Suppose $f : U\rightarrow S$ is a flat separated finitely presented morphism whose fibers are equidimension 1, and which admits lots of sections. I'm looking for additional conditions (avoiding properness) that guarantee that $f$ is locally quasi-projective (that is, $X$ admits an $f$-ample invertible sheaf locally on $S$).
For example, let's assume that in addition $f$ has reduced and connected geometric fibers, and $f$ admits disjoint sections $\sigma_1,\ldots,\sigma_n$ which meets every irreducible component of every fiber. Is this enough to show that $f$ is locally quasi-projective? If not, can one find additional mild criteria (avoiding properness) that guarantee local-quasi-projectivity?
If $f$ were also proper, then we can check quasi-projectiveness/ampleness on fibers, noting that positive $\iff$ ample for curves. However the result that you can check ampleness on fibers really uses properness, so I'm wondering if this can be circumvented in our case, since our flatness assumption excludes the typical nonproper counterexample involving a blowup).
