By Spencer Bloch, Igor V. Dolgachev, William Fulton

**Contents: V.A. Alexeev:** Theorems approximately solid divisors on log Fano kinds (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 *c*21 = 10 and *c*2 = 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 tools 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 C*n*.- **B. ****Moishezon, M. Teicher:** Braid workforce approach in advanced geometry, II: from preparations of traces and conics to cuspidal curves.- **D.Yu. Nogin:** Notes on unparalleled vector bundles and helices.- **M. Saito:** Hodge conjecture and combined causes II.- **C. Seeley, S. Yau:** Algebraic tools within the research of simple-elliptic singularities.- **R. Smith, R. ****Varley:** Singularity idea utilized to ***- divisors.- **A.N. ****Tyurin:** A moderate generalization of the concept of Mehta- Ramanathan.- **F.L. Zak:** a few homes of twin forms and their functions in projective geometry.- **Yu.G. Zarhin:** Linear irreducible Lie algebras and Hodge constructions.

**Read Online or Download Algebraic Geometry: Proceedings of the US-USSR Symposium held in Chicago, June 20–July 14, 1989 PDF**

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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**

Blow-up of Y), and is sufficient SkX' c Y'UZ', the if X Z by to for (X',Y') with centre Z' prove and by inductive (X',Y'UZ'). case Sk+iX), (proper pre-image the assertion assumption, Moreover, and In general it (X,Y). for the Theorem is true Theorem for is 38 Bibliography i. Danilov, Nauk, The 33(1978), Russian Math. 2. Deligne, geometry no. of toric 2(200), varieties, 85-134 Uspekhi (English Mat. transl: Surveys 33(1978)). de Hodge. II, Publ. Math. IHES, 40(1972), Theorie de Hodge. III, Publ.

By using vector bundles, we reprove the following result of F. Cossec [Co]: Theorem 1. In the notation above /A-~-Fi+id ¢ 0 4p ff and only f f A is a Reye polarization. PROOF. F i = 3 and C z = -2. Assume A is a Reye polarization. Then S lies on a non-singular quadric. Let rq be the plane containing F i. F j = Fi+I0°Fj = 1, the planes n i, ~i+10 and ~j lie in the same family, hence there exists a hyperplane containing any pair of them. This hyperplane cuts out a unique curve in I-Fi-Fi+10+A I. Let us prove the converse.

We know thatlP(F(E)) ~. IP3. For general section s of E its zero cycle lies on a trisecant in the embedding by IA+KI. This defines a rational map IP3 ~ X. Since E is unique, this map is dominant. R e m a r k 2 ([CV]). Assume that A is a Cayley polarization. Then the union of trisecants of S is isomorphic tothe quartic hypersurface of singular quadrics in the 5-dimensional linear system of quadrics parametrized by [hi*. Corollary 3. Let S b e an Enriques surface o f degree 10 in IPS and C be its smooth hyperplane section.