The birational geometry of moduli of cubic surfaces and cubic surfaces with a line

TL;DR

This paper analyzes the geometric structure of moduli spaces of cubic surfaces with a line, explicitly determining the cones of effective and nef divisors on their compactifications.

Contribution

It provides explicit descriptions of the effective and nef cones for the moduli space of cubic surfaces with a line and extends these results to unmarked cubic surfaces.

Findings

Effective and nef cones on the toroidal compactification are explicitly determined.

Cones for the moduli space of unmarked cubic surfaces are computed.

Results enhance understanding of the birational geometry of these moduli spaces.

Abstract

We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubics surfaces.