Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A canonicity criterion for toric varieties and the classification of canonical 4-simplices

Marco Ghirlanda

Abstract

Based on the Reid-Shepherd-Barron-Tai criterion for canonical and terminal quotient singularities, we characterize canonicity and terminality of a toric variety in terms of its local class group actions. Specializing it to the Picard number one setting, we arrive at a classification algorithm for canonical and terminal fake weighted projective spaces in any dimension. In dimension four it gives, up to isomorphism, 710450 canonical fake weighted projective spaces. We take a look at the corresponding Calabi-Yau hypersurfaces, compute the Fine interior of the associated canonical simplices, and discuss the results.

A canonicity criterion for toric varieties and the classification of canonical 4-simplices

Abstract

Based on the Reid-Shepherd-Barron-Tai criterion for canonical and terminal quotient singularities, we characterize canonicity and terminality of a toric variety in terms of its local class group actions. Specializing it to the Picard number one setting, we arrive at a classification algorithm for canonical and terminal fake weighted projective spaces in any dimension. In dimension four it gives, up to isomorphism, 710450 canonical fake weighted projective spaces. We take a look at the corresponding Calabi-Yau hypersurfaces, compute the Fine interior of the associated canonical simplices, and discuss the results.
Paper Structure (4 sections, 12 theorems, 47 equations, 1 algorithm)

This paper contains 4 sections, 12 theorems, 47 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Characterizing canonicity and terminality
  3. Classifying fake weighted projective spaces
  4. Fine interiors

Key Result

Theorem 1.1

Let $X$ be a $\mathbb{Q}$-factorial projective toric variety. Then $X$ is canonical if and only if $\mathrm{age}(h) \ge 1$ for all $1 \neq h\in H_i$, $i=1,\dots,s$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Lemma 2.3
  • proof
  • proof : Proof of Construction \ref{['constr:slice']}
  • proof
  • Proposition 2.5
  • proof
  • Example 2.6
  • ...and 21 more