Many constructions in algebraic geometry require one to choose a point
outside a countable union of subvarieties. Over $\C$ this is always
possible. Over a countable field, a countable union of subvarieties
can cover all the closed points. Let $k$ be a finitely generated
field of characteristic zero and let $\kbar$ be an algebraic closure.
Let $A$ be a semiabelian variety defined over $k$, and let $\End(A)$
be the ring of endomorphisms of $A$ over $\kbar$. Let $X\subset A$ be
a subvariety of smaller dimension. We show that $\Union_{f\in
\End(A)} f(X(\kbar))$ does not equal $A(\kbar)$. Bogomolov and
Tschinkel show that the above is false for $k$ equal to an algebraic
closure of a finite field, and use the result to show that on any
Kummer surface over such $k$, the union of all rational curves covers
all of the closed points. We give further examples of such problems.