Moduli spaces of quadratic complexes and their singular surfaces

TL;DR

This paper constructs and relates moduli spaces of quadratic line complexes and their singular surfaces, revealing that most cosingular complexes form curves, thus advancing the understanding of their geometric classification.

Contribution

It introduces explicit moduli spaces for quadratic complexes and their singular surfaces, and studies the morphism connecting these spaces, providing new insights into their geometric structure.

Findings

The moduli space of quadratic line complexes is constructed.

A morphism from complexes to singular surfaces is established.

Most varieties of cosingular complexes are curves.

Abstract

We construct the coarse moduli space $\cM_{qc}(\sigma)$ of quadratic line complexes with a fixed Segre symbol $\sigma$ as well as the moduli space $\cM_{ss}(\sigma)$ of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism $\pi: \cM_{qc}(\sigma) \ra \cM_{ss}(\sigma)$. Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.