On the divisorial contractions to curves of threefolds

TL;DR

This paper proves that divisorial contractions to curves in terminal threefolds are weighted blow-ups under certain embeddings and classifies these blow-ups when the curve is smooth, advancing the understanding of threefold birational geometry.

Contribution

It establishes that all divisorial contractions to curves are weighted blow-ups and provides a classification for smooth curves, clarifying their structure in threefolds.

Findings

Divisorial contractions are weighted blow-ups under suitable embeddings.

Classification of weighted blow-ups for smooth curves in threefolds.

Enhanced understanding of the structure of threefolds in algebraic geometry.

Abstract

We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth.