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

Abstract

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z Z supported at general points of P 2 \mathbf {P}^2 . We relate this problem to the open geometric problem of determining the splitting type of the pullback of Ω P 2 \Omega _{\mathbf {P}^2} 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 P 2 \mathbf {P}^2 . 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.