Betti numbers for fat point ideals in the plane: a geometric approach

TL;DR

This paper explores the graded Betti numbers of fat point schemes in the plane, linking algebraic invariants to geometric properties of rational curves, proposing a conjecture, and verifying it in many cases.

Contribution

It introduces a conjecture relating Betti numbers to geometric splitting types and proves it in numerous cases, advancing understanding of fat point ideals in algebraic geometry.

Findings

Conjecture for Betti numbers in all but one degree for general fat point schemes.

Verification of the conjecture in a broad range of cases.

Development of a new computational approach for Betti numbers and splitting types.

Abstract

We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.