By Aleksandr Pukhlikov

Birational tension is a impressive and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that convinced usual households of algebraic kinds (for instance, 3-dimensional quartics) belong to a similar class kind because the projective area yet have significantly diversified birational geometric homes. particularly, they admit no non-trivial birational self-maps and can't be fibred into rational types by way of a rational map. The origins of the speculation of birational pressure are within the paintings of Max Noether and Fano; besides the fact that, it used to be merely in 1970 that Iskovskikh and Manin proved birational superrigidity of quartic three-folds. This booklet supplies a scientific exposition of, and a accomplished advent to, the speculation of birational pressure, offering in a uniform manner, rules, options, and effects that up to now might purely be present in magazine papers. the hot quick growth in birational geometry and the widening interplay with the neighboring components generate the turning out to be curiosity to the rigidity-type difficulties and effects. The e-book brings the reader to the frontline of present examine. it truly is basically addressed to algebraic geometers, either researchers and graduate scholars, yet is usually available for a much broader viewers of mathematicians acquainted with the fundamentals of algebraic geometry

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3. The Sarkisov theorem on conic bundles. Let S be a smooth projective variety of dimension dim S ≥ 2, ρ : E → S an (algebraic) vector bundle of rank 3, ρ : P(E) → S its projectivization, that is, a locally trivial P2 -bundle over S. A hypersurface V ⊂ P(E), equipped with the natural projection π : V → S, π = ρ | V , is called a conic bundle over S, if every ﬁbre π −1 (s) ⊂ P2 = ρ−1 (s) is a conic in P2 . Let E be the sheaf of sections of the dual bundle E ∗ → S. Then P(E) coincides with the projective bundle P(E) in the sense of Grothendieck (Proj, see [Hart]).

2 does not depend on the theorem on resolution of singularities and for this reason holds in any characteristic. With the exceptional divisor E ⊂ V + a uniquely determined sequence of blow X ups is associated. Let X be a projective (possibly singular) variety, ψ : V + a birational map, contracting E to a subvariety B = ψ(E) ⊂ X of codimension ≥ 2, whereas B ⊂ Sing Xis not contained entirely in the set of singular points. −1 (B) the Let σB : X(B) → X be the blow up of the subvariety B, E(B) = σB exceptional divisor.

Assume that every mobile linear system Σ on V with the zero virtual threshold of canonical adjunction, cvirt (Σ) = 0, is the pullback of a system on the base: Σ = π ∗ Λ, where Λ is some mobile linear system on S. Then any birational map (6) π V ↓ S χ V ↓ S , π where π : V → S is a ﬁbration into varieties of negative Kodaira dimension, is ﬁbrewise, that is, there exists a rational dominant map ρ : S S , making the diagram (6) commutative, π ◦ χ = ρ ◦ π. In other words, π ≥ π in the sense of the order on the set of rationally connected structures: π is the least element of RC(V ).