The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type (Q2220241)

Consider the Sylvester-nondegenerate cubic surface given in \(\mathbb{P}^4\) by the equations \[\sum_{a=0}^4{y_a}=\sum_{a=0}^4{\lambda_ay_a^3}=0\] for certain \(\lambda_a\ne 0\), and denote by \(H\) its Hessian surface. The minimal resolution of \(H\) is a \(K3\) surface \(X\), admitting an involution \(\sigma\) acting freely on \(X\), except possibly for some exceptional divisors coming from ordinary nodes of \(H\). The paper focuses on the smooth quotient \(S\) of \(X\) by \(\sigma\), which can either be an Enriques surface or a Coble surface. More specifically, the authors describe explicitly the automorphism group of the surface \(S_t\) of the \(1\)-dimensional family parametrized by \((\lambda_0,\ldots,\lambda_4)=(1,1,1,1,t)\), \(t\ne 0\), and they show that for any \(t\ne 1\), the group of automorphisms \(\mathrm{Aut}(S_t)\) is isomorphic to the semi-direct product \(G\) of the free product of four groups of order \(2\) with the symmetric group \(\mathfrak{S}_4\). In particular \(\mathrm{Aut}(S_t)\), \(t\ne 1\), does not depend on whether \(S_t\) is an Enriques or a Coble surface. The generators of \(\mathrm{Aut}(S_t)\) are obtained via explicit elliptic fibrations on \(S_t\), and the proof that \(\mathrm{Aut}(S_t)\cong G\) for \(t\ne 1\) relies on a close inspection of the nef cone of the surface \(S_t\). In the last part of the paper the authors provide some different reincarnations of the group \(G\), as a lattice in \(\mathrm{PGL}_2(\mathbb{Q}_3)\) and as a subgroup of isometries of Euclidean \(3\)-space generated by the isometries of a regular tetrahedron and the reflections across its facets.