Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On rational double points over nonclosed fields

Christian Liedtke, Matthew Satriano

TL;DR

This work provides a complete framework for rational double point (RDP) singularities over perfect fields by realizing them as quotients ${\mathbb A}^2_k/G$ with $G$ a finite linearly reductive subgroup scheme of ${\mathbf{SL}}_{2,k}$. It develops a descent theory for classical subgroups and their representations, connects them to the McKay correspondence via graphs, and uses normalizers to classify twists through Galois cohomology. The paper then delivers explicit equations for all split and twisted RDPs across the ADE families, including detailed characteristic-dependent forms and splitting fields, and identifies when no nontrivial twists occur (notably for $E_7$ and $E_8$ over perfect fields). The results unify group-scheme descent, McKay theory, and invariant theory to produce concrete, model-level descriptions of RDPs on nonclosed fields, with potential broad implications for singularity theory and arithmetic geometry.

Abstract

We compute the equations of all rational double point singularities and we determine their types over perfect ground fields $k$ that arise as quotient singularities by finite linearly reductive subgroup schemes of $\textrm{SL}_{2,k}$.

On rational double points over nonclosed fields

TL;DR

This work provides a complete framework for rational double point (RDP) singularities over perfect fields by realizing them as quotients with a finite linearly reductive subgroup scheme of . It develops a descent theory for classical subgroups and their representations, connects them to the McKay correspondence via graphs, and uses normalizers to classify twists through Galois cohomology. The paper then delivers explicit equations for all split and twisted RDPs across the ADE families, including detailed characteristic-dependent forms and splitting fields, and identifies when no nontrivial twists occur (notably for and over perfect fields). The results unify group-scheme descent, McKay theory, and invariant theory to produce concrete, model-level descriptions of RDPs on nonclosed fields, with potential broad implications for singularity theory and arithmetic geometry.

Abstract

We compute the equations of all rational double point singularities and we determine their types over perfect ground fields that arise as quotient singularities by finite linearly reductive subgroup schemes of .

Paper Structure

This paper contains 23 sections, 15 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Descending the classical subgroups and their representations
  3. McKay graphs
  4. Twisted subgroup schemes
  5. Outlook
  6. Classifying twisted forms using the normalizer: Theorem \ref{['thm:NG-and-twists']}
  7. Twisted forms of classical groups: Theorem \ref{['thm:gp-sch-over-k-main-features']}
  8. Definitions and properties of our group schemes over $k$
  9. Showing all representations descend
  10. Normalizers of group schemes
  11. Normalizers and the McKay graph: Theorem \ref{['thm:NGmodGAutMcKay']}
  12. Equations of twisted RDP singularities: Theorem \ref{['tm:twisted-invariants']}
  13. Types $A$ and $B$
  14. $\mathop{\mathrm{char}}\nolimits k\neq2$
  15. $\mathop{\mathrm{char}}\nolimits k=2$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $k$ be a perfect field of characteristic $p\geq0$. Let $G\subset{\mathbf{SL}}_{2,k}$ be a finite linearly reductive group scheme. Then, ${{\mathbb A}}^2_k/G$ is given by one of the following equations: More precisely:

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • proof : Proof of Theorem \ref{['thm:NG-and-twists']}
  • Corollary 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • ...and 21 more