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.

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.

We shall proceed to the study of threefold Mori fibre spaces. This chapter is an introduction to the Sarkisov program, which factorises a birational map of Mori fibre spaces as a composite of elementary maps called Sarkisov links. Corti established it in dimension three. The program twists the birational map into an isomorphism by means of the Noether-Fano inequality. Whereas Hacon and McKernan obtained a new program in an arbitrary dimension, we stick to the traditional approach which fits better into the explicit study. The program stems from the pioneer work of Iskovskikh and Manin on the irrationality of a smooth quartic threefold. We review the rationality problem and discuss the notions of rationality, stable rationality and unirationality. A rational variety has a large number of Mori fibre spaces in its birational class. Oppositely, a variety birational to essentially only one Mori fibre space is said to be birationally rigid. We explain a general strategy for applying the Sarkisov program to the rationality and birational rigidity problem, and demonstrate this by an example due to Corti and Mella which has exactly two birational structures of a Mori fibre space.

This chapter outlines the general theory of the minimal model program. The program outputs a representative of each birational class which is minimal with respect to the numerical class of the canonical divisor. It grew out of the surface theory with allowing mild singularities. For a given variety, it produces a minimal model or a Mori fibre space after finitely many birational transformations which are divisorial contractions and flips. The program is formulated in the logarithmic framework where we treat a pair consisting of a variety and a divisor. It functions subject to the existence and termination of flips. Hacon and McKernan with Birkar and Cascini proved the existence of flips in an arbitrary dimension. The termination of threefold flips follows from the decrease in the number of divisors with small log discrepancy. Shokurov reduced the termination in an arbitrary dimension to certain conjectural properties of the minimal log discrepancy. It is also important to analyse the representative output by the program. For a minimal model, we expect the abundance which claims the freedom of the linear system of a multiple of the canonical divisor.