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.

The final chapter treats minimal threefolds. We explain the abundance for threefolds due to Miyaoka and Kawamata depending on the numerical Kodaira dimension. The initial step is to prove the non-vanishing which means the existence of a global section of some pluricanonical divisor. If the irregularity is not zero, then the Albanese map provides enough geometric information. In the case of irregularity zero, Miyaoka applied the generic semi-positivity via positive characteristic. We derive abundance from non-vanishing after replacing the threefold by a special divisorially log terminal pair. Birational minimal models are connected by flops and have the same Betti and Hodge numbers. In dimension three, they have the same analytic singularities. One can expect the finiteness of minimal models ignoring the marking map. This is a part of Kawamata and Morrison's cone conjecture for Calabi-Yau fibrations. We explain Kawamata's work on the conjecture for threefold fibrations with non-trivial base. In dimension three, there exists a uniform number for l such that the l-th pluricanonical map is birational to the Iitaka fibration. We find this number explicitly in the case of general type.

A Mori fibre space of relative dimension two is a del Pezzo fibration. We address the problem of finding a good representative in each equivalence class of threefold del Pezzo fibrations. In the case of degree at least three, Corti realised a standard model embedded anti-canonically into a projective space bundle. We explain the construction by Kollár for degree three as a semistable model. The stability theory completes the existence of standard models for degree two and one. Mori and Prokhorov proved that the multiplicity of a fibre is bounded by six. We also discuss the rationality problem of del Pezzo fibrations to a rational curve. The total space is always rational when the degree is at least five. We exhibit Alexeev's work for degree four. He established a criterion for rationality in terms of topological Euler characteristic. The lower degree case is treated from the point of view of birational rigidity. Pukhlikov proposed the K2-condition and proved that it is sufficient for birational rigidity. We compare it with Grinenko's K-condition. For a fibration of degree one from a smooth threefold, the K-condition turns out to be equivalent to the birational superrigidity.