Divisors of bielliptic surfaces and embeddings in p4

Abstract

Bielliptic surfaces (also called "hyperelliptic surfaces") are defined to be minimal algebraic surf aces of Kodaira dimension O and irregularity 1. They play a special role in the birational classification of surfaces. The first part of this paper gives an explicit description of the cohomology group H 2(S, Z) for a bielliptic surface S. In the second part the author proves the existence of smooth bielliptic surfaces in P4 • The proof relies on Reider's criterion for very-ampleness. In fact, a complete characterization of polarized bielliptic surfaces in P4 is given. These surfaces add to the very short list of known irregular surfaces in P4 , the other two being the abelian surfaces of Horrocks-Mumford and the elliptic quintic scrolls.