The resurgence of ideals of points and the containment problem

TL;DR

This paper investigates the containment problem for symbolic and ordinary powers of ideals, providing complete solutions for specific cases like points on a conic or with up to nine points, and exploring ideals with symbolic powers equal to ordinary powers.

Contribution

It offers new results on the containment problem for ideals defining 0-dimensional subschemes, especially in geometric configurations and for ideals with symbolic powers equal to ordinary powers.

Findings

Complete solutions for points on a conic

Results for up to 9 general points

Examples of ideals with symbolic powers equal to ordinary powers

Abstract

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our previous paper (arXiv:0706.3707) we give a complete solution in terms of the least degrees of nonzero elements of the symbolic powers, under the condition that the ideal is generated in a single degree and such that that degree is the regularity of the ideal (see Corollary 1.2). Using geometric methods, we also give complete solutions for ideals of points in the projective plane, when those points lie on an irreducible conic or when the points are general and there are at most 9 points. Finally, we give new examples of ideals all of whose symbolic powers are ordinary powers of the ideal.