Algebraic Geometry: Proceedings of the US-USSR Symposium by Spencer Bloch, Igor V. Dolgachev, William Fulton

By Spencer Bloch, Igor V. Dolgachev, William Fulton

Contents: V.A. Alexeev: Theorems approximately stable divisors on log Fano types (case of index r >n - 2).- D. Arapura: Fano maps and basic groups.- A. Bertram, L. Ein, R. Lazarsfeld: Surjectivity of Gaussian maps for line bundles of huge measure on curves.- V.I. Danilov: De Rham advanced on toroidal variety.- I. Dolgachev, I. Reider: On rank 2 vector bundles with c21 = 10 and c2 = three on Enriques surfaces.- V.A.Iskovskih: in the direction of the matter of rationality of conic bundles.- M.M. Kapranov: On DG-modules over the De Rham advanced and the vanishing cycles functor.- G. Kempf: extra on computing invariants.- G. Kempf: powerful equipment in invariant theory.- V.A. Kolyvagin: at the constitution of the Shafarevich-Tate groups.- Vic.S. Kulikov: at the basic staff of the supplement of a hypersurface in Cn.- B. Moishezon, M. Teicher: Braid crew strategy in advanced geometry, II: from preparations of strains and conics to cuspidal curves.- D.Yu. Nogin: Notes on remarkable vector bundles and helices.- M. Saito: Hodge conjecture and combined factors II.- C. Seeley, S. Yau: Algebraic equipment within the examine of simple-elliptic singularities.- R. Smith, R. Varley: Singularity thought utilized to ***- divisors.- A.N. Tyurin: A moderate generalization of the concept of Mehta- Ramanathan.- F.L. Zak: a few houses of twin forms and their functions in projective geometry.- Yu.G. Zarhin: Linear irreducible Lie algebras and Hodge constructions.

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Extra info for Algebraic Geometry: Proceedings of the US-USSR Symposium held in Chicago, June 20–July 14, 1989

Example text

1. ,¢mmRs. Let x: V - o S be a standard conic bundle over a surface S ( not necessary rational) with discriminant curve C. Lemma 1 (cf. [41 prop. 2). For any integer m -> l the divisor C+mK s is effective ff and only if for any standard conic bundle n': V' ~ S' birationally equivalent to n: V --+ S the divisor C'+mKs,,whereC' is the discriminant curve for n', is effective. In other words, the property of the divisor C+K S to be effective (or not effective) is a birational invariant in the class of birational standard forms.

Conversely, assume that condition (i') is satisfied for a standard conic bundle x: V --~ S over a rational surface S. If C = O, then condition (i) is obviously satisfied for any free pencil of genus 0 on S. Assume that C # O. Then the arithmetic genus Pa(C) >_ 1, In fact, ff Pa(C) = 0, then, since C is reduced and connected, there exists a smooth irreducible component Z c C such that Zo(C-Z) = 1. However this is impossible because the points of intersection Zc3(C-Z) are the branch points of the non-trivial double covering 7,o--~ Z induced by the covering ~: C --~ C, and, by Hurwitz formula their number is even.

Every Enriques surface S adndts a Fano polarizationA. The complete linear ,system [A[ defines a birational map S ~ IPs whose image is a surface with at most double rationalpoints as its singularities. PROOF. This is proven in [CD1] under the assumption that char(k) -- 0. 7 i n [ C D l ] . f _>3 for all vectors f with f2 = 0. 1 from toc. cit. we find a nef divisor A with Az = 10 and Aof _>3 for all f with f2 >_3. This is a Fano polarization. The property of the map given by the linear system IA] follows from Corollary 2 of appendix to Chapter 4 of[CD1].

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