The moduli space of cactus flower curves and the virtual cactus group
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929347367010304 |
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| author | Ilin, Aleksei Kamnitzer, Joel Li, Yu Przytycki, Piotr Rybnikov, Leonid |
| author_facet | Ilin, Aleksei Kamnitzer, Joel Li, Yu Przytycki, Piotr Rybnikov, Leonid |
| contents | The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ 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 $ \ft_n $, we have the space $ F_n = \C^n \setminus Δ/ \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, 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 $ \overline \ft_n(\BR) $ and $ \overline F_n(\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 $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06880 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The moduli space of cactus flower curves and the virtual cactus group Ilin, Aleksei Kamnitzer, Joel Li, Yu Przytycki, Piotr Rybnikov, Leonid Algebraic Geometry Group Theory Representation Theory The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ 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 $ \ft_n $, we have the space $ F_n = \C^n \setminus Δ/ \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, 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 $ \overline \ft_n(\BR) $ and $ \overline F_n(\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 $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group. |
| title | The moduli space of cactus flower curves and the virtual cactus group |
| topic | Algebraic Geometry Group Theory Representation Theory |
| url | https://arxiv.org/abs/2308.06880 |