On the birationality of the bicanonical map of a surface section of a threefold

Let (M,L) be a polarized threefold of log-general type. The birationality of the bicanonical map of a smooth surface S ∈ |L| is studied. This problem was previously considered and partially solved by the first and fourth author, who gave a satisfactory classification unless h(OM) = 0 and pg(S) = 3, 4, 5. This paper focuses on the remaining cases which are the hardest, settling the problem. Introduction and statement of the result Let M be a smooth projective threefold and let L be a very ample line bundle on M. In this paper we consider the problem of describing the pair (M,L) in the case when there exists at least one smooth surface Ŝ ∈ |L| such that Ŝ is of general type, and the bicanonical map associated to |2Kb S| is not birational. If a surface of general type Ŝ is a very ample (or even merely, an ample) divisor on a smooth threefoldM, then it follows from [3, (7.9.1)] that either: 1. M is a P-bundle over a smooth surface Y with Ŝ a meromorphic section; or 2. there exists a fibering p : M → Y of M onto a smooth surface Y with the general fiber isomorphic to P and p|b S a generically two-to-one morphism; or 3. (M,L) is of log-general type, i.e., for sufficiently large N , the linear system |N(KM + L)| gives a birational map. The first class is trivial in the sense that every smooth Ŝ ∈ |L| is birational to the base surface Y , and conversely, given any smooth surface Y , some surface birational to it is a very ample divisor as in this class. The second class contains very special varieties, while the third class contains the overwhelming majority of threefolds with ∗2000 Mathematics Subject Classification. Primary 14N30, 14J30, 14E05, 14C20; Secondary 14J29, 14J10.