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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.06718
- arXiv:
- arXiv:2009.06718
- Bibcode:
- 2020arXiv200906718T
- Keywords:
-
- Mathematics - Number Theory;
- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- 11C20;
- 11G07;
- 14H42;
- 14H50;
- 14H55;
- 32A08
- E-Print:
- 16 pages. To appear in The Ramanujan Journal. Portions of this work previously appeared as arXiv:1911.01350 which was split during refereeing