Let (X, Y ) be a smooth non-projective Moishezon compactification of C 3 with b 2 (X) = 1. Then Y is a non-normal irreducible divisor on X with K X = −rY (r = 1, 2). In this paper, we mainly study the case where Y is not nef, that is, there is a curve C such that (Y · C) X < 0. First, we investigate the structure of the boundary divisor Y (Theorem 1) under the mild assumption that b 3 (X) = 0. Next we define the invariant δ(X) and compute them for the examples with non-nef boundaries (Theorems 2 and 3).