Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

Abstract

Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $π_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $π_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $π_1(\mathcal{C})$ ``remembers'' the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that $\mathcal{C}$ has no nontrivial biholomorphic automorphisms as complex analytic orbifold.