Fix an algebraically closed field $k$ and a positive integer $d$. My question is, what is the number of birational classes of dimension $d$, projective varieties over $k$ with geometric genus 0? If it makes the question answerable over fields where we don't know resolution of singularities in higher dimensions, what if we impose smooth?
Obviously for $d=1$ there is a unique birational class. If $d=2$ I know for example that $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$ are not isomorphic, but they are of course birational. Are there explicit examples of two non-birational geometric genus 0 surfaces? Is there some nice way of enumerating the birational classes?
Thanks
