Coble surfaces: projective models and automorphisms with related topics

TL;DR

This paper studies complex Coble surfaces, showing their isotropic sequences can be extended, providing a specific birational quintic model, and characterizing their involutions as lifts of Bertini involutions.

Contribution

It introduces a birational quintic model for unnodal complex Coble surfaces and characterizes their involutions as lifts of Bertini involutions.

Findings

Isotropic sequences can be extended to length 10.

Existence of a specific birational quintic model in P^3.

Involutions on the surface are lifts of Bertini involutions.

Abstract

In this work, we want to show several properties of an unnodal, complex Coble surface $X$ with irreducible boundary curve $C \in |-2 K_X|$. Namely, we show that every isotropic sequence ${\cal E}_1, \ldots, {\cal E}_r$ with $r \le 8$ and ${\cal E}_i {\cal E}_j = 1 - \delta_{i, j}$ can be extended to a sequence of length $10$. Moreover, such a surface admits a birational quintic model $\overline X \subset \mathbb{P}^3$, with equation $\alpha X_0 X_1^2 X_2^2 + \beta X_0 X_1^2 X_3^2 + \gamma X_0 X_2^2 X_3^2 + X_1 X_2 X_3 q = 0$, where $q$ is a quadric form. Finally, we use this birational model to show that every biregular involution $i : X \to X$ on such a Coble surface is the lift of a Bertini involution.