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Blow-ups of three-dimensional toric singularities

S. A. Kudryavtsev

Abstract

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions $f\colon (Y,E)\to (X\ni P)$ are described provided that $\Exc f=E$ is an irreducible divisor, $(X\ni P)$ is a toric terminal singularity, $f(E)$ is a toric subvariety and $Y$ has canonical singularities.

Blow-ups of three-dimensional toric singularities

Abstract

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions are described provided that is an irreducible divisor, is a toric terminal singularity, is a toric subvariety and has canonical singularities.

Paper Structure

This paper contains 6 sections, 45 theorems, 55 equations.

Table of Contents

  1. Preliminary results and facts
  2. Toric blow-ups
  3. Three-dimensional blow-ups. Case of curve
  4. Toric log surfaces
  5. Non-toric three-dimensional blow-ups. Case of point
  6. Main theorems. Case of point

Key Result

Proposition 1.1

Koetal Let $f_i\colon Y_i\to X$ be two divisorial contractions of normal varieties, where $\operatorname{Exc} f_i=E_i$ are irreducible divisors and $-E_i$ are $f_i$-ample divisors. If $E_1$ and $E_2$ define the same discrete valuation of the function field $\mathcal{K}(X)$, then the contractions $f_

Theorems & Definitions (108)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Example 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • proof
  • Definition 1.7
  • Remark 1.8
  • ...and 98 more