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

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.

• The paper proves that the minimal invariant locus coincides with the semistable reduction translation locus for all points under reductive group actions on projective schemes.
• It introduces a convex analytic formulation of order functions that links geometric invariant theory stability to the metric geometry of Bruhat-Tits buildings.
• The work extends results from arithmetic dynamics to higher dimensions, providing concrete criteria for non-emptiness and computability of invariant loci.

Semistable Reductions, Minimality of Invariants, and Group Scheme Actions on Projective Schemes

Introduction and Motivation

The paper "Semistable reductions and minimalities of invariants for group scheme actions on projective schemes" (2604.25659) investigates the relationship between geometric invariant theory (GIT) and Bruhat-Tits buildings in the study of flat projective schemes with group scheme actions over non-archimedean fields. Specifically, it concerns reductive group scheme actions on projective schemes and examines semistability and associated minimality conditions through explicit loci defined in the (arithmetic) translation spaces derived from the group schemes.

This work generalizes crucial components developed in the context of arithmetic dynamics (notably for $\PGL_{n+1}$, $\GL_{n+1}$, and $\SL_{n+1}$ actions on parameter spaces of endomorphisms), extends foundational results for $n=1$ to higher dimensions, and abstracts minimal invariant loci and semistable reduction translation loci from dynamical formulations to general group-theoretic and geometric frameworks.

Framework and Definitions

Let $K$ be an algebraically closed, complete, non-archimedean field, with non-trivial valuation, and $O_K$ its ring of integers. Consider a flat, projective $O_K$-scheme $X$ with an ample, $G$-linearized line bundle $L$, and a reductive $\GL_{n+1}$0-group scheme $\GL_{n+1}$1 acting on $\GL_{n+1}$2.

Key objects are:

1. The translation space $\GL_{n+1}$3, a combinatorial variant of the Bruhat-Tits building associated to $\GL_{n+1}$4;
2. For each $\GL_{n+1}$5-point $\GL_{n+1}$6:
   • The minimal invariant locus $\GL_{n+1}$7 in $\GL_{n+1}$8, defined as the minimal locus of certain 'order functions' determined by $\GL_{n+1}$9-invariant global sections;
   • The semistable reduction translation locus $\SL_{n+1}$0, consisting of classes $\SL_{n+1}$1 such that $\SL_{n+1}$2 specializes to a semistable $\SL_{n+1}$3-point under the reduction map;
3. The augmented Euclidean building $\SL_{n+1}$4, which is a $\SL_{n+1}$5-space and provides a natural metric structure extending $\SL_{n+1}$6.

All GIT (semi)stability notions are used in the strongest technical sense (e.g., via Hilbert-Mumford criterion), and the lifting of maximal tori through reduction is fundamental for the global analysis of semistable loci.

Main Results

Coincidence of Minimal Invariant and Semistable Reduction Translation Loci

A principal result is the coincidence of the minimal invariant locus and the semistable reduction translation locus for arbitrary reductive group actions on projective schemes under mild completeness conditions:

For all $\SL_{n+1}$7, $\SL_{n+1}$8. Furthermore, when the stable reduction translation locus $\SL_{n+1}$9 is non-empty (i.e., the reduction of $n=1$0 is stable), $n=1$1 and $n=1$2 is a singleton.

In dynamical parlance, this generalizes the equality of the minimal resultant locus and the semistable reduction locus to higher-dimensional parameter spaces and more general group actions.

Non-emptiness of Loci for Complete Buildings

If the Euclidean augmentation $n=1$3 is metrically complete—a condition satisfied, e.g., for spherically complete $n=1$4 and certain group-theoretic types—then for $n=1$5 in the stable locus, $n=1$6 is guaranteed to be non-empty. This extends Rumely's non-emptiness result for $n=1$7 and highlights the robust connection between metric properties of the building and GIT-theoretic stability.

Structure and Computation in Dynamics

For explicit parameter spaces arising in dynamics (e.g., $n=1$8, acted on by various group schemes), detailed criteria and calculations are given for minimal and semistable loci. The construction applies Schur functors and explicit GIT criteria (cf. the stability of plane cubics under group actions) to compute these sets in concrete examples.

Methods and Technical Contributions

The analysis hinges on several technical devices:

• A convex analytic formulation for the order functions underlying the minimal invariant locus, which are extended naturally to the Euclidean building. This connects GIT-theoretic notions to the metric and convex geometry of buildings.
• The application of the Hilbert-Mumford criterion in an explicit, combinatorial way through weight decompositions for group tori and the computation of associated weight polytopes.
• Rigorous control over lifts of maximal tori from special fibers to the generic point, facilitating comparison between reduction-theoretic and minimality-theoretic constructs.
• Utilization of convexity and completeness results in $n=1$9-spaces to deduce non-emptiness and minimal structure properties for the loci.

The paper systematically organizes the translation between geometric invariant theory, arithmetic geometry, and the combinatorial geometry of buildings, resolving minimality and reduction questions in the general setting of group schemes and projective schemes.

Implications and Future Directions

Practically, these results yield precise, verifiable criteria for GIT (semi)stability and minimality in parameter spaces subjected to group actions, with direct applications to arithmetic dynamics, moduli problems, and the theory of invariants. The identification and classification of minimal invariant loci in buildings facilitate efficient reduction algorithms and moduli interpretations for classifying orbits under group schemes.

Theoretically, the bridging of GIT, reduction theory, and the metric geometry of buildings sets groundwork for new directions in geometric representation theory, non-archimedean geometry, and moduli space compactifications. The explicit description of loci via convex analytic methods may suggest further extensions to non-split or non-reductive group schemes, higher-level stratifications, and connections to Berkovich and analytic geometry.

Questions regarding the metric completeness of $K$0 for general $K$1 and $K$2, and the consequences for the structure and computation of minimal loci, are highlighted as sources for further research and possible generalization.

Conclusion

The paper rigorously demonstrates that, for reductive group actions on projective schemes over non-archimedean fields, the minimal invariant locus in the associated translation space coincides with the semistable reduction translation locus, and is non-empty under a completeness assumption on the associated Euclidean building. The analytic and geometric results presented unify previous work from arithmetic dynamics and geometric invariant theory, and provide a structural foundation for explicit computations and further theoretical developments in the study of orbits, reductions, and invariants under group scheme actions.