Counterexamples to containment problems for fat points schemes in the projective plane via multiplier ideals

TL;DR

This paper investigates the containment problem for fat points schemes in the projective plane, providing sharp bounds on Waldschmidt constants and identifying cases where symbolic and ordinary powers of ideals do not contain each other.

Contribution

It introduces new lower bounds on Waldschmidt constants and constructs counterexamples to known containment relations for point ideals in the plane.

Findings

Established sharp lower bounds on Waldschmidt constants.

Constructed explicit counterexamples to containment $I^{(2r-1)} ot\subseteq I^r$.

Enhanced understanding of the relationship between symbolic and ordinary powers in algebraic geometry.

Abstract

In this note we address the relation between symbolic and ordinary powers of the ideal of a reduced set or points in projective space: the so-called containment problem. In particular, we obtain sharp lower bounds on the Waldschmidt constants of ideals of points in the plane not satisfying $I^{(2r-1)} \subseteq I^r$.