Counting 3-uple Veronese surfaces

TL;DR

This paper counts the number of 3-Veronese surfaces passing through 13 general points, using a novel construction on the Hilbert scheme and Atiyah-Bott localization to solve an enumerative geometry problem.

Contribution

It introduces a new approach to count 3-Veronese surfaces via a construction on the Hilbert scheme and localization techniques, extending classical enumerative methods.

Findings

Count of 3-Veronese surfaces through 13 points obtained

Development of a space of 'complete triangles' on the Hilbert scheme

Application of Atiyah-Bott localization in enumerative geometry

Abstract

This paper culminates in the count of the number of 3-Veronese surfaces passing through 13 general points. This follows the case of 2-Veronese surfaces discovered by Coble in the 1920's. One important element of the calculation is a direct construction of a space of "complete triangles." Our construction is different from the classical ordered constructions of Schubert, Collino and Fulton, as it occurs directly on the Hilbert scheme of length 3 subschemes of the plane. We transport the enumerative problem into a 26-dimensional Grassmannian bundle over our space of complete triangles, where we perform Atiyah-Bott localization. Several important questions arise, which we collect at the end of the paper.