Fano-Mori contractions of high length on projective varieties with terminal singularities

TL;DR

This paper classifies certain birational contractions on projective varieties with terminal singularities, showing they are weighted blow-ups under specific conditions, and provides a classification of divisorial contractions with bounded fiber dimension.

Contribution

It proves that specific extremal contractions are weighted blow-ups and classifies divisorial contractions with bounded fiber dimension on varieties with terminal singularities.

Findings

Contractions with R.(K_X+(n-2)L)<0 are weighted blow-ups of smooth points.

Classified divisorial contractions with R.(K_X+rL)<0 and fibers of dimension ≤ r+1.

Provided a framework for understanding extremal contractions in the context of terminal singularities.

Abstract

Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.