Random hypersurfaces and embedding curves in surfaces over finite fields

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in an unspecified 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.