  We prove the general diagram method theorem valid for the quite large class
of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also
Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds
with Q-factorial terminal singularities (Preprint alg-geom/9311007).
  As an application, we get the following result about 3-dimensional Calabi-Yau
manifolds X: Assume that the Picard number \rho (X) > 93. Then one of two cases
(i) or (ii) holds:
  (i) there exists a small extremal ray on X;
  (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X)
and the cubic intersection hypersurface W_X have a common point; here, we don't
claim that h is rational!).
  As a corollary, we get: Let X be a 3-dimensional Calabi-Yau manifold. Assume
that the nef cone NEF(X) is finite polyhedral and X does not have a small
extremal ray. Then there exists a rational nef element h with h^3=0 if
\rho(X)>93.
  To prove these results about Calabi-Yau manifolds, we also use one result of
V.V. Shokurov on the length of divisorial extremal rays (see Appendix by V.V.
Shokurov). Thus, one should consider the results on Calabi-Yau 3-folds above as
our common results with V.V. Shokurov.
  We also discuss generalization of results above to so called Q-factorial
models of Calabi-Yau 3-folds, which sometimes permits to involve non-polyhedral
case and small extremal rays to the game.
