Suppose $\mathcal{L}$ is a base-point free ample invertible sheaf on a complex surface $S$ inducing a finite morphism $S\to \mathbb{P}^2$ (so $\mathcal{L}$ has 3 global sections). I am wondering if there is a connection between the degree of this map and the Hilbert polynomial of $\mathcal{L}$? I have some other hypotheses on $S$ if needed.
Is it correct that the degree of the map will be ($2!$ times) the leading coefficient of the Hilbert polynomial?
If this is not true, can you say anything else about the Hilbert polynomial of $\mathcal{L}$ from the existence of such a map?
Any suggestions/hints are greatly appreciated!