Configurations of 10 points and their incidence varieties

TL;DR

This paper classifies the components of incidence varieties for configurations of up to 10 points in the projective plane, revealing their birational types and using computer algebra to analyze special arrangements called superfigurations.

Contribution

It provides a detailed classification of realization spaces for point configurations up to 10 points, identifying their birational types and introducing a computational approach for superfigurations.

Findings

Each realization space component is birational to a projective space, genus 1 curve, or K3 surface.

Classification of 163 superfigurations using computer algebra.

Complete description of incidence varieties for configurations of up to 10 points.

Abstract

Incidence varieties are spaces of $n$-tuples of points in the projective plane that satisfy a given set of collinearity conditions. We classify the components of incidence varieties and realization moduli spaces associated to configurations of up to 10 points, up to birational equivalence. We show that each realization space component is birational to a projective space, a genus 1 curve, or a K3 surface. To do this, we reduce the problem to a study of 163 special arrangements called superfigurations. Then we use computer algebra to describe the realization space of each superfiguration.