist is false. The specific problem, though, remains open

math.AC↗
arXiv2017

Regularity of Line Configurations

We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of the coordinate ring of the arrangement. We also prove that every (d-1)-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth $\ge 3$. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang.

math.AG↗
arXiv2017

On the diameter of an ideal

We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.

math.AC↗
arXiv2017

Regulating Hartshorne's connectedness theorem

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity $\le 3$, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.

math.AG↗
arXiv2016

F-thresholds, integral closure and convexity

The purpose of this note is to revisit the results of arXiv:1407.4324 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 allows 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.

math.AC↗
arXiv2015

When does depth stabilize early on?

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the above fact holds true.

math.AC↗
arXiv2015

An alternative algorithm for computing the Betti table of a monomial ideal

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 numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.

math.AC↗
arXiv2014

Multiplicities of Classical Varieties

The $j$-multiplicity plays an important role in the intersection theory of Stückrad-Vogel cycles, while recent developments confirm the connections between the $ε$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $ε$-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

math.AC↗
arXiv2014

Connectivity of pseudomanifold graphs from an algebraic point of view

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings.

math.CO↗
arXiv2014

Linear syzygies, flag complexes, and regularity

We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.

math.AC↗
arXiv2014

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 other 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 five. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed.

math.AC↗
arXiv2013

The $F$-pure threshold of a determinantal ideal

The $F$-pure threshold is a numerical invariant of prime characteristic singularities, that constitutes an analogue of the log canonical thresholds in characteristic zero. We compute the $F$-pure thresholds of determinantal ideals, i.e., of ideals generated by the minors of a generic matrix.

math.AC↗
arXiv2013

An intriguing ring structure on the set of d-forms

The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block stable ideals (a class of ideals containing the Borel fixed ones) of K[x_1,\ldots,x_n] generated in degree d. As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.

math.AC↗