A remark on toric foliations

TL;DR

This paper investigates the structure of toric foliations on projective Q-factorial toric varieties, establishing conditions under which extremal contractions are projective space bundles and identifying the foliation as a relative tangent sheaf.

Contribution

It provides a new criterion linking extremal ray length and foliation rank to the structure of extremal contractions in toric varieties.

Findings

Extremal contraction is a projective space bundle under certain conditions.

The foliation corresponds to the relative tangent sheaf of the contraction.

The result characterizes the geometry of toric foliations with specific extremal properties.

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.