  We use Poonen's closed point sieve to prove two independent results. First,
we show that the obvious obstruction to embedding a curve in a smooth surface
is the only obstruction over a perfect field, by proving the finite field
analogue of a Bertini-type result of Altman and Kleiman. Second, we prove a
conjecture of Vakil and Wood on the asymptotic probability of hypersurface
sections having a prescribed number of singularities.
