Let $f_1,\cdots,f_s$ be elements of $\mathbb{C}[x_1,\cdots,x_n]$ and let $V$ be the variety that these polynomials define in $\mathbb{C}^n$. Under what conditions on the $f_1,\cdots,f_s$ can we construct a surjective map $\mathbb{C}^n \rightarrow V$?

This question appeared under bounty for 50 points here: http://math.stackexchange.com/questions/346583/constructing-a-projection-onto-a-variety

Edit: There is no requirement for the map to have any structure (at this point). The only requirement is surjectivity. What i want to achieve is given any $x \in \mathbb{C}^n$, i want to be able to modify its coordinates suitably so that to achieve an element of the variety. Every element of the variety should have a pre-image under the above process.

Second Edit: The existence of a surjective set-theoretic map might be trivial. However, how do we find such a map? Additionally, what possibilities for existence do we have if we assume that the map preserves structure?(e.g. morphism of varieties)