Math. Z. 213, 509-521 (1993) Mathematische Zeitschrift ~# Springer-Verlag1993 Smooth Enriques surfaces in p4 and exceptional bundles Sonia Brivio Dottorato di Ricerca, Dipartimento di Matematica Politecnico di Torino, Corso Duca degli Abruzzi, 24, 1-10129 Torino, Italy Received 29 January 1992; in final form 5 June 1992 Introduction The purpose of this paper is to point out how to use exceptional vector bundles to construct smooth Enriques surfaces in p4. More precisely, we show that the existence of a (non minimal) smooth Enriques surface S, of degree 10 in e4, implies the existence of a stable and exceptional rank-2 vector bundle e$ on the minimal model X of S. Note that the smoothness of S directly implies the stability of g. Moreover, as we will see, X must contain a smooth rational curve, so that in particular X is a special Enriques surface. Exceptional vector bundles on Enriques surfaces have been recently studied by various authors, in particular these bundles have the following properties: c2 = t and c~ = 4 t-2, and fixing the determinant the moduli space consists of a single point. Hoil Kim showed that these vector bundles exist if and only if the surface is nodal, see [-K]. Reider and Dolgachev studied the case t= 3, showing that there exists essentially a unique such a bundle with ample determi- nant; this bundle is called Reye bundle, because it embeds the surface into the Grassmannian of lines of p3 as a Reye congruence. We generalize this result to the case t =4: we show that again there exists essentially a unique bundle g with ample determinant on a generic Enriques surface X. As a second result in this paper, we use this bundle to give a simple construction of a smooth non minimal Enriques surface in p4 of degree 10. The construction is the following: let rt: P(g)-+X be the obvious projection and s a generic section of h~ s defines a quasi-section S=P(g) which is the blowing up of X in 4 points. The linear system [Oe~)(S)| n* Ox(Kx)[ is 4-dimensional and its restriction to S yields an embedding of S into p4. More geometrically, the section s defines a 4-secant plane to the embedding of X into p7 by detg | Ox(Kx). Projecting X from this plane we obtain S. We have to point out that a different construction of S appears in [D-E-S], where the authors use degeneration loci of maps between two vector bundles of p4 and Macauley system, to produce a very rich list of examples of smooth surfaces in p4.