A Fano variety is defined by the ampleness of the anti-canonical divisor. Kollár, Miyaoka and Mori proved that Fano varieties of fixed dimension form a bounded family. In the singular case, Birkar settled the boundedness known as the Borisov-Alexeev-Borisov conjecture. The general elephant conjecture holds for Gorenstein Fano threefolds thanks to Shokurov and Reid. Without the Gorenstein condition, there exist counter-examples. Iskovskikh established a classification of Fano threefolds with Picard number one. His approach is founded upon the work of Fano, who studied an anti-canonically embedded Fano threefold by projecting it doubly from a line. Mukai provided a biregular description by means of vector bundles. There exist 95 families of terminal Q-Fano threefold weighted hypersurfaces. Corti, Pukhlikov and Reid concluded that a general Q-Fano threefold in each of these families is birationally rigid. Finally we describe the relation between birational rigidity and K-stability. The K-stability was introduced for the problem of the existence of a Kähler-Einstein metric. If a Q-Fano threefold in one of the 95 families is birationally superrigid, then it is K-stable.

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.

One of the landmarks in birational geometry is the attainment of the existence of threefold flips by Mori. We elucidate his approach in detail in this chapter. Passing through the analytic category and the flop of the double cover, we reduce the existence to the general elephant conjecture on an irreducible extremal neighbourhood. The study of an extremal neighbourhood is performed with numerical invariants defined in terms of filtrations of the structure sheaf and the dualising sheaf. Locally at a point, the inverse image of the curve by the index-one cover turns out to be planar. We divide singular points into types according to this structure. Then we classify the set of singular points by deforming the neighbourhood. It is easy to prove that the general elephant is Du Val when it does not contain the exceptional curve. The hard case when it contains the curve requires a really delicate analysis of how the curve is embedded in the threefold. As discussed in the preceding chapter, an extremal neighbourhood is considered to be a one-parameter deformation of a principal prime divisor on it. We describe the associated surface morphism and build a threefold flip from it.

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.

A Mori fibre space of relative dimension one is a conical fibration and it is birational to a standard conic bundle. The object in this chapter is a Q-conic bundle, namely a Mori fibre space from a threefold to a surface. The analytic germ along a fibre is analogous to an extremal neighbourhood but the higher direct image of the canonical sheaf may not vanish. Mori and Prokhorov achieved a classification of Q-conic bundle germs and the general elephant conjecture when the central fibre is irreducible. This implies Iskovskikh's conjecture that the base surface is Du Val of type A. We also discuss the rationality problem of threefold standard conic bundles over a rational surface. Beauville, after Mumford, proved that the intermediate Jacobian is isomorphic to the Prym variety of the double cover of the discriminant divisor. By virtue of Shokurov's analysis, we have a complete criterion for rationality when the base is relatively minimal. Sarkisov proved that a standard conic bundle is birationally superrigid if a certain pseudo-effective threshold is at least four. In contrast, the rationality in dimension three is nearly paraphrased as having the threshold less than two.

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.