  Pop proved that a smooth curve C over an ample field K that has a K-rational
point has |K| many K-rational points. We strengthen this result by showing that
there are |K| many K-rational points that do not lie in a given proper
subfield, even after applying a rational map. As a consequence we gain insight
into the structure of existentially definable subsets of ample fields. In
particular, we prove that a perfect ample field has no existentially definable
proper infinite subfields.
