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.

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.

We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k< 0.9997*2^n/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k< 0.998*3^n/(2n+1) (the constant 1 being the optimal value)

We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in P N . In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.