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Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunately, I don't know how to compute the cohomology of my $S$; I only know that it has 'many holes' in it (i.e. it is 'very far from being projective'), and it is finite over a variety $X$ with $NS(X)=0$ and 'very many holes'.

Any advice would be very welcome! In particular, are there any special methods that can simplify the calculation of $H^2(S)$ in this situation?

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Do you have some particular example in mind? Is it possible to describe your variety? –  Dmitri Dec 30 '12 at 11:58
    
I am trying to study schemes that are finite over products of spectra of function fields, such that the base and one of the multipliers are algebraically closed. –  Mikhail Bondarko Dec 30 '12 at 12:20

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