Smoothness of stabilisers in generic characteristic

Technical Summary: Smoothness of Stabilisers in Generic Characteristic

Problem Statement
The paper addresses the smoothness of stabilisers (centralisers and normalisers) for algebraic group actions in positive characteristic. While it is well-established that for split reductive groups acting on themselves or their Lie algebras, centralisers of elements are smooth for "very good" primes, these results often rely on specific classifications or restrictive hypotheses. The authors aim to generalize these findings to arbitrary finitely presented affine group schemes $G$ acting on separated schemes $X$ of finite type over a commutative unital ring $R$. Specifically, they seek to prove that there exists a threshold prime $p_0$ such that for any algebraically closed field $k$ of characteristic $p \ge p_0$, the centraliser of any closed subscheme $Y \subseteq X_k$ is smooth. They also investigate the normaliser of $Y$, noting that the bound for the normaliser may depend on $Y$, whereas the bound for the centraliser depends only on the complexity of the action.

Methodology
The authors employ a novel combination of first-order model theory and computational algebraic geometry, specifically:

1. The Lefschetz Principle: They utilize a version of the Lefschetz principle from model theory, which asserts that a first-order sentence true in algebraically closed fields of characteristic 0 is also true in algebraically closed fields of sufficiently large positive characteristic.
2. Gröbner Basis Techniques: To apply the Lefschetz principle, the authors must express geometric properties (such as smoothness and ideal membership) as first-order sentences in the language of rings ($L_{ring}$). They achieve this by quantifying over Gröbner bases of bounded degree.
3. Complexity Measures: The authors introduce precise notions of "complexity" for schemes, morphisms, and group actions. This involves bounding the degrees of polynomials defining ideals, the degrees of maps in comorphisms, and the degrees of structure maps in Hopf algebras (comultiplication, antipode, counit).
   • They define $d$-bounded Gröbner bases and show that properties like being a Gröbner basis, the dimension of a scheme, and ideal membership can be encoded by first-order formulas with a number of free variables depending only on the bound $d$.
   • They introduce $G$-complexity and $(G, \Delta)$-complexity to bound the interaction between the group action and the subschemes.

Key Contributions and Results
The paper establishes the following main theorems:

• Theorem 1.1 (Smooth Centralisers): Let $G$ be an algebraic $R$-group and $X$ a separated $G$-scheme of finite type over $R$. There exists a prime $p_0$ such that for any $R$-field $k$ of characteristic $p \ge p_0$, the centraliser $C_{G_k}(Y)$ is smooth for every closed subscheme $Y \subseteq X_k$. Crucially, $p_0$ depends only on the complexity of $G$ and $X$, not on the specific subscheme $Y$.
• Theorem 1.2 (Smooth Normalisers): Let $G$ and $X$ be as above, and let $Y$ be a specific closed subscheme of $X$. There exists a prime $p_1$ (depending on $Y$) such that for $p \ge p_1$, the normaliser $N_{G_k}(Y_k)$ is smooth. The authors note that the dependence of $p_1$ on $Y$ is necessary and cannot be removed in the general case.
• Corollary 1.3 (Modules): For a finitely generated projective $G$-module $V$, there exists a prime $p_V$ such that for $p \ge p_V$, the centralisers and normalisers of any $k$-subspace $W \subseteq V_k$ are smooth.
• Theorem 5.2 (Modular Kostant-Kirillov-Souriau): The authors apply their main results to symplectic geometry. They prove that for an algebraic group $G$ over an algebraically closed field of sufficiently large characteristic $p$, the Kostant-Kirillov-Souriau (KKS) theorem holds: the dual of the Lie algebra $\mathfrak{g}^*$ decomposes into a disjoint union of symplectic varieties, where each variety is a coadjoint orbit. This relies on the fact that for large $p$, the stabilisers of points in $\mathfrak{g}^*$ are smooth, ensuring the separability of the orbit map.

Significance and Scope
The paper claims to be the first to successfully carry out the program suggested by Schoutens (2000) of using the Lefschetz principle combined with Gröbner bases to solve specific problems in algebraic geometry regarding generic smoothness.

• Generality: The results extend beyond the affine case to quasi-projective varieties and other schemes of finite type. They apply to arbitrary algebraic groups over rings, not just split reductive groups over fields.
• Uniformity: A significant finding is that the bound for the smoothness of centralisers is uniform across all closed subschemes of a given complexity, whereas normalisers require a bound dependent on the specific subscheme.
• Limitations: The authors explicitly note that the results fail without complexity bounds. They provide counterexamples where the action is not of bounded complexity (e.g., involving Frobenius twists or infinite disjoint unions of modules) where stabilisers remain non-smooth regardless of the characteristic.
• Application: The primary application demonstrated is the extension of the KKS theorem to large positive characteristics, providing a structural decomposition of the Poisson variety $\mathfrak{g}^*$ into symplectic coadjoint orbits, a result that generally fails in small characteristics due to non-separable orbit maps.

The paper does not propose new experimental applications or future research directions beyond the immediate mathematical implications of these structural theorems. Its contribution lies in establishing a robust framework for transferring smoothness properties from characteristic 0 to large positive characteristics via model-theoretic and algorithmic algebraic methods.