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 $C\in \mathrm{\mid }-2K_X\mathrm{\mid }$. Namely, we show that every isotropic sequence ${\mathcal{E}}_1,\dots ,{\mathcal{E}}_r$ with $r\le 8$ and ${\mathcal{E}}_i{\mathcal{E}}_j=1-{\delta }_{i,j}$ can be extended to a sequence of length $10$. Moreover, such a surface admits a birational quintic model $\bar{X}\subset {\mathbb{P}}^3$, 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.