Configurations of 10 points and their incidence varieties

2025/07/13 by Kelly Isham, Isham, Kelly, Nathan O. Kaplan +9
Computer Science · Engineering · Mathematics · #14J10 #14N20 #51A20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2507.09829

openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Line configurations and K3 surfaces
• An algorithm for counting arcs in higher-dimensional projective space
• Mnev-Sturmfels universality for schemes
• Curves of genus 2 and Desargues configurations
• Universality theorems for configuration spaces of planar linkages

Related

• Incidence strata of affine varieties with complex multiplicities
• Moduli Spaces of Arrangements of 10 Projective Lines with Quadruple Points
• Varieties of Lines in 3-Space
• The Cremona-Richmond Configuration revisited and generalized
• A version of Kapranov's Chow quotient and smooth moduli space of point configurations in \PP²