On curve sections of rank two reflexive sheaves

Let F be a normalized rank 2 reflexive sheaf on P-3 with Chern classes cl,c z, cs. Let a be the least integer such that 0 not equal (HF)-F-0(alpha) and beta be the smallest integer such that (HF)-F-0(n) has sections whose zero scheme is a curve for all n greater than or equal to beta. We show that if To is the largest root of the cubic polynomial P(T) = T-3 - (6c(2) + 6 alphac(1) + 6 alpha (2) + 1)T + 3(2 alpha + c(1))(c (2) + c(1)alpha + alpha (2)) then beta less than or equal to T-0 - alpha - c(1) - 1. There are applicati ons to the smallest degree of a. surface containing a curves which are the zero schemes of sections of (HF)-F-0(alpha).