Let $f:X\rightarrow Y$ be a birational map of projective varieties. Is known that if $f$ is a small modification then $h^0(f_*(D))=h^0(D)$ for any $D\in Pic(X)$.
Exists a relation between $h^0(D)$ and $h^0(f_*(D))$ in general?
Zero-cohomology of birational varieties
Joaquín Moraga
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