Birational maps of 3-folds

TL;DR

This paper demonstrates that complex birational transformations of 3-dimensional algebraic varieties can be decomposed into simpler, well-understood operations such as flops, blow-downs, and divisorial contractions, aiding classification efforts.

Contribution

It introduces a factorization approach for 3-fold terminal flips and contractions into elementary steps, advancing the understanding of their structure.

Findings

3-fold terminal flips can be factored into flops and contractions.

Divisorial contractions to points with minimal discrepancies are included in the factorization.

The approach simplifies the analysis of 3-fold birational maps.

Abstract

We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.