TITLE: Can every algebraic variety of dimension $n$ be covered by $n+1$ affine opens?
QUESTION [11 upvotes]: 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?)

REPLY [14 votes]: No. Example 4.9 of Roth and Vakil 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 for the projective case, and the discussion in Section 22.4.15 for the quasi-projective case.