Discriminants of cubic curves and determinantal representations
Number Theory
2022-05-04 v2 Algebraic Geometry
Complex Variables
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.
Keywords
Cite
@article{arxiv.2009.06718,
title = {Discriminants of cubic curves and determinantal representations},
author = {Manh Hung Tran},
journal= {arXiv preprint arXiv:2009.06718},
year = {2022}
}
Comments
16 pages. To appear in The Ramanujan Journal. Portions of this work previously appeared as arXiv:1911.01350 which was split during refereeing