96120: The degree of the linear orbit of a cubic surface

TL;DR

This paper computes the degree of the orbit closure of a general cubic surface under the action of the projective linear group, using numerical algebraic geometry methods, providing a key invariant in algebraic geometry.

Contribution

It introduces a numerical approach to determine the degree of the orbit closure of cubic surfaces under group action, a problem previously approached through theoretical methods.

Findings

The degree of the orbit closure is explicitly computed as 96120.

Numerical algebraic geometry effectively computes geometric invariants of group orbits.

The method can be applied to similar problems in algebraic geometry.

Abstract

The projective linear group \(\pgl(\comp,4)\) acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry.