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Q pY ' " -X >(X,Y) ,niX,y ) be such a d e s i n g u l a r i z a t i o n a) (X,Y) f:(X',Y') there can prove Proposition.

It follows from the previous Proposition that an ample Fano polarization defines an embedding S ~ IP5 onto a surface of degree 10 (a Fano model o f an Enriques surface). For example, if S does not contain smooth rational curves (it is called unnodal in this case), every Fano polarization is ample. Another example is a Reye congruence which is defined as the surface o f lines in lP ~ which are contained in a subpencil of a fixed web of quadrics satisfying a certain condition of regularity. This surface is a nodal Enriques surface and its Plficker embedding is defined by an ample Fano polarization.

C o s s e c , I. Dolgachev, Enriques surfaces I. Birkhauser. 1989 F. C o s s e c , 1. Dotgachev, Enriques surfaces II (in preparation). F. C o s s e t , Reye congruences, Trans. Amer. Math. Soc. 280 (1983), 737-751. A. Conte, A. (to appear in Trans. Amer. Math. ) H. Kim, Vector bundles on Enriques surfaces. D. thesis. Univ. of Mich. 1990. S. Kuloshov, A theorem on existence of exceptional bundles on surfaces of type K3, Izv. Akad. Is-a] Nauk SSSR, Ser. , 53, (1989), 363-378. N. Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in characteristic p [re] (preprint).

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