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General elephants for threefold extremal contractions with one-dimensional fibers: exceptional case

Shigefumi Mori, Yuri Prokhorov

Abstract

Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f : (X, C) \to (Z, o)$ such that $C = f^{-1} (o)_{\mathrm{red}}$ and $-K_X$ is $f$-ample. Assume that each irreducible component of $C$ contains at most one point of index $>2$. We prove that a general member $D\in |{-}K_X|$ is a normal surface with Du Val singularities.

General elephants for threefold extremal contractions with one-dimensional fibers: exceptional case

Abstract

Let be a germ of a threefold with terminal singularities along a connected reduced complete curve with a contraction such that and is -ample. Assume that each irreducible component of contains at most one point of index . We prove that a general member is a normal surface with Du Val singularities.

Paper Structure

This paper contains 15 sections, 22 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Low index cases
  4. Extension techniques
  5. Review of KM92
  6. Case $\mathrm{\left(IC\right)}$
  7. Case $\mathrm{\left(IIB\right)}$
  8. Case $\mathrm{\left(k3A\right)}$
  9. Case $\mathrm{\left(kAD\right)}$
  10. Subcase \ref{['scase:kAD:2']}.
  11. Subcase \ref{['scase:kAD:1']}
  12. Case $\mathrm{\left(k2A_2\right)}$
  13. Proof of the main theorem
  14. Examples
  15. $\mathbb{Q}$-conic bundles with irreducible central fiber

Key Result

Theorem 1.1

Let $(X,\, C)$ be an extremal curve germ with irreducible central curve $C$. Then a general member $D\in |{-}K_X|$ is a normal surface with Du Val singularities.

Theorems & Definitions (44)

  • Theorem 1.1: KM92, MP:cb3
  • Theorem 1.2: MP:cb2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • ...and 34 more