If an equidimensional variety $V$ of dimension $m$ is locally a set-theoretic complete intersection (i.e., it can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ for a large enough $N$; we consider set-theoretic intersections here) then it can also be covered by complements to 
smooth affine varieties $A_i$ of dimension $N$ of unions of $N-m$ of  open affine subvarieties  of $A_i$. I wonder: is the converse true also? What can be said about the versions of these statements for $N-m+c$ instead of $N-m$ (for some $c>0$; cf. http://mathoverflow.net/questions/191210/how-would-you-call-a-variety-that-is-locally-a-complete-intersection-up-to-defec)? I don't know whether the answers depend on the base field.