On the containment problem for fat points

Abstract

Given an ideal $I$, the containment problem is concerned with finding the values $m$ and $r$ such that the $m$-th symbolic power of $I$ is contained in its $r$-th ordinary power. A central issue related to this is determining the resurgence for ideals $I$ of fat points in projective space. In this paper we obtain complete results for the resurgence of fat point schemes $m_0{P}_0+m_1{P}_1+m_2{P}_2$ in ${\mathbb{P}}^N$ for any distinct points $P_0,P_1,P_2$, and, when the points $P_0,\dots ,P_n$ are collinear, we extend this result to fat point schemes $m_0{P}_0+\cdots +m_n{P}_n$. As a by-product of our determining the resurgence for all three points fat point ideals, we give new examples of ideals with symbolic defect zero. In case the points are noncollinear, a three fat points ideal can be regarded as a monomial ideal, but it is typically not square-free.