On the Monodromy and Galois Group of Conics Lying on Heisenberg Invariant Quartic K3 Surfaces

TL;DR

This paper investigates the algebraic and Galois-theoretic properties of conics on Heisenberg invariant quartic K3 surfaces, revealing the structure of their moduli space and monodromy groups.

Contribution

It analyzes the field of definition and monodromy group of conics on these surfaces, showing the moduli space has 10 irreducible components.

Findings

The field of definition of conics is characterized.

The monodromy group of the conics is determined.

The moduli space has 10 irreducible components.

Abstract

In "Curves on Heisenberg invariant quartic surfaces in projective 3-space", Eklund showed that a general $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant quartic K3 surface contains at least $320$ conics. In this paper we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant quartic K3 surface with a marked conic has $10$ irreducible components.