The paper investigates the non-vanishing of H1(E(n)), where E is a
(normalized) rank two vector bundle over any smooth irreducible threefold X
of degree d such that Pic(X) \cong \ZZ. If ϵ is the integer
defined by the equality ωX=OX(ϵ), and α is the least
integer t such that H0(E(t))=0, then, for a non-stable E (α≤0) the first cohomology module does not vanish at least between the
endpoints 2ϵ−c1 and −α−c1−1. The paper also shows
that there are other non-vanishing intervals, whose endpoints depend on
α and also on the second Chern class c2 of E. If E is stable the
first cohomology module does not vanish at least between the endpoints
2ϵ−c1 and α−2. The paper considers also the case of a
threefold X with Pic(X) \ne \ZZ but Num(X) \cong \ZZ and gives similar
non-vanishing results.Comment: 18 page