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We show that the first order structure whose underlying universe is $\mathbbC$ and whose basic relations are all algebraic subset of $\mathbb C^2$ does nothave quantifier elimination. Since an algebraic subset of $\mathbb C ^2$ needseither to be of dimension $\leq 1$ or to have a complement of dimension $\leq1$, one can restate the former result as a failure of quantifier eliminationfor planar complex algebraic curves. We then prove that removing the planarityhypothesis suffices to recover quantifier elimination: the structure with theuniverse $\mathbb C$ and a predicate for each algebraic subset of $\mathbb C^n$of dimension $\leq 1$ has quantifier elimination.

Some (non-)elimination results for curves in geometric structures