Federico Pieroni | Searcharxiv

arXiv2026

On Coble surfaces and their automorphisms

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.

math.AG↗

arXiv2026

Coble surfaces: projective models and automorphisms with related topics

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.

math.AG↗