  A Heegaard diagram for a 3-manifold M is a closed, oriented surface S
together with a pair (X, Y) of compact 1-manifolds in S whose components serve
as attaching curves for the 2-handles of the two sides of a Heegaard splitting
for M. The diagram is positive if X and Y can be oriented so that the
intersection number <X,Y>_p = +1 at each point p in their intersection. Every
(compact, orientable) 3-manifold can be represented by a positive diagram, but
the argument for this suggests that the minimal genus, phg(M), for a positive
diagram may be much larger than the minimal genus,hg(M), among all diagrams.
This paper investigates this situation for the class of closed orientable
Seifert manifolds over an orientable base. We show that phg(M) = hg(M) for most
of these manifolds with phg(M) never more than hg(M)+2. The cases phg(M) >
hg(M) are determined and occur when the minimal genus splitting is horizontal.
The arguments provide an alternate proof distinguishing these from vertical
splittings.
