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On Chow groups of toric schemes over a DVR

Ana María Botero

TL;DR

This work gives an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR and of their special fibers.

Abstract

We give an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR. As an application, we compute the Chow groups of several toric schemes over a DVR and of their special fibers.

On Chow groups of toric schemes over a DVR

TL;DR

This work gives an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR and of their special fibers.

Abstract

We give an explicit combinatorial presentation of the Chow groups of a toric scheme over a DVR. As an application, we compute the Chow groups of several toric schemes over a DVR and of their special fibers.
Paper Structure (16 sections, 13 theorems, 92 equations, 6 figures)

This paper contains 16 sections, 13 theorems, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Outline of the paper
  4. Conventions
  5. Toric schemes over $S$
  6. SCR polyhedral complexes
  7. Toric schemes over a DVR
  8. Combinatorial characterization
  9. Torus orbits
  10. ${\mathbb T}_S$-invariant divisors on toric schemes
  11. Chow groups of toric schemes
  12. Chow (homology) groups
  13. Chow groups of toric schemes
  14. Chow groups of the special fiber
  15. Examples
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k\geq 0$ be an integer. Then the Chow group ${\operatorname{CH}}_k({ \mathcal{X}}/S)$ is generated by torus invariant horizontal and vertical cycles of $S$-absolute dimension $k$. A collection of generators is given by the family of horizontal cycles $\left[V(\sigma)\right]$ for $\sigma$ a $(n-

Figures (6)

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  • ...and 1 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 41 more