Symbolic powers and the Containment Problem

The symbolic powers of an ideal I relate to the primary decomposition of the ordinary powers of I, and consist, under certain circumstances, of the functions that vanish up to order n in the corresponding variety. The Containment Problem of symbolic and ordinary powers consists of determining which symbolic powers are contained in which ordinary powers. Given a radical ideal in a regular ring, there is a uniform answer to this question by Ein--Lazarsfeld--Smith, Hochster--Huneke and Ma--Schwede, but that is not necessarily best possible. In particular, a question of Huneke remains open for prime ideals, and a generalization proposed by Harbourne has been shown to fail for certain radical ideals. In this dissertation, we study the Containment Problem for regular rings containing a field, presenting versions of Harbourne's Conjecture that do hold.

We show that Harbourne's Conjecture holds when R/I is F-pure, and prove stronger containments in the case when R/I is strongly F-regular, results that are joint work with Craig Huneke. We also answer Huneke's question positively for monomial curves in dimension 3, provide evidence that Harbourne's Conjecture might always hold eventually, and study the (non-)relationship between the Containment Problem and the finite generation of the symbolic Rees Algebra. Finally, we discuss algorithms for computing symbolic powers over a polynomial ring.