  We show that the first order structure whose underlying universe is $\mathbb
C$ and whose basic relations are all algebraic subset of $\mathbb C^2$ does not
have quantifier elimination. Since an algebraic subset of $\mathbb C ^2$ needs
either to be of dimension $\leq 1$ or to have a complement of dimension $\leq
1$, one can restate the former result as a failure of quantifier elimination
for planar complex algebraic curves. We then prove that removing the planarity
hypothesis suffices to recover quantifier elimination: the structure with the
universe $\mathbb C$ and a predicate for each algebraic subset of $\mathbb C^n$
of dimension $\leq 1$ has quantifier elimination.
