Moduli Spaces of Arrangements of 10 Projective Lines with Quadruple Points

TL;DR

This paper classifies the moduli spaces of arrangements of 10 projective lines with quadruple points, revealing their potential disconnectedness and providing explicit equations for certain arrangements.

Contribution

It offers a detailed classification of these moduli spaces and introduces defining equations for arrangements with reducible moduli spaces after complex conjugation.

Findings

Moduli spaces can have 3 or 4 disconnected components.

Explicit equations are provided for arrangements with reducible moduli spaces.

Some arrangements' moduli spaces remain reducible after quotienting by complex conjugation.

Abstract

We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We also present defining equations to those arrangements whose moduli spaces are still reducible after taking quotients of complex conjugations.