  Let the threefold X be a general smooth conic bundle over the projective
plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this
paper we prove the existence of two natural families C(+) and C(-) of curves on
X, such that the Abel-Jacobi map F sends one of these families onto a copy of
the theta divisor (Theta), and the other -- onto the jacobian J(X). The general
curve C of any of these two families is a section of the conic bundle
projection, and our approach relates such C to a maximal subbundle of a rank 2
vector bundle E(C) on C, or -- to a minimal section of the ruled surface
P(E(C)). The families C(+) and C(-) correspond to the two possible types of
versal deformations of ruled surfaces over curves of fixed genus g(C). As an
application, we find parameterizations of J(X) and (Theta) for certain classes
of Fano threefolds, and study the sets Sing(Theta) of the singularities of
(Theta).
