Searcharxiv⌕ Search

arXiv subjects

Christopher A. Francisco

Publications and source records attributed to Christopher A. Francisco.

At least 19 recordsLinked to original sources
arXiv2020

Computing generalized Frobenius powers of monomial ideals

Generalized Frobenius powers of an ideal were introduced in work of Hernández, Teixeira, and Witt as characteristic-dependent analogs of test ideals. However, little is known about the Frobenius powers and critical exponents of specific ideals, even in the monomial case. We describe an algorithm to compute the critical exponents of monomial ideals and use this algorithm to prove some results about their Frobenius powers and critical exponents. Rather than using test ideals, our algorithm uses techniques from linear optimization.

math.AC↗
arXiv2019

Partial coloring, vertex decomposability, and sequentially Cohen-Macaulay simplicial complexes

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

math.AC↗
arXiv2018

Asymptotic resurgence via integral closures

Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed). For a monomial ideal the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.

math.AC↗
arXiv2017

The Rees algebra of a two-Borel ideal is Koszul

Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

math.AC↗
arXiv2016

Free and Non-free Multiplicities on the $A_3$ Arrangement

We give a complete classification of free and non-free multiplicities on the $A_3$ braid arrangement. Namely, we show that all free multiplicities on $A_3$ fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory.

math.AC↗
arXiv2013

Powers of squarefree monomial ideals and combinatorics

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.

math.AC↗
arXiv2011

Generalizing the Borel property

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.

math.AC↗
arXiv2010

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely algebraic characterizations of perfect graphs, independent of the Strong Perfect Graph Theorem; the first characterization is in terms of the sets Ass(R/J^s), while the second characterization is in terms of the saturated chain condition for associated primes.

math.AC↗
arXiv2010

Borel generators

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.

math.AC↗
arXiv2010

A conjecture on critical graphs and connections to the persistence of associated primes

We introduce a conjecture about constructing critically (s+1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture, we prove that the statement is true if we also assume that χ_f(G), the fractional chromatic number of the graph G, satisfies χ(G) -1 < χ_f(G) <= χ(G). We give an algebraic proof of this result.

math.AC↗
arXiv2010

Associated primes of monomial ideals and odd holes in graphs

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of $J(G)^2$.

math.AC↗
arXiv2009

Splittings of monomial ideals

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

math.AC↗
arXiv2008

Whiskers and sequentially Cohen-Macaulay graphs

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G \cup W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G \cup W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G \backslash S is not sequentially Cohen-Macaulay, then G \cup W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.

math.AC↗
arXiv2007

Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P^3 defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph to each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.

math.AC↗
arXiv2006

Some families of componentwise linear monomial ideals

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we study ideals of the form I=m_{J_1}^{a_1} \cap ... \cap m_{J_s}^{a_s}. These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s <= 3 or when J_i \cup J_j = [n] for all i \neq j. When s >= 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s=2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k)=0, our work also yields new cases in which this conjecture holds.

math.AC↗
arXiv2006

Sequentially Cohen-Macaulay Edge Ideals

Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

math.AC↗
arXiv2005

On the componentwise linearity and the minimal free resolution of a tetrahedral curve

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this work by showing that they are "almost always" componentwise linear, i.e. their homogeneous ideals have the property that for any d, the degree d component of the ideal generates a new ideal whose minimal free resolution is linear. The one type of exception is clearly spelled out and studied as well. The main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves, and the connection to the tetrahedral curves that are minimal in their even liaison classes. With this preparation, we also describe the minimal free resolution of a tetrahedral curve, and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution. Finally, we begin the study of the generic initial ideal (gin) of a tetrahedral curve. We produce the gin for arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class.

math.AC↗