cO_{X}(D))$ is isomorphic to $\CC[x,y,z]_{w,d}$, the $w$-homogeneous polynomials of degree $d := \deg_w({\left \lfloor D \right \rfloor})$.

It is a well-known result for integral Weil divisors. The general rational case follows from the fact that by definition $\cO_X(D) = \cO_X({\left \lfloor D \right \rfloor})$. The isomorphism converts $(h)$ into the $w$-homogeneous polynomial defined by $(h) + {\left \lfloor D \right \rfloor} \geq 0$.

\[prop:H0YD\] For any $D \in \operatorname{Weil}_{\QQ}(X)$, $m_i \in \QQ$, and $D' \in \operatorname{Weil}_{\QQ}(Y)$, one has:

1.  \[item1-lemma-h0\] Under the previous isomorphism $H^0(Y,\cO_Y(\pi^{*}D + \sum_{i=1}^{s} m_i E_i))$ can be identified with the subspace $\{ F \in \CC[x,y,z]_{w,d} \mid \operatorname{mult}_{E_i} \pi^{*} F + m'_i \geq 0, \forall i \}$, where $m'_i := m_i + \operatorname{mult}_{E_i} \pi^{*} {\left \{ D \right \}}$, $i=1,\ldots,s$.

2.  \[item2-lemma-h0\] The cohomology $H^0(Y,\cO_Y(D'))$ consists of all $w$-homogeneous polynomials $F$ of degree $\deg_w {\left \lfloor \pi_{*} D' \right \rfloor}$ such that $\operatorname{mult}_{E_i} \pi^{*} F \geq \operatorname{mult}_{E_i} ( \pi^{*} {\left \lfloor \pi_{*} D' \right \rfloor} - D' )$ for all $i=1,\ldots,s$.

The statement \[item2-lemma-h0\] is a direct consequence of \[item1-lemma-h0\] after writing $D'$ in the form $\pi^{*}D + \sum_{i=1}^{s} m_i E_i$ for $D=\pi_{*} D'$ and $m_i = \operatorname{mult}_{E_i}(D'-\pi^{*} D)\in\QQ$.

In order to prove \[item1-lemma-h0\], let us consider $h \in K(Y)$. According to Proposition \[prop:H0D\], its associated divisor $(h)$ belongs to $H^0(Y,\cO_Y(\pi^{*}D + \sum_{i=1}^{s} m_i E_i))$ if and only if $$\label{eq:condition-h}
(h) + \pi^{*}D + \sum_{i} m_i E_i \geq 0.$$ Since $K(Y)$ and $K(X)$ are isomorphic under $\pi$, the global section $h$ can be written as $h = \pi^{*} h'$ for some $h' \in K(X)$. The condition from  in terms of $h'$ is $(h') + D \geq 0$, or equivalently $h \in H^0(X,\cO_X(D))$, and $$\label{eq:condition-h'}
\pi^{*} ((h')+D) + \sum_i m_i E_i = 