I am working on a particular case of the following problem.
Let $X$ be a projective algebraic surface, $L$ a base point free invertible sheaf on $X$ and $\varphi:X\rightarrow \mathbb{P}^n$ the morphism induced by the complete linear system $|L|$. We denote by $W$ the image of $\varphi$. I was wondering if there is any way to compute de dimension of $W$.
This is what I have thought: If we take $m>>0$ then the dimension of $W$ coincides with the degree on $m$ of the Hilbert polynomial $h^0(\mathcal{O}_W(m))$ of $W$. Imagine that we are able to compute $h^0(L^{\otimes m})$. Since $\varphi^*\mathcal{O}_W(1)=L$, we obtain $h^0(\varphi^*\mathcal{O}_W(m))$.
So, my question is: Is there any relation between $h^0(\varphi^*\mathcal{O}_W(m))$ and $h^0(\mathcal{O}_W(m))$ (or at least if we impose certain conditions on our surface or our sheaf)? In case there is not, how could we approach this problem?