On Coble surfaces and their automorphisms

TL;DR

This paper studies automorphisms of complex Coble surfaces, identifying two families with specific automorphism properties and analyzing their moduli space characteristics.

Contribution

It characterizes two families of Coble surfaces with automorphism T acting trivially on boundary C, revealing one family is entirely fixed and the other is of codimension 3.

Findings

First family of Coble surfaces is nodal with T acting as identity.

Second family is small, with codimension 3 in the moduli space.

In the first family, T equals the identity on the entire surface.

Abstract

Given a complex Coble surface $X$ with irreducible boundary $C$, we consider a specific automorphism $T : X \to X$, initially defined by Pompilj. We show that there are two families of Coble surfaces satisfying the condition $T|_C = \mathbb{1}_C$. Every Coble surface $X$ in the first family in nodal, and moreover the stronger equality $T = \mathbb{1}_X$ holds. Meanwhile, the second family is ''small'', since it has codimension $3$ in the moduli space of Coble surfaces.