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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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birational invariants of projective surfaces

I am studying Castelnuovo's rationality criterion for surfaces. Let $S$ be a projective surface and $K$ a canonical divisor on $S$. Let's use the notation $h^i(S,\mathcal{O}_S)=dimH^i(S,\mathcal{O}_S …