Discriminants of cubic curves and determinantal representations

TL;DR

This paper explores a novel approach to expressing the discriminant of smooth plane cubic curves using determinantal representations, linking algebraic forms with theta functions through matrix representations.

Contribution

It introduces a new method to derive the classical discriminant formula via determinantal representations of cubic forms.

Findings

Discriminant expressed as a product of theta functions

Determinantal representations connect algebraic and analytic aspects

Provides a new perspective on classical algebraic geometry results

Abstract

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach to obtain this classical result by using determinantal representations. More precisely, one can represent a non-singular cubic form as the determinant of a matrix whose elements are linear forms. Theta functions naturally appear in this representation and thus in the discriminant of the cubic.