Quartic Curves and Their Bitangents
Algebraic Geometry
2012-01-04 v2 Symbolic Computation
Abstract
A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.
Keywords
Cite
@article{arxiv.1008.4104,
title = {Quartic Curves and Their Bitangents},
author = {Daniel Plaumann and Bernd Sturmfels and Cynthia Vinzant},
journal= {arXiv preprint arXiv:1008.4104},
year = {2012}
}
Comments
26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8, other minor changes