Quartic Curves and Their Bitangents

TL;DR

This paper explores the geometric and algebraic properties of smooth quartic curves, focusing on their representations as symmetric determinants, sums of squares, and their associated Cayley octads, with algorithms for computation and geometric insights.

Contribution

It introduces exact algorithms for computing Cayley octads and Steiner complexes from bitangents, linking classical geometry with modern spectrahedral representations.

Findings

Algorithms for computing Cayley octads from bitangents

Expressions of Vinnikov quartics as spectrahedra

Equations for the variety of Cayley octads

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.