Are symbolic powers highly evolved?

TL;DR

This paper explores the structural properties of symbolic powers of ideals, proposing conjectures to explain existing results and proving them in specific cases, especially for general points in the projective plane.

Contribution

It introduces new conjectures linking symbolic powers of ideals to geometric and algebraic properties, and proves these conjectures in certain cases, advancing understanding of symbolic powers.

Findings

Conjectures explaining the degree of forms in ideals of fat points

Proofs of conjectures for general points in the projective plane

Connections between symbolic powers and the Eisenbud-Mazur Conjecture

Abstract

Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in the projective plane. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.