The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915132751216640 |
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| author | Jackson, Joshua Swinarski, David |
| author_facet | Jackson, Joshua Swinarski, David |
| contents | Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a ``worst'' destabilizing 1-parameter subgroup $λ^{*}$. It is natural to ask: what are the worst 1-PS for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric rational curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_00458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity Jackson, Joshua Swinarski, David Algebraic Geometry 14L24 (Primary) 14H20, 14Q05 (Secondary) Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a ``worst'' destabilizing 1-parameter subgroup $λ^{*}$. It is natural to ask: what are the worst 1-PS for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric rational curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples. |
| title | The worst destabilizing 1-parameter subgroup for toric rational curves with one unibranch singularity |
| topic | Algebraic Geometry 14L24 (Primary) 14H20, 14Q05 (Secondary) |
| url | https://arxiv.org/abs/2502.00458 |