By David Mumford
From the experiences: "Although a number of textbooks on smooth algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the pink ebook of sorts and schemes, now as sooner than essentially the most very good and profound primers of contemporary algebraic geometry. either books are only precise classics!" Zentralblatt
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Extra resources for Algebraic Geometry I: Complex Projective Varieties
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.
Algebraic Geometry I: Complex Projective Varieties by David Mumford