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.

Every threefold divisorial contraction that contracts the divisor to a curve is the usual blow-up about the generic point of the curve. It is uniquely described as the symbolic blow-up as far as it exists. The general elephant conjecture is settled by Kollár and Mori when the fibre is irreducible. On the assumption of this conjecture, the symbolic blow-up always exists as a contraction from a canonical threefold. We want to determine whether it is further terminal. Tziolas analysed the case when the extraction is from a smooth curve in a Gorenstein terminal threefold, and Ducat did when it is from a singular curve in a smooth threefold. They follow the same division into cases based upon the divisor class of the curve in the Du Val section. Tziolas describes the symbolic blow-up as a certain weighted blow-up, whilst Ducat realises it by serial unprojections. The unprojection is an operation to construct a new Gorenstein variety from a simpler one. The contraction can be regarded as a one-parameter deformation of the birational morphism of surfaces cut out by a hyperplane section. In reverse, one can construct a threefold contraction by deforming an appropriate surface morphism.

Singularity is an obstacle to the treatment of algebraic varieties but at the same time enriches the geometry. Since a terminal threefold singularity is isolated, it is often more flexible to treat it in the analytic category. Artin's algebraisation theorem, Tougeron's implicit function theorem and the Weierstrass preparation theorem are fundamental analytic tools. Taking quotient produces singularities. We clarify the notion of quotient and define the weighted blow-up in the context of which cyclic quotient singularities appear. We furnish a complete classification of terminal threefold singularities due to Reid and Mori. First we deal with singularities of index one and next we describe those of higher index by taking the index-one cover. It turns out that the general member of the anti-canonical system of a terminal threefold singularity is always Du Val. This insight is known as the general elephant conjecture and plays a leading role in the analysis of threefold contractions. Reid established an explicit formula of Riemann-Roch type on a terminal projective threefold. We also discuss canonical threefold singularities and bound the index by means of the above formula.