Divisorial Mori contractions of submaximal length

TL;DR

This paper characterizes the structure of divisorial Mori contractions of submaximal length, showing they are birational to either a projective bundle or a quadric bundle, extending previous results on maximal length cases.

Contribution

It extends the classification of Mori contractions by analyzing the case of submaximal length, revealing their birational models as projective or quadric bundles.

Findings

Exceptional locus is birational to a projective bundle or quadric bundle.

Generalizes previous maximal length results to submaximal length cases.

Provides new insights into the structure of divisorial Mori contractions.

Abstract

A result due to Cho, Miyaoka, Shepherd-Barron [CMSB] and Kebekus [Ke] provides a numerical characterization of projective spaces. More recently, Dedieu and H\"oring [DH] gave a characterization of smooth quadrics based on similar arguments. As a relative version of [CMSB] and [Ke], H\"oring and Novelli proved in [HN] that the locus covered by positive-dimensional fibres in a Mori contraction of maximal length is a projective bundle up to birational modification. We change the length hypothesis and we prove that the exceptional locus of a divisorial Mori contraction of submaximal length is birational either to a projective bundle, or to a quadric bundle.