By Spencer Bloch, Igor V. Dolgachev, William Fulton
Contents: V.A. Alexeev: Theorems approximately reliable divisors on log Fano types (case of index r >n - 2).- D. Arapura: Fano maps and primary groups.- A. Bertram, L. Ein, R. Lazarsfeld: Surjectivity of Gaussian maps for line bundles of enormous 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: potent tools in invariant theory.- V.A. Kolyvagin: at the constitution of the Shafarevich-Tate groups.- Vic.S. Kulikov: at the primary workforce of the supplement of a hypersurface in Cn.- B. Moishezon, M. Teicher: Braid workforce procedure 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 learn of simple-elliptic singularities.- R. Smith, R. Varley: Singularity conception utilized to ***- divisors.- A.N. Tyurin: A mild generalization of the concept of Mehta- Ramanathan.- F.L. Zak: a few homes of twin types and their functions in projective geometry.- Yu.G. Zarhin: Linear irreducible Lie algebras and Hodge constructions.
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Extra resources for Algebraic Geometry: Proceedings of the US-USSR Symposium held in Chicago, June 20–July 14, 1989
Let E be as in the previous theorem. Then h°(E) = 4, h ' ( E ) = h q E ) = O. 4. Stability. Recall that a vector bundle E i s H - s t a b l e (resp. H-semi-stable), where H is a divisor, if for every line subbtmdle L in E L-H < ½ q ( E ) - H (resp. L,H _<½cl(E)-H). Theorem 3. Let E be a rank 2 vector bundle on an Enriques surface S with q ( E ) = A and ~ ( E ) = 3. The following assertions are equivalent: (i) E is A-semi-stable; (ii) E is isomorphic to the Reye bundle. (iii) E is A-stable. PROOF.
Hence H°(S,E(-Fi)) ~ 0 proving our claim. Let L =Cgs(D) be an invertible subsheaf of E with the maximal degree A,D. A = 3. (A-D) + deg(Z') = c2(E) = 3. A) _<0. From this it follows D 2 ->0 with equality holding if and only if D,A = 3, Z' = O. A = 3. D = Fj for some j (Lemma 1), and we obtain the following exact sequence: 0 --~ (9 s(F) -~ E --, ~ S ( A - F ) - , 0 Thus E is either isomorphic to (9 s(F)ffK9 s(A-F) or E is a non-trivial extension. In the latter case Ext'((gS(A-F},(gs(F)) ~ H'(S,(gs(2FyA)) -=-H'(S,(gs(A-FFFj')) ~0.
The Cayley polarization maps S onto a surface in IP 5 isomorphic to the surface of reducible quadrics in a 5-dimensional linear system of quadrics in IP 5. In this note we will study rank 2 vector bundles E on S with Cl(E) = A and c2(E) = 3, where A is an ample divisor on S with A~ = i0. If A is a Reye polarization, we may assume that S lies in the Grassmann variety G(2,4) in its Plficker embedding. Then an example of such a bundle is the restriction of the universal quotient bundle on G(2,4). One of the motivation for this work was to verify whether this bundle is stable.
Algebraic Geometry: Proceedings of the US-USSR Symposium held in Chicago, June 20–July 14, 1989 by Spencer Bloch, Igor V. Dolgachev, William Fulton