A Universal Formula For Counting Cubic Surfaces

TL;DR

This paper derives a universal formula using equivariant geometry to count how often a general cubic surface appears in a family, providing exact degrees and counts for specific geometric configurations.

Contribution

It introduces a universal counting formula for cubic surfaces and applies it to compute orbit closure degrees and hyperplane section counts.

Findings

PGL(4) orbit closure degree of a generic cubic surface is 96120

A general cubic surface appears 42120 times as a hyperplane section of a cubic 3-fold

Provides a new universal counting method for cubic surfaces

Abstract

Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic 3-fold.