Stacks, Monodromy and Symmetric Cubic Surfaces

TL;DR

This paper explores the influence of symmetries on monodromy groups in enumerative geometry, using moduli stacks to better understand the structure of cubic surfaces with symmetric configurations.

Contribution

It introduces a stack-based framework to analyze monodromy groups affected by symmetries, providing new insights into cubic surfaces and their lines.

Findings

Monodromy groups are significantly shaped by symmetry constraints.

Moduli stacks offer a powerful approach to study geometric monodromy.

Examples demonstrate the impact of symmetry on cubic surface configurations.

Abstract

We investigate monodromy groups arising in enumerative geometry, with a particular focus on how these groups are influenced by prescribed symmetries. To study these phenomena effectively, we work in the framework of moduli stacks rather than moduli spaces. This perspective proves broadly useful for understanding and constructing monodromy. We illustrate these ideas through several examples, with special attention to the 27 lines on a cubic surface, assuming the surface admits a given symmetry group.