Star configuration points and plane curves

2siLet ℓ1,...,ℓ1 be l lines in ℙ2 such that no three lines meet in a point. Let X(l) be the set of points {ℓi ∩ ℓj {divides} 1 ≤ i < j ≤ l} ⊆ ℙ2. We call X(l) a star configuration. We describe all pairs (d, l) such that the generic degree d curve in ℙ2 contains an X(l). Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems. © 2011 American Mathematical Society.openopenCarlini E.; van Tuyl A.Carlini, E.; van Tuyl, A

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.Comment: minor changes: added Cor. 3.10 and some references. To appear in Proc. Amer. Math. So

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 \chi_f(G), the fractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <= \chi(G). We give an algebraic proof of this result.Comment: 11 pages; Minor changes throughout the paper; to appear in Discrete Math

Regularity and h-polynomials of toric ideals of graphs

For all integers 4 ≤ r ≤ d, we show that there exists a finite simple graph G = Gr,d with toric ideal IG ⊂ R such that R/IG has (Castelnuovo-Mumford) regularity r and h-polynomial of degree d. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function Hof a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function Hof a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. Fo r some families of valid Hilbert functions, we are also able to show that His the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines