  We prove a semiample generalization of Poonen's Bertini Theorem over a finite
field that implies the existence of smooth sections for wide new classes of
divisors. The probability of smoothness is computed as a product of local
probabilities taken over the fibers of the morphism determined by the relevant
divisor. We give several applications including a negative answer to a question
of Baker and Poonen by constructing a variety (in fact one of each dimension)
which provides a counterexample to Bertini over finite fields in arbitrarily
large projective spaces. As another application, we determine the probability
of smoothness for curves in Hirzebruch surfaces, and the distribution of points
on those smooth curves.
