Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A remark on toric foliations

Osamu Fujino, Hiroshi Sato

Abstract

If a toric foliation on a projective Q-factorial toric variety has an extremal ray whose length is longer than the rank of the foliation, then the associated extremal contraction is a projective space bundle and the foliation is the relative tangent sheaf of the extremal contraction.

A remark on toric foliations

Abstract

If a toric foliation on a projective Q-factorial toric variety has an extremal ray whose length is longer than the rank of the foliation, then the associated extremal contraction is a projective space bundle and the foliation is the relative tangent sheaf of the extremal contraction.
Paper Structure (3 sections, 2 theorems, 29 equations)

This paper contains 3 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on toric varieties
  3. Proof of Theorem \ref{['theorem1.3']}

Key Result

Theorem 1.2

Let $X=X(\Sigma)$ be a $\mathbb Q$-factorial toric variety with its fan $\Sigma$ in the lattice $N\simeq \mathbb Z^n$. Then there exists a one-to-one correspondence between the set of toric foliations on $X$ and the set of complex vector subspaces $V\subset N_{\mathbb C}:=N\otimes _{\mathbb Z} \math holds, that is, the first Chern class of $\mathscr F_V$ is $\sum _{\rho\subset V}D_\rho$, where $D_

Theorems & Definitions (6)

  • Definition 1.1: Foliations and toric foliations
  • Theorem 1.2: see pang
  • Theorem 1.3: Main Theorem
  • proof : Proof of Theorem \ref{['theorem1.3']}
  • Claim
  • proof : Proof of Claim