3 - Divisorial Contractions to Points

Summary

In this chapter, we furnish a systematic classification of threefold divisorial contractions which contract the divisor to a point, mainly due to the author. The classification is founded on a numerical one obtained by the singular Riemann-Roch formula, which makes a list of the basket of fictitious singularities. The list consists of a series of ordinary types and several exceptional types. The discrepancy in the case of exceptional type is very small. We establish the general elephant conjecture for the divisorial contraction by a delicate analysis of a tree of rational curves realised as the intersection of a certain surface with the exceptional divisor. We further describe the general elephant as a partial resolution of the Du Val singularity. The singular Riemann-Roch formula computes the dimensions of parts in lower degrees of the graded ring for the contraction restricted to the exceptional divisor. We recover the graded ring from these numerical data and nearly conclude that the divisorial contraction is a certain weighted blow-up of the cyclic quotient of a complete intersection inside a smooth fivefold. Examples are collected in accordance with the classification.