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

TL;DR

This paper investigates the fundamental group of the moduli space of smooth cubic surfaces, showing it retains information about nodal surfaces and has no nontrivial automorphisms.

Contribution

It proves the divisor subgroup of the fundamental group is characteristic and deduces the automorphism group of the moduli space is trivial.

Findings

The divisor subgroup of the fundamental group is characteristic.

The moduli space has no nontrivial biholomorphic automorphisms.

The group-theoretic property reflects geometric features of cubic surfaces.

Abstract

Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $\pi_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $\pi_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $\pi_1(\mathcal{C})$ ``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.