ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

Abstract

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.