General elements in anticanonical systems of threefolds with divisorial contractions and applications to classification

TL;DR

This paper studies threefolds with divisorial contractions, showing that general anticanonical elements have Du Val singularities and providing bounds on discrepancies for certain contractions, aiding classification efforts.

Contribution

It establishes that general anticanonical elements have Du Val singularities and bounds discrepancies for contractions to compound D_n or E_n points, advancing classification theory.

Findings

General anticanonical elements have Du Val singularities.

Discrepancies to compound D_n or E_n points are at most four.

Descriptions of divisorial contractions to compound A_n points.

Abstract

We treat threefolds with divisorial contractions whose exceptional divisors contract to compound Du Val points. We prove that general elements in their anticanonical systems around the exceptional divisors have at worst Du Val singularities. As applications to classification, we describe divisorial contractions to compound A_n points, and moreover we conclude that discrepancies of divisorial contractions to compound D_n or E_n points are at most four.