Monodromy in the space of symmetric cubic surfaces with a line

TL;DR

This paper investigates the monodromy and Galois groups of lines on symmetric cubic surfaces, revealing that symmetry constraints lead to a Klein 4-group Galois group and explicit formulas for lines.

Contribution

It establishes the structure of the moduli space as an arithmetic quotient and computes the Galois group for symmetric cubic surfaces, linking symmetry to algebraic solutions.

Findings

Galois group is the Klein 4-group for symmetric cubic surfaces

Explicit formulas for lines on symmetric cubic surfaces are derived

Moduli space is an arithmetic quotient of the complex hyperbolic line

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.