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

Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek

TL;DR

The paper computes explicit birational descriptions for moduli spaces of cubic surfaces, both with and without a marked line, by determining the effective and nef cones on toroidal and Kirwan compactifications tied to ball quotient models. It leverages a rich network of identifications (GIT, Naruki/NMostow Deligne–Mostow, Hassett spaces) to relate divisors across covers and contractions, deriving precise cone descriptions and canonical class formulas. A key outcome is that the unmarked toroidal and Kirwan models are not $K$-equivalent, evidenced by their differing top self-intersections of the canonical divisor, while the marked-line case reveals a cohesive, tractable nef/ effective cone structure via explicit boundary and Eckardt divisors. Together, these results illuminate the birational geometry of cubic surface moduli, clarifying how boundary components, discriminants, and special loci govern the global divisor theory and birational maps between compactifications.

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.

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

TL;DR

The paper computes explicit birational descriptions for moduli spaces of cubic surfaces, both with and without a marked line, by determining the effective and nef cones on toroidal and Kirwan compactifications tied to ball quotient models. It leverages a rich network of identifications (GIT, Naruki/NMostow Deligne–Mostow, Hassett spaces) to relate divisors across covers and contractions, deriving precise cone descriptions and canonical class formulas. A key outcome is that the unmarked toroidal and Kirwan models are not -equivalent, evidenced by their differing top self-intersections of the canonical divisor, while the marked-line case reveals a cohesive, tractable nef/ effective cone structure via explicit boundary and Eckardt divisors. Together, these results illuminate the birational geometry of cubic surface moduli, clarifying how boundary components, discriminants, and special loci govern the global divisor theory and birational maps between compactifications.

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.
Paper Structure (6 sections, 16 theorems, 73 equations, 2 figures)

This paper contains 6 sections, 16 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The set-up, and the intersection theory
  3. The moduli space of cubics with a line
  4. The effective cones
  5. The nef cones
  6. Intersection theory on the Kirwan blowup

Key Result

Theorem 1.1

The cones of nef and effective divisors on the toroidal compactification $\overline\mathcal{M}={\overline{\mathcal{B}_4/\Gamma}}$ are given by and

Figures (2)

  • Figure 1: The images of boundary divisors under the map $h$
  • Figure 2: Two 1-dimensional boundary strata in $\widetilde{\mathcal{M}}_{0,7}$ contracted to points under $h$

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4: Schock
  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • proof : Proof of \ref{['cor:M']}
  • Remark 3.1
  • ...and 25 more