Download Algebraic geometry V. Fano varieties by A.N. Parshin, I.R. Shafarevich, Yu.G. Prokhorov, S. Tregub, PDF

By A.N. Parshin, I.R. Shafarevich, Yu.G. Prokhorov, S. Tregub, V.A. Iskovskikh

The purpose of this survey, written by way of V.A. Iskovskikh and Yu.G. Prokhorov, is to supply an exposition of the constitution thought of Fano forms, i.e. algebraic vareties with an abundant anticanonical divisor. Such types certainly seem within the birational type of sorts of unfavorable Kodaira size, and they're very as regards to rational ones. This EMS quantity covers assorted ways to the type of Fano types similar to the classical Fano-Iskovskikh "double projection" approach and its differences, the vector bundles procedure as a result of S. Mukai, and the strategy of extremal rays. The authors speak about uniruledness and rational connectedness in addition to fresh development in rationality difficulties of Fano kinds. The appendix includes tables of a few sessions of Fano forms. This publication should be very important as a reference and study consultant for researchers and graduate scholars in algebraic geometry.

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We start from the Euler sequence ⊕(n+1) 0 → OPn (−1) → OPn → TPn (−1) → 0. 44 1. HOLOMORPHIC VECTOR BUNDLES AND THE GEOMETRY OF Pn The (n − 1) st exterior power of this sequence is ⊕(n+1) (*) 0 → Λn−2 TPn (−1) ⊗ OPn (−1) → OPn n−1 → Λn−1 TPn (−1) → 0. Furthermore, Λn−1 TPn (−1) ∼ = Ω1Pn (1) ⊗ det TPn (−1) ∼ = Ω1Pn (2). Let E = ((Λn−2 TPn (−1)) ⊗ OPn (−1))∗ . The sequence which is dual to (*) is ⊕(n+1) 0 → TPn (−2) → OPn 2 → E → 0, which shows that E is globally generated of rank n+1 n r= −n= . 2 2 For n ≥ 3 we have r ≥ n; thus there is an exact sequence ⊕(r−n) 0 → OPn →E→E →0 with a holomorphic n-bundle E .

Wm are linearly independent, the sections swi have no common zeros. ,swm ) OPn −−−−−→ TPn (−1)⊕(m+1) . , (2) ⊕(m+1) 0 → OPn → TPn →E→0 is exact. In order to determine the degree of homogeneity of E, we investigate the restrictions of this (n(m + 1) − 1)-bundle E to k-dimensional projective subspaces P(W ) ⊂ Pn . Claim: Let W0 = Cw0 + · · · + Cwm ⊂ Cn+1 be the subspace spanned by the vectors w0 , . . , wm and let W ⊂ Cn+1 be a (k + 1)-dimensional subspace. i) If W0 ⊂ W , then ⊕[(n−k)(m+1)−1] E|P(W ) ∼ = TP(W ) (−1)⊕(m+1) ⊕ OP(W ) ii) If W0 ⊂ W , then ⊕(n−k)(m+1) E|P(W ) ∼ = E ⊕ OP(W ) with E a bundle over P(W ) such that h0 (P(W ), E ∗ ) = 0.

Ar ). SE = MaE is the set of jump lines. An r-bundle E over Pn which has no jump line does not necessarily have to split. ). 3. For r < n every uniform r-bundle over Pn splits as a direct sum of line bundles. Proof. We prove the theorem by induction over r. For r = 1 there is nothing to prove. Suppose the assertion is true for all uniform r -bundles with 1 ≤ r < r, r < n. If E is a uniform r-bundle, then we can without restriction assume that E has the splitting type aE = (a1 , . . , ar ), a1 ≥ · · · ≥ ar with a1 = · · · = ak = 0, ak+1 < 0.

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