Toric Fano contractions associated to long extremal rays

TL;DR

This paper proves that in toric Fano contractions, if the extremal ray's length exceeds the fiber's dimension, the contraction results in a projective space bundle, revealing a geometric structure under specific conditions.

Contribution

It establishes a new criterion linking the length of extremal rays to the structure of toric Fano contractions, specifically identifying when they form projective space bundles.

Findings

Extremal rays with length greater than fiber dimension lead to projective space bundles.

Provides a geometric characterization of certain toric Fano contractions.

Enhances understanding of the structure of extremal rays in toric geometry.

Abstract

We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.