  Let X/S be a quasi-projective morphism over an affine base. We develop in
this article a technique for proving the existence of closed subschemes H/S of
X/S with various favorable properties. We offer several applications of this
technique, including the existence of finite quasi-sections in certain
projective morphisms, and the existence of hypersurfaces in X/S containing a
given closed subscheme C, and intersecting properly a closed set F.
  Assume now that the base S is the spectrum of a ring R such that for any
finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied
if R is the ring of integers of a number field, or the ring of functions of a
smooth affine curve over a finite field. We prove in this context a moving
lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective
and flat over S. We also show the existence of a finite surjective S-morphism
to the projective space P_S^d for any scheme X projective over S when X/S has
all its fibers of a fixed dimension d.
