Discriminants of cubic curves and determinantal representations

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.

16 pages. To appear in The Ramanujan Journal. Portions of this work previously appeared as arXiv:1911.01350 which was split during refereeing