Plane quartics: the matrix of bitangents

TL;DR

This paper explores the construction of a matrix representing bitangents of a plane quartic, building on classical and recent results to reconstruct the curve from its bitangent configuration.

Contribution

It addresses the inverse problem of constructing the bitangent matrix from the 28 bitangents, extending previous work on parametrizations and reconstructions.

Findings

Established a method to construct the bitangent matrix from 28 bitangents.

Connected classical results with modern matrix parametrizations.

Provided insights into the inverse problem of curve reconstruction.

Abstract

Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special 7-tuples of bitangents such that the six points at which any subtriple of bitangents touches the quartic do not lie on the same conic in the projective plane. In 2005 Lehavi proved that a smooth plane quartic can be explicitly reconstructed from its 28 bitangents; this result improved Aronhold's method of recovering the curve. In a 2011 paper Plaumann, Sturmfels and Vinzant introduced an eight by eight symmetric matrix parametrizing the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse's result for which every smooth quartic curve has exactly 36 equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e. the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic.