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arxiv: 2604.25659 · v2 · pith:ZRN3NSXCnew · submitted 2026-04-28 · 🧮 math.AG · math.DS

Semistable reductions and minimalities of invariants for group scheme actions on projective schemes

Rin Gotou , Y\^usuke Okuyama This is my paper

Pith reviewed 2026-05-19 17:56 UTC · model grok-4.3

classification 🧮 math.AG math.DS
keywords semistable reductionminimal invariantsgroup scheme actionsprojective schemesnon-archimedean fieldsBruhat-Tits buildingtranslation space
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The pith

For every point on a projective scheme under a reductive group action, the minimal invariant locus coincides with the semistable reduction translation locus.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines two loci inside the translation space attached to a reductive group scheme G acting on a flat projective scheme X over the ring of integers of a complete non-archimedean field K. It proves that the minimal invariant locus MinInvLoc_x and the semistable reduction translation locus SSRL_x are identical for every K-point x. The same loci are shown to be non-empty once K satisfies an additional mild completeness condition. The coincidence already supplies a new statement in higher-dimensional dynamical settings, while the non-emptiness statement recovers the one-dimensional result of Rumely in the spherical-complete case.

Core claim

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, the minimal invariant locus MinInvLoc_x and the semistable reduction translation locus SSRL_x in the translation space BT_G(K) coincide, and under a mild completeness assumption they are non-empty.

What carries the argument

The pair of loci MinInvLoc_x and SSRL_x inside the translation space BT_G(K), a variant of the Bruhat-Tits building for G_K, whose equality and non-emptiness are established for every K-point x.

If this is right

  • The coincidence supplies a new statement for dynamical systems in dimensions greater than one.
  • Non-emptiness recovers Rumely's one-dimensional result at least when K is spherically complete.
  • The two loci can be treated interchangeably when studying semistable reductions of invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identification may streamline calculations of minimal invariants in higher-dimensional arithmetic geometry.
  • Similar loci could be defined and compared for actions of non-reductive groups or over fields with weaker valuation properties.
  • The result suggests a uniform way to locate minimal models inside Bruhat-Tits buildings for projective schemes.

Load-bearing premise

The field K satisfies a mild completeness assumption beyond being algebraically closed, complete, non-archimedean, and non-trivially valued.

What would settle it

An explicit K-point x on such an X for which MinInvLoc_x differs from SSRL_x, or for which both loci are empty even when the mild completeness condition holds.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that for a reductive group scheme G acting on a flat projective scheme X over the ring of integers of an algebraically closed, complete, non-archimedean, non-trivially valued field K, the minimal invariant locus MinInvLoc_x and the semistable reduction translation locus SSRL_x coincide inside the variant Bruhat-Tits translation space BT_G(K) for every K-point x of X. Under a mild additional completeness assumption on K, both loci are shown to be non-empty. The coincidence is presented as new in higher dimensions relative to the dynamical results of Szpiro--Tepper--Williams and Rumely, while the non-emptiness recovers Rumely's one-dimensional result at least in the spherical-complete case.

Significance. If the identification and non-emptiness hold, the work supplies a concrete link between minimal invariants and semistable reductions via Bruhat-Tits geometry for group-scheme actions on projective schemes. The extension of the coincidence statement to higher dimensions and the conditional non-emptiness result constitute a genuine advance over the cited one-dimensional dynamical literature.

minor comments (3)
  1. The abstract refers to a 'mild completeness assumption' on K without stating its precise content; this should be formulated explicitly in the introduction or in the statement of the main theorem.
  2. Notation for the variant Bruhat-Tits space BT_G(K) and the two loci MinInvLoc_x, SSRL_x is introduced in the abstract but should be recalled with a brief reminder of their definitions at the beginning of the section containing the main theorem.
  3. The manuscript would benefit from a short comparison table or diagram illustrating how the new loci relate to classical semistable loci or to the original Rumely construction in dimension one.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. We are pleased that the referee views the identification of MinInvLoc_x with SSRL_x and the conditional non-emptiness result as a genuine advance over the one-dimensional dynamical literature.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines MinInvLoc_x and SSRL_x independently as loci inside the variant Bruhat-Tits space BT_G(K) for a reductive group scheme action on a projective scheme X. The central claim is a theorem establishing their coincidence for every K-point x, with non-emptiness following from an explicitly stated mild completeness assumption on K. No equations, self-citations, or constructions in the abstract or context reduce the equality to a definitional identity, fitted parameter, or imported ansatz. The derivation chain treats the loci as distinct objects whose relationship is proven rather than presupposed, making the result self-contained against external benchmarks in algebraic geometry and non-archimedean geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper introduces two new loci whose definitions rest on the standard structure of Bruhat-Tits buildings for reductive groups over non-archimedean fields and the flatness of the projective scheme; no free parameters or invented physical entities are apparent from the abstract.

axioms (2)
  • standard math The translation space BT_G(K) is a variant of the Bruhat-Tits building associated to the reductive group scheme G over the non-archimedean field K.
    Invoked when defining the loci inside BT_G(K).
  • domain assumption X is a flat projective scheme over the ring of K-integers with G acting on it.
    Setup assumption stated in the abstract for the action and the K-points.
invented entities (2)
  • MinInvLoc_x no independent evidence
    purpose: Minimal invariant locus for the G-action at the K-point x
    Newly defined object whose properties are studied.
  • SSRL_x no independent evidence
    purpose: Semistable reduction translation locus in BT_G(K)
    Newly defined object whose properties are studied.

pith-pipeline@v0.9.0 · 5715 in / 1319 out tokens · 55397 ms · 2026-05-19T17:56:36.555784+00:00 · methodology

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