By Aleksandr Pukhlikov
Birational pressure is a outstanding and mysterious phenomenon in higher-dimensional algebraic geometry. It seems that yes usual households of algebraic types (for instance, third-dimensional quartics) belong to an analogous class sort because the projective area yet have considerably diversified birational geometric houses. particularly, they admit no non-trivial birational self-maps and can't be fibred into rational types via a rational map. The origins of the speculation of birational pressure are within the paintings of Max Noether and Fano; in spite of the fact that, it was once in simple terms in 1970 that Iskovskikh and Manin proved birational superrigidity of quartic three-folds. This booklet provides a scientific exposition of, and a complete advent to, the speculation of birational stress, proposing in a uniform manner, rules, thoughts, and effects that to this point may simply be present in magazine papers. the new swift growth in birational geometry and the widening interplay with the neighboring components generate the becoming curiosity to the rigidity-type difficulties and effects. The publication brings the reader to the frontline of present study. it really is essentially addressed to algebraic geometers, either researchers and graduate scholars, yet can also be available for a much broader viewers of mathematicians accustomed to the fundamentals of algebraic geometry
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Extra resources for Birationally Rigid Varieties: Mathematical Analysis and Asymptotics
1). Namely, to prove (super)rigidity, one should be able either to exclude or to untwist every geometric discrete valuation as a maximal singularity. The exclusion part is the main one of the two. It is most diﬃcult and usually takes a lot of eﬀort. The untwisting part is much easier but for some varieties can be quite non-trivial, as we demonstrate in Section 6. Sections 2, 4 and 5 develop various techniques of excluding maximal singularities: the technique of counting multiplicities (leading to the crucial 4n2 -inequality) is explained in Section 2, the more recent technique based on the connectedness principle of Shokurov and Koll´ ar is developed in Section 4 and various geometric methods making use of the particular projective embedding of the variety under consideration are discussed in Section 5 by the example of the complete intersection V2·3 ⊂ P5 .
That is why the breakthrough that was made in [CG, IM] was of such great importance. Note that each of the papers [CG, IM] generated a new direction of further research: the so-called transcendental method and the method of maximal singularities, respectively. By means of the transcendental method very precise results on non-rationality of three-dimensional conic bundles were obtained, and the Clemens degeneration techniques made it possible to prove non-rationality of generic (in the complex analytic sense) pencils of cubic surfaces and many classes of Fano three-folds [Clem74, Clem82, Clem83, Bar, Tyu79].
It is more convenient to use the presentation of the variety P(E) as a projective bundle P(E) in the sense of Grothendieck, where E is a locally free sheaf of sections of the dual bundle E ∗ , E = 3i=0 OP1 (ai ), and we may assume that a0 = 0 ≤ a1 ≤ a2 ≤ a3 . A hypersurface V ⊂ P(E) equipped with the natural projection π = ρ | V : V → P1 , is a ﬁbration into cubic surfaces over P1 , if every ﬁbre π −1 (t) ⊂ P3 = ρ−1 (t) is a cubic surface. The hypersurface V is given by a section of the invertible sheaf L⊗3 ⊗ ρ∗ OP1 (l) for some l ∈ Z, where L is the tautological sheaf in the sense of Grothendieck.
Birationally Rigid Varieties: Mathematical Analysis and Asymptotics by Aleksandr Pukhlikov