Focal Varieties of Curves of Genus 6 and 8

TL;DR

This paper establishes a Torelli-type theorem for genus 6 and 8 curves, demonstrating they can be reconstructed from their Brill-Noether varieties, and describes the focal varieties for these curves.

Contribution

It provides a new Torelli-type theorem for genus 6 and 8 curves using Brill-Noether varieties and characterizes their focal varieties.

Findings

Genus 6 curves have focal varieties that are hypersurfaces.

Curves of genus 6 and 8 can be reconstructed from their Brill-Noether varieties.

The focal variety of a general, canonical, nonhyperelliptic genus 6 curve is a hypersurface.

Abstract

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.