Let $f:X\dashrightarrow Y$ be a birational map of smooth projective varieties, i.e., there exist open subsets $U_1, \subset X$ and $U_2 \subset Y$ such that $f|_{U_1} : U_1 \rightarrow U_2$ is an isomorphism.

There is an homomorphism induced by $f$ given by

$$ f_* : Div(X) \rightarrow Div(Y) , D\mapsto \overline{ f(D|_{U_1}) }$$ Is known that if $f$ is a small modification then $h^0(f_*(D))=h^0(D)$ for any divisor $D$ of X.

Exists a relation between $h^0(D)$ and $h^0(f_*(D))$ in general for a divisor $D$ of X?

If $D$ is a divisor of $X$ and $\mathcal{L}_D$ is the associated line bundle we consider the following notation $h^0(D) = h^0(X, \mathcal{L}_D)$.