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Semistable reduction over thick log points

Alexander E. Motzkin, Michael Temkin

Abstract

We establish a version of a semistable reduction theorem over a log point with a non-trivial nilpotent structure. In order to do this we extend the classical desingularization theories to non-reduced schemes with generically principal nilradical.

Semistable reduction over thick log points

Abstract

We establish a version of a semistable reduction theorem over a log point with a non-trivial nilpotent structure. In order to do this we extend the classical desingularization theories to non-reduced schemes with generically principal nilradical.
Paper Structure (38 sections, 22 theorems, 8 equations)

This paper contains 38 sections, 22 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Log smoothness over thick points
  3. Main results
  4. Outline of the paper
  5. Conventions
  6. Desingularization on principally thick manifolds
  7. Hironaka's resolution of non-reduced schemes
  8. Thick manifolds
  9. Hironaka's theorem
  10. Strong resolution
  11. Principally thick manifolds
  12. Other characterizations
  13. Desingularization of generically nilprincipal varieties
  14. Resolution on ptms
  15. Principalization on ptms
  16. ...and 23 more sections

Key Result

Theorem 1.2.2

Let $k$ be a field of characteristic zero and $B={\rm Spec}(k[\pi]/(\pi^n))$. Then there exists a procedure ${\mathcal{F}}$ compatible with smooth morphisms $Y\to X$ which obtains as an input a generically smooth morphism of finite type $X\to B$ and outputs a modification ${\mathcal{F}}(X)\colon X'\

Theorems & Definitions (59)

  • Remark 1.1.1
  • Remark 1.1.2
  • Remark 1.2.1
  • Theorem 1.2.2
  • Remark 1.2.3
  • Definition 2.1.2
  • Lemma 2.1.3
  • proof
  • Definition 2.1.5
  • Theorem 2.1.6: Hironaka
  • ...and 49 more