Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An algorithm for the minimal model program in dimension three

Takehiko Yasuda

Abstract

We construct an algorithm for the minimal model program in dimension three over the field of algebraic numbers. As auxiliary results, we also construct algorithms for computing bigraded global Hom modules and for computing Stein factorization.

An algorithm for the minimal model program in dimension three

Abstract

We construct an algorithm for the minimal model program in dimension three over the field of algebraic numbers. As auxiliary results, we also construct algorithms for computing bigraded global Hom modules and for computing Stein factorization.
Paper Structure (13 sections, 18 theorems, 90 equations, 4 algorithms)

This paper contains 13 sections, 18 theorems, 90 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Monograded varieties and bigraded varieties
  3. Monograded varieties and bigraded varieties
  4. Segre isomorphisms
  5. B2M projections and graph morphisms
  6. Homogeneous morphisms
  7. Nefness of canonical divisors
  8. Bigraded global Hom modules
  9. Stein factorization
  10. Computing a homogeneous coordinate ring of $Z$
  11. Computing the graph of $h\colon Y\to Z$
  12. Contraction morphisms
  13. Flips and the minimal model program

Key Result

Theorem 1.1

There is an algorithm, Algorithm algo:MMP, to run the minimal model program in dimension three over $\overline{\mathbb{Q}}$, the field of algebraic numbers. Namely, when a normal $\mathbb{Q}$-factorial projective variety of dimension 3 with only log terminal singularities is given as an input, then that is one of the minimal model program sequences starting from $X$. In particular, either $X_{n}$

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • ...and 30 more