Abstract. We develop tools to study the problem of containment of symbolic powers I (m) in powers I r for a homogeneous ideal I in a polynomial ring k[P N ] in N + 1 variables over an arbitrary algebraically closed field k. We obtain results on the structure of the set of pairs (r, m) such that I (m) ⊆ I r . As corollaries, we show that I 2 contains I (3) whenever S is a finite generic set of points in P 2 (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of [ELS] and [HH1] are optimal for every fixed dimension and codimension.

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in P 2 of fixed degree d, and it is known to be true in many cases. We propose a new problem, namely to understand the number of conditions imposed by a general union of fat points on the incomplete linear system defined by the condition of passing through a given finite set of points Z (not general). Motivated by work of Di Gennaro-Ilardi-Vallès and Faenzi-Vallès, we give a careful analysis for the case where there is a single general fat point, which has multiplicity d − 1. There is an expected number of conditions imposed by this fat point, and we study those Z for which this expected value is not achieved. We show, for instance, that if Z is in linear general position then such unexpected curves do not exist. We give criteria for the occurrence of such unexpected curves and describe the range of values of d for which they occur. Unexpected curves have a very particular structure, which we describe, and they are often unique for a given set of points. In particular, we give a criterion for when they are irreducible, and we exhibit examples both where they are reducible and where they are irreducible. Furthermore, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we obtain a new interpretation of the splitting type of a line arrangement. Finally, we use our results to establish a Lefschetz-like criterion for Terao's conjecture on the freeness of line arrangements. 2

Abstract. We relate properties of linear systems on X to the question of when I r contains I (m) in the case that I is the homogeneous ideal of a finite set of distinct points p 1 , . . . , p n ∈ P 2 , where X is the surface obtained by blowing up the points. We obtain complete answers for when I r contains I (m) when the points p i lie on a smooth conic or when the points are general and n ≤ 9.

Abstract. 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.