Threefold extremal contractions of type (IIA), I

TL;DR

This paper classifies threefold extremal contractions of type (IIA) in algebraic geometry, focusing on germs with terminal singularities along a curve and analyzing the structure via a normal divisor.

Contribution

It provides a classification of threefold extremal contractions of type (IIA) based on the properties of a normal divisor containing the curve.

Findings

Classification of (IIA) type contractions in terms of the divisor H

Identification of conditions for normality of H

Structural insights into threefold contractions with terminal singularities

Abstract

Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA) and that a general member $H\in |O_X|$ containing $C$ is normal. We classify such germs in terms of $H$.