Counting divisorial contractions with centre a $cA_n$-singularity

TL;DR

This paper refines the classification of 3-dimensional divisorial contractions centered at $cA_n$-singularities, linking local analytic types to global algebraic contractions and revealing the abundance of certain contractions.

Contribution

It simplifies existing classifications, describes global contractions from local data, and demonstrates the uncountable diversity of divisorial contractions with discrepancy at least 2.

Findings

Simplified classification of divisorial contractions with $cA_n$-singularity.

Established correspondence between local analytic and global algebraic contractions.

Proved existence of uncountably many such contractions when discrepancy ≥ 2.

Abstract

First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a $cA_n$-singularity, also called compound $A_n$ singularity. Next, we describe the global algebraic divisorial contractions corresponding to a given local analytic equivalence class of divisorial contractions with centre a point. Finally, we consider divisorial contractions of discrepancy at least 2 to a fixed variety with centre a $cA_n$-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.