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Let $S, T$ be algebraic surfaces over $k=\mathbb{C}$, and $\phi: S \longrightarrow T$ a surjective morphism. Furthermore we have the numerical invariants: \begin{align*} q(S) &:= \dim H^1(S, \mathcal{O}_S)\\ P_n &:= \dim H^0(S, nK) \end{align*} where $K$ is a canonical divisor.

Now from the literature, i understand that we have \begin{align*} q(S) &\geq q(T)\\ P_n(S) &\geq P_n(T) \end{align*} which makes intuitively a lot of sense, since i imagine we could somehow pull back elements of the cohomology groups from $T$ to those of $S$.

Could someone explain the principle behind this? It might be really easy; i know just the very basics of sheaf cohomology and am a little stressed for my exam on monday. =)

(By the way, since i mention an exam i feel the need to state: this is not homework)


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