On the graded resolution of ideals of a few general fat points of Pn

AbstractWe compute the graded Betti numbers of the ideal of “few” (at most n) fat points of Pn with support in general position, under the assumption that the multiplicities are all equal except one, which has to be at least twice as much.We do so by performing a sequence of splittings and applying the graded version given by Fatabbi (J. Algeb. 242 (2001) 92) of a result of Eliahou and Kervaire (J. Algeb. 129 (1990) 1).In doing so, we also prove that the ideal of at most n general fat points of Pn is always splittable and we compute the graded Betti numbers of the product of two ideals whose generators involve disjoint sets of indeterminates

On the hadamard product of degenerate subvarieties

We consider generic degenerate subvarieties Xi ⊂ Pn. We determine an integer N, depending on the varieties, and for n≥N we compute dimension and degree formulas for the Hadamard product of the varieties Xi. Moreover, if the varieties Xi are smooth, their Hadamard product is smooth too. For n < N, if the Xi are generically di-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus

A good leaf order on simplicial trees

Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of H\`a and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry, Birkhauser volume (2013

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February 2015. Comments are welcome. Revised version corrects some typos, updates the references, and clarifies some hypotheses. To appear in the Journal of Algebraic Combinatoric

Regularity Index of Fat Points in the Projective Plane

AbstractIn this paper we determine a new upper bound for the regularity index of fat points of P2, without requiring any geometric condition on the points. This bound is intermediate between Segre′s bound, that holds for points in the general position, and the more general bound, that is attained when the points are collinear: in fact, both of these bounds can be recovered as particular cases. Furthermore, our bound cannot, in general, be sharpened: in fact, it is attained if there are either many collinear points or collinear points with high multiplicities