The Hessian Discriminant

TL;DR

This paper expresses the Hessian discriminant of cubic surfaces using fundamental invariants, providing a new way to compute and understand these invariants for smooth cubics of rank 6.

Contribution

It offers a novel expression of the Hessian discriminant in terms of fundamental invariants, addressing a specific open question in cubic surface theory.

Findings

Expressed the Hessian discriminant via fundamental invariants.

Provided a method to compute invariants for smooth cubics of rank 6.

Answered a previously open question in the field.

Abstract

We express the Hessian discriminant of a cubic surface in terms of fundamental invariants. This answers Question 15 from the \emph{27 questions on the cubic surface}. We also explain how to compute the fundamental invariants for smooth cubics of rank 6.