chicago - O n r ank 2 vector b u n d l e s with ~ Igor...

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On rank 2 vector bundles with ~ = 10 and c 2 = 3 on Enriques surfaces. Igor Dolgachev and Igor Reider* University of Michigan, Ann Arbor, Mi 48109, USA University of Oklahoma, Norman, OK, 73019, USA 1. lnlxoduction, Let S be an Enriques surface over an algebraically dosed field k and A be a numerically effective divisor on S with A 2 = 10, It is known that one can always find such A with the property that the linear system 1A[ gives a birational map onto a surface of degree 10 in IP 5 with at most double rational points as singularities. We will assume that there exists an ample A. Then S can he embedded by IAI into IP 5. A is said to be a Reye polarization if the image of S ties on a quadric. A is a said to be a Cavley polarization it" A+K S isa Reye polarization. It can be proven leD2] that S admits a Reye polarization ff and only if it is nodal, i.e. contains smooth rational curves (nodalcurves). Note that every Enriques surface lying on a quadric is isomorphic to a Reye congruence of lines inlP -~ (char(k) ~ 2). 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. The formula for the dimension of the moduli space of stable vector bundles shows that one expects the existence of at most finitely many isomorphisms classes of stable E's as above. We will see that the existence of at least one such E depends very much on the properly of the polarization A. More precisely, we show that, if it exists, then it is unique and A is a Reye polarization. As an application we give a characterization of the Cayley polarizations A by the condition that the variety of trisecants of (S,A) is three-dimensional. By other means this result was obtained by A. Conte and A. Verra [CV]. 1. Fano polarizations of Enriques surfaces. A numerically effective (nef) divisor A on an Enriques surface S is called a Fano polarizationif A 2 = 10 and A.F _> 3 for every nef divisor F with F 2 = 0. Proposition 1. Every Enriques surface S adndts a Fano polarizationA. The complete linear ,system [A[ defines a birational map S ~ IP s 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. The description of all vectors x in Pic(S) with x 2 = 10 is given in Corollary 2.5.7 in[CDl]. From this it follows that there exists a vector x with x 2 = 10 and x.f _> 3 for all vectors f with f2 = 0. Applying Theorem 3.2.1 from toc. cit. we find a nef divisor A with A z = 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]. The assumption of characteristic 0 can
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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chicago - O n r ank 2 vector b u n d l e s with ~ Igor...

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