Suppose $X$ is a complete algebraic variety of dimension $n$. Must there exist an affine covering with $n+1$ pieces? 


(For a projective variety in $\mathbf{P}^m$, we can always project it to some subspace $\mathbf{P}^n$ with $n$ equals the dimension of X, by a composition of projection from points. Since projection is affine morphism, we are done. For complete varieties, we have Chow lemma, thus $X$ is dominated by some projective variety, but can we find a such a covering or not?)



Can every algebraic variety of dimension $n$ be covered by $n+1$ affine opens?
No. Example 4.9 of [Roth and Vakil](http://arxiv.org/abs/math/0406384) shows that, for any $m$, there is a singular, integral complete $3$-fold which cannot be covered by $m$ open affines. The authors mention as an open problem whether there is a smooth example. If you don't require varieties to be integral, Example 4.8 gives a simpler construction, which the authors credit to Jason Starr.


There was also some discussion in comments about whether this is true for projective or quasi-projective varieties. The answer is yes in both cases. See the proof of Theorem 18.2.6 in [Vakil's textbook](http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf) for the projective case, and the discussion in Section 22.4.15 for the quasi-projective case.


