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Monodromy in the space of symmetric cubic surfaces with a line

Thomas Brazelton, Sidhanth Raman

TL;DR

This work studies the monodromy of lines on cubic surfaces restricted by $S_4$ symmetry. By combining moduli-theoretic uniformization (via a complex hyperbolic line) with Eisenstein lattice and representation-theoretic techniques, the authors show the symmetric moduli space is an arithmetic quotient and the monodromy group of the 27 lines over the symmetric locus is the Klein four group $K_4$, lying inside the Weyl group $W(E_6)$. They further show the incidence fiber splits into 12 components under $K_4$ and provide explicit radical formulas for the lines, defined over a Klein four extension $\mathbb{Q}(\sqrt{\alpha},\sqrt{\beta})$. The analysis reveals that symmetry constrains the Galois group from the generic $S_4$ to $K_4$, while Hodge-theoretic constraints limit monodromy in the parameter space to a group of order 96, which is not sufficient to determine the symmetric monodromy. The results illuminate how equivariant geometry and period maps interact with classical line geometry, offering a framework for studying monodromy in other symmetry-restricted enumerative problems and yielding concrete formulas for lines in the symmetric case.

Abstract

We explore the enumerative problem of finding lines on cubic surfaces defined by symmetric polynomials. We prove that the moduli space of symmetric cubic surfaces is an arithmetic quotient of the complex hyperbolic line, and determine constraints on the monodromy group of lines on symmetric cubic surfaces arising from Hodge theory and geometry of the associated cover. This interestingly fails to pin down the entire Galois group. Leveraging computations in equivariant line geometry and homotopy continuation, we prove that the Galois group is the Klein 4-group. This means that, despite a general cubic surface admitting no formula in radicals for its lines, an $S_4$-symmetric cubic does; we work out these formulas explicitly. This is the first computation in what promises to be an interesting direction of research: studying monodromy in classical enumerative problems restricted by a finite group of symmetries.

Monodromy in the space of symmetric cubic surfaces with a line

TL;DR

This work studies the monodromy of lines on cubic surfaces restricted by symmetry. By combining moduli-theoretic uniformization (via a complex hyperbolic line) with Eisenstein lattice and representation-theoretic techniques, the authors show the symmetric moduli space is an arithmetic quotient and the monodromy group of the 27 lines over the symmetric locus is the Klein four group , lying inside the Weyl group . They further show the incidence fiber splits into 12 components under and provide explicit radical formulas for the lines, defined over a Klein four extension . The analysis reveals that symmetry constrains the Galois group from the generic to , while Hodge-theoretic constraints limit monodromy in the parameter space to a group of order 96, which is not sufficient to determine the symmetric monodromy. The results illuminate how equivariant geometry and period maps interact with classical line geometry, offering a framework for studying monodromy in other symmetry-restricted enumerative problems and yielding concrete formulas for lines in the symmetric case.

Abstract

We explore the enumerative problem of finding lines on cubic surfaces defined by symmetric polynomials. We prove that the moduli space of symmetric cubic surfaces is an arithmetic quotient of the complex hyperbolic line, and determine constraints on the monodromy group of lines on symmetric cubic surfaces arising from Hodge theory and geometry of the associated cover. This interestingly fails to pin down the entire Galois group. Leveraging computations in equivariant line geometry and homotopy continuation, we prove that the Galois group is the Klein 4-group. This means that, despite a general cubic surface admitting no formula in radicals for its lines, an -symmetric cubic does; we work out these formulas explicitly. This is the first computation in what promises to be an interesting direction of research: studying monodromy in classical enumerative problems restricted by a finite group of symmetries.

Paper Structure

This paper contains 30 sections, 45 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Paper structure
  4. Acknowledgements
  5. Notation
  6. Moduli constructions
  7. Parameter space of cubic surfaces
  8. Marked parameter space of cubic surfaces
  9. Parameter space of symmetric cubic surfaces
  10. Symmetry and stability
  11. Reviewing Allcock--Carlson--Toledo
  12. Basic cohomology knowledge
  13. The module structure on cohomology
  14. Moduli of framed cubic surfaces
  15. The period mapping
  16. ...and 15 more sections

Key Result

Theorem 1.1

There are analytic isomorphisms of orbifolds $\mathcal{S} \cong P\Gamma \backslash (\mathbb{C}\mathbb{H}^1 - \mathcal{H}^{S_4})$ and $\mathcal{S}^s \cong P\Gamma \backslash \mathbb{CH}^1$, where $\Gamma < \mathrm{U}(1,1)$ is an arithmetic lattice. Moreover the inclusion of moduli spaces $\mathcal{S}

Figures (1)

  • Figure 1: Left: the Cayley nodal cubic surface. Right: the tricuspidal cubic surface

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.4: Hilbert, hilbertVollen
  • Lemma 2.5: AllcockCarlsonToledo
  • Proposition 2.6
  • Definition 2.7
  • ...and 83 more