Automorphisms and Hodge isometries of surfaces with $K^2=p_g=1$

Abstract

Let $S$ be a minimal complex surface of general type with $K_S^2=p_g(S)=1$. By a theorem of Catanese, the canonical model of $S$ is a complete intersection of two sextics in $\mathbf{P}(1,2,2,3,3)$ whose equations admit a normal form, and the global Torelli theorem fails for these surfaces. We describe $\operatorname{Aut}(S)$ in terms of the coefficients of the normal form, prove that $|\operatorname{Aut}(S)|\leq 72$, and classify the surfaces attaining the bound: two surfaces with singular canonical model and a one-parameter family $X_r$ of smooth surfaces. For the surfaces $X_r$ we compute a basis of the primitive cohomology consisting of eigenvectors of a subgroup $\mathbf{Z}/6\mathbf{Z} \times \mathbf{Z}/6\mathbf{Z}$ of $\operatorname{Aut}(X_r)$, given by residues of explicit polynomials, together with the intersection form in this basis. We then show that the natural map from $\operatorname{Aut}(X_r)$ to the group of Hodge isometries of the primitive lattice modulo $\pm1$ is injective but has image of index at least $331\,776$, so that the Torelli principle for automorphisms fails for these surfaces.