Counting Divisorial Contractions with Centre a $c{A}_n$-Singularity

Abstract

First, we simplify the existing classification due to Kawakita and Yamamoto of $3$-dimensional divisorial contractions with centre a $c{A}_n$-singularity, also called a 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 $c{A}_n$-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.