Plane quartics and heptagons

TL;DR

This paper explores the relationship between plane heptagons and quartic curves, establishing a precise count of heptagons associated with a general quartic using advanced algebraic geometry techniques.

Contribution

It proves that a general plane quartic corresponds to exactly 864 complex heptagons, combining intersection theory and classical algebraic geometry methods.

Findings

A general plane quartic is the adjoint of exactly 864 heptagons.

The number 864 is both an upper and lower bound for the count.

Explicit analysis of the Klein quartic supports the bounds.

Abstract

Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly 864 distinct complex heptagons. This number had been numerically computed by Kohn et al. We use intersection theory and the Scorza correspondence for quartics to show that 864 is an upper bound, complemented by a lower bound obtained through explicit analysis of the famous Klein quartic.