Semistable reductions and minimalities of invariants for group scheme actions on projective schemes
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| Format: | Preprint |
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2026
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| _version_ | 1866914568600551424 |
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| author | Gotou, Rin Okuyama, Yûsuke |
| author_facet | Gotou, Rin Okuyama, Yûsuke |
| contents | Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\operatorname{SSRL}_x$ in the translation space $\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumption, also their non-emptiness. In the dynamical setting which has been studied by Szpiro--Tepper--Williams and Rumely, the coincidence result is already new in higher dimensions, and the non-emptiness result includes Rumely's $1$-dimensional result at least in the spherical complete case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25659 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semistable reductions and minimalities of invariants for group scheme actions on projective schemes Gotou, Rin Okuyama, Yûsuke Algebraic Geometry Dynamical Systems Primary 14L24, Secondary 14L30, 32P05 Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\operatorname{SSRL}_x$ in the translation space $\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumption, also their non-emptiness. In the dynamical setting which has been studied by Szpiro--Tepper--Williams and Rumely, the coincidence result is already new in higher dimensions, and the non-emptiness result includes Rumely's $1$-dimensional result at least in the spherical complete case. |
| title | Semistable reductions and minimalities of invariants for group scheme actions on projective schemes |
| topic | Algebraic Geometry Dynamical Systems Primary 14L24, Secondary 14L30, 32P05 |
| url | https://arxiv.org/abs/2604.25659 |