Genus three curves and 56 nodal sextic surfaces

TL;DR

This paper presents a new, simplified construction method for sextic surfaces with 56 nodes, starting from non-hyperelliptic genus three curves, including explicit examples derived from the Klein curve.

Contribution

It introduces an alternative, straightforward construction approach for 56-nodal sextic surfaces using genus three curves, expanding on prior complex methods.

Findings

Successfully constructed explicit equations for 56-nodal sextic surfaces

Demonstrated the method with the Klein curve example

Provided a simpler approach compared to previous constructions

Abstract

Catanese and Tonoli showed that the maximal cardinality for an even set of nodes on a sextic surface is 56 and they constructed such nodal surfaces. In this paper we give an alternative, rather simple, construction for these surfaces starting from a non-hyperelliptic genus three curve. We illustrate our method by giving explicitly the equation of such a sextic surface starting from the Klein curve.