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Infinitesimal rational actions

Bianca Gouthier

Abstract

For any $k$-group scheme of finite type $G$ acting rationally on a $k$-variety $X$, if the action is generically free then the dimension of $\mathrm{Lie}(G)$ is upper bounded by the dimension of the variety. We show that this is the only obstruction when $k$ is a perfect field and $G$ is infinitesimal commutative trigonalizable. We also give necessary conditions to have faithful rational actions of infinitesimal commutative trigonalizable group schemes on varieties.

Infinitesimal rational actions

Abstract

For any -group scheme of finite type acting rationally on a -variety , if the action is generically free then the dimension of is upper bounded by the dimension of the variety. We show that this is the only obstruction when is a perfect field and is infinitesimal commutative trigonalizable. We also give necessary conditions to have faithful rational actions of infinitesimal commutative trigonalizable group schemes on varieties.
Paper Structure (14 sections, 35 theorems, 166 equations)

This paper contains 14 sections, 35 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Finite group schemes
  3. Finite commutative group schemes
  4. The socle of a finite group scheme
  5. Trigonalizable group schemes
  6. Young diagrams for commutative unipotent group schemes of height one
  7. Algebraic description of infinitesimal commutative unipotent group schemes
  8. Actions of finite group schemes
  9. Actions and rational actions
  10. Actions of finite group schemes and module algebras
  11. Nilpotent derivations and p-basis
  12. Generically free rational actions
  13. Proof of the main Theorem
  14. Faithful rational actions

Key Result

Proposition 1.2

Let $G$ be an infinitesimal $k$-group scheme with commutative trigonalizable Frobenius kernel $\ker(F_G)\simeq \ker(F_G)^u\times_k\ker(F_G)^d$ and $X$ be a $k$-variety of dimension $n$. If there exists a faithful rational $G$-action on $X$, then $s=dim_k(\mathop{\mathrm{Lie}}\nolimits (\ker(F_{G})^d

Theorems & Definitions (115)

  • Conjecture 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1: Absolute Frobenius
  • Definition 2.2: Relative Frobenius
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5: Lie algebra
  • Remark 2.6
  • ...and 105 more