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arXiv · 2507.09829

Configurations of 10 points and their incidence varieties

Kelly Isham·Nathan Kaplan·Sam Kimport·Rachel Lawrence·Luke Peilen·Max Weinreich
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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.

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Kelly Isham, Nathan Kaplan, Sam Kimport, Rachel Lawrence, Luke Peilen, Max Weinreich. 2025-07-13. Configurations of 10 points and their incidence varieties. https://arxiv.org/abs/2507.09829

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