Schedule for: 17w5027 - Ordinary and Symbolic Powers of Ideals

A welcome drink will be served at the hotel.

Consider a plane curve of degree d with points $p_1,\ldots,p_s$ of multiplicities $m_1=mult_{p_1}(C),\ldots,m_s=mult_{p_s}(C)$. Define $Q(C,p_1,\ldots,p_s)$ to be $d^2-\sum_im_i^2$ and say that $C$ is negative curve of $Q<0$. A well known conjecture states that there are no integral plane curves $C$ with $Q(C,p_1,\ldots,p_s)<-1$ for generic points $p_i$. More generally, suppose we have points $q_1,\ldots,q_r$ in addition to the generic points $p_i$. Can there be integral negative curves $C$ with $Q(C,p_1,\ldots,p_s,q_1,\ldots,q_r)<-1$ if $mult_{p_i}(C)>0$ for some $i$? If so, what can be said about the occurrence of such curves? Work of R. De Gennaro, D. Faenzi, G. Ilardi and J. Valles shows in general it can happen so the problem now is to classify such occurrences. I'll discuss some joint work with D. Cook II, J. Migliore and U. Nagel giving partial results.

In this talk we present some selected known results and new questions on edge ideals and their powers. We introduce and examine a new family of edge ideals associated to vertex-weighted oriented graphs that has been recently studied by E. Reyes, Y. Pitones, and J. Toledo.

This talk will be in part historical, and in part recent progress, on the containment problem for the ordinary and symbolic powers of self-radical ideals. If I is a self-radical ideal, the problem asks which symbolic power of I is contained in which ordinary power of I. We will in particular discuss two problems: the uniformity problem in complete local domains, and Harbourne’s conjecture for graded ideals in polynomial rings. In particular, the talk will present the recent work of Eloisa Grifo and myself proving Harbourne’s conjecture in the case the quotient ring is F-pure or dense F-pure type.

We investigate symbolic powers of ideals that arise from combinatorial data. Examples include monomial ideals and ideals defining certain flats in the intersection lattice of a hyperplane arrangement. I will survey what has been done towards elucidating the asymptotic behavior of the symbolic powers of such ideals. Several invariants have been introduced and studied in this context, including the Waldschmidt constant and the resurgence. We give bounds on these asymptotic invariants. This is based on joint works with Bocci-Cooper-Guardo-Harbourne-Janssen-Nagel-Van Tuyl-Vu, Dumnicki-Harbourne-Nagel-Szemberg-Tutaj Gasińska and Bauer-Di Rocco-Harbourne-Huizenga-Szemberg.

We begin by discussing general necessary and sufficient conditions for finite generation of symbolic algebras. Then we focus on the case of symbolic algebras of rational monomial curves in three dimensional affine space, including the role of negative curves, and the differences between positive characteristic and characteristic zero. We give examples of finite generation and non finite generation. We end by discussing some open problems.

In this talk I will show how to estimate degrees of generators of local cohomology modules. I will also survey several applications to Rees algebras, to hyperplane sections, to symbolic powers, and to ideals of Pfaffians. This is joint work with Andy Kustin and Bernd Ulrich.

Let $R$ be a polynomial ring in $r$ variables. Let $L$ be a general linear form. A result of Stanley and of Watanabe says that if I is an ideal generated by $r$ powers of linearly independent linear forms (e.g. the variables), and if $j$ and $k$ are any positive integers, then multiplication by $L^k$ from the degree $j$ component of $R/I$ to the degree $j+k$ component has maximal rank. This leads to the question of what happens when we have powers of more than $r$ linear forms. Over the last decade or so several papers have addressed different aspects of this problem, most often focusing on the case $k=1$ (i.e. studying the so-called Weak Lefschetz Property). In the last half year or so, progress has been made on the problem when $k > 1$. I will give an overview of the history of this problem, talk about recent work in collaboration with R. Miró-Roig and with U. Nagel, and mention open problems and directions for new research.

Information on the Hilbert function of a finite set of fat points is relevant in a variety of studies such as interpolation, linear systems of rational varieties, or the weak Lefschetz property. In particular, one is interested in the least degree from which on the Hilbert function is predictable or, equivalently, its Castelnuovo-Mumford regularity. An optimal upper bound, named after Segre, was conjectured by Trung and, independently, by Fatabbi and Lorenzini in 2001. In joint work with Bill Trok, we recently established this conjecture. Furthermore, we derive an alternate regularity bound that improves the Segre bound in some cases, for example, if the support contains a subset of general points. Among the arguments is a new partition result for matroids.