Extremal contractions from 4-dimensional manifolds to 3-folds

TL;DR

This paper classifies 2-dimensional fibers of extremal contractions from smooth projective 4-folds to 3-folds, showing they are connected by short chains of rational curves and that the anti-canonical system is free over the contraction.

Contribution

It provides a classification of fibers in extremal contractions from 4-folds to 3-folds, including properties of rational chains and anti-canonical linear systems.

Findings

Any two points in such a fiber are joined by a chain of rational curves of length at most 2.

The linear system |-K_X| is g-free.

The fibers are classified and their geometric properties are established.

Abstract

Let $g : X \to Y$ be the contraction of an extremal ray of a smooth projective 4-fold $X$ such that $\dim Y=3$. Then $g$ may have a finite number of 2-dimensional fibers. We shall classify those fibers. Especially we shall prove that any two points of such a fiber is joined by a chain of rational curves of length at most 2 with respect to $-K_X$, and that $|-K_X|$ is $g\text{-free}$.