The Alpha Problem & Line Count Configurations

Abstract

Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\mathbb{P}}^n$. The conjectures involve both containment statements and bounds for the initial degree in which there is a non-zero form in an ideal. Working with initial degrees, we verify two of these conjectures for special line count configurations in projective 2-space over an algebraically closed field of characteristic 0.

This version contains an alternate proof of the main combinatorial identity that was suggested by an anonymous referee. v3 also fixes some typos

References in corpus (4)

• Are symbolic powers highly evolved?
• Containment results for ideals of various configurations of points in P^N
• Containment Counterexamples for ideals of various configurations of points in ${\mathbf{P}}^N$
• Symbolic powers of ideals of generic points in P^3