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The Integral Chow Ring of Smooth Non-strict Toric Stacks

Sota Suzuki

Abstract

We extend the Stanley-Reisner ring to the non-strict stacky fans introduced by Geraschenko and Satriano. We then prove that this ring is isomorphic to the integral Chow ring of the smooth non-strict toric stack defined by the given non-strict stacky fan. This generalizes known results for the integral Chow rings of less general toric stacks.

The Integral Chow Ring of Smooth Non-strict Toric Stacks

Abstract

We extend the Stanley-Reisner ring to the non-strict stacky fans introduced by Geraschenko and Satriano. We then prove that this ring is isomorphic to the integral Chow ring of the smooth non-strict toric stack defined by the given non-strict stacky fan. This generalizes known results for the integral Chow rings of less general toric stacks.

Paper Structure

This paper contains 8 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cox constructions
  4. Non-strict toric stacks and non-strict stacky fans
  5. Integral Chow ring of smooth non-strict toric stacks
  6. Proof of the main theorem
  7. The integral Chow ring of fantastacks
  8. Examples

Key Result

Theorem 1.0.2

Let $(\Sigma,\beta\colon L\to N)$ be a non-strict stacky fan such that $X_\Sigma$ is smooth and has no torus factors. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (25)

  • Definition 1.0.1
  • Theorem 1.0.2
  • Theorem 2.1.1
  • Theorem 2.1.2
  • proof
  • Lemma 2.1.3
  • Definition 2.2.1
  • Definition 2.2.2
  • Remark 2.2.3
  • Remark 2.2.4
  • ...and 15 more