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Toward the classification of threefold extremal contractions with one-dimensional fibers

Shigefumi Mori, Yuri Prokhorov

TL;DR

The paper addresses the problem of classifying threefold extremal contractions with terminal singularities along a reducible central fiber $C$. It develops a local-to-global framework using index-one covers, $C$-laminal ideals, and $\ell$-exact sequences, combined with deformation methods to glue irreducible-component models. The main result is a near-complete table describing admissible configurations (types of components, non-Gorenstein points, and contraction type: flipping, divisorial, or $\mathbb{Q}$-conic bundle), with most cases realized via explicit constructions or deformations. This advances the threefold minimal model program by clarifying how reducible central fibers can occur and how their global contractions are organized, aligning with the General Elephant spirit of Reid’s program and providing constructive methods for many extremal germs.

Abstract

An extremal curve germ is a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve~$C$ such that there exists a $K_X$-negative contraction $f : X \to Z$ with~$C$ being a fiber. We give a rough classification of extremal curve germs with reducible central curve~$C$.

Toward the classification of threefold extremal contractions with one-dimensional fibers

TL;DR

The paper addresses the problem of classifying threefold extremal contractions with terminal singularities along a reducible central fiber . It develops a local-to-global framework using index-one covers, -laminal ideals, and -exact sequences, combined with deformation methods to glue irreducible-component models. The main result is a near-complete table describing admissible configurations (types of components, non-Gorenstein points, and contraction type: flipping, divisorial, or -conic bundle), with most cases realized via explicit constructions or deformations. This advances the threefold minimal model program by clarifying how reducible central fibers can occur and how their global contractions are organized, aligning with the General Elephant spirit of Reid’s program and providing constructive methods for many extremal germs.

Abstract

An extremal curve germ is a germ of a threefold with terminal singularities along a connected reduced complete curve~ such that there exists a -negative contraction with~ being a fiber. We give a rough classification of extremal curve germs with reducible central curve~.
Paper Structure (10 sections, 38 theorems, 148 equations, 1 table)

This paper contains 10 sections, 38 theorems, 148 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Case $\mathrm{\left(IC\right)}$+ $\mathrm{\left(k2A\right)}$
  5. Cases $\mathrm{\left(k2A\right)}$+$\mathrm{\left(kAD\right)}$ and $\mathrm{\left(k2A\right)}$+$\mathrm{\left(k3A\right)}$.
  6. (Case $\mathrm{\left(kAD\right)}$ with $m\ge 5$)
  7. Proof of Theorem \ref{['thm:main']}
  8. Examples
  9. Construction
  10. General facts on extremal curve germs

Key Result

Theorem 1.1

Let $(X,C)$ be an extremal curve germ with reducible central curve where the $C_i$ are irreducible components. Then one of the cases in Table tab:table below is possible, where $\operatorname{Sing}^{\mathrm{nG}}(X)$ is the set of non-Gorenstein points of $X$, the notation $n\times(\mathrm{\textasteriskcentered})$ means that $(X,C)$ has exactly $n$ components of typ

Theorems & Definitions (90)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.3.1: Mori:flip
  • Definition 2.3.2: Mori:flip
  • Proposition 2.3.3: cf. KM92
  • Proposition 2.3.4
  • proof : Sketch of the proof
  • Proposition 2.3.5
  • proof : Sketch of the proof
  • ...and 80 more