On the minimal free resolution for fat point schemes of multiplicity at most 3 in P^2

TL;DR

This paper proves that fat point schemes in P^2 with multiplicities up to 3 and sufficiently many points have minimal free resolutions matching the expected minimal generators, confirming their optimal algebraic structure.

Contribution

It establishes the minimal free resolution for fat point schemes of multiplicity at most 3 in P^2 with many points, confirming their expected algebraic properties.

Findings

Number of generators is as small as possible in each degree.

Fat point schemes have maximal Hilbert function.

Schemes have the expected minimal free resolution.

Abstract

Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.