2023/08/14 by Aleksei Ilin, Ilin, Aleksei, Joel Kamnitzer +7 · 3 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2308.06880
The space \ftn = \Cn/\C of n points on the line modulo translation has a natural compactification \ftn as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within \ftn , we have the space Fn = \Cn ∖ Δ/ \C of n distinct points. We introduce a natural compatification Fn along with a map Fn → \ftn , whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both \ftn and Fn, are special fibers of 1-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus 0 curves with n+2 marked points. We find combinatorial models for the real loci \ftn(\BR) and Fn(\BR) . Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to Fn(ℝ) gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.