Subfields of ample fields. Rational maps and definability

Pop proved that a smooth curve C over an ample field K with C(K)≠∅ has |K| many rational points. We strengthen this result by showing that there are |K| many rational points that do not lie in a given proper subfield, even after applying a rational map. This has several consequences. For example, we gain insight into the structure of existentially definable (i.e. diophantine) subsets of ample fields.