# Branched maps from a Riemann surface to projective space

Consider a compact Riemann surface $$S$$. Much is known about the behaviour of maps $$f:S\to\mathbb{CP}^1$$. In particular, we know that given that $$f$$ has ramification $$e(p)$$ at each $$p\in S$$, we can use the Riemann-Hurwitz formula to find the degree of $$f$$, namely $$\text{deg}(f)=1-g+\sum_{p\in S}\frac{e(p)-1}{2}\ .$$ Moreover, we can say a lot about the moduli space of branched covers of the sphere by a surface $$S$$. For instance, if $$X$$ has genus $$g$$ and $$n$$ marked points $$z_i\in S$$, $$n$$ marked points $$x_i\in\mathbb{CP}^1$$, and $$n$$ nontrivial ramification indices $$w_i=e(z_i)$$, there exists a branched cover $$f:S\to\mathbb{CP}^1$$ with $$f(z_i)=x_i$$ at only finitely many points in the moduli space $$\mathcal{M}_{g,n}$$ as we vary the surface $$S$$ and the points $$z_i$$. In this sense, once one fixes the indices $$w_i$$ and the points $$x_i$$ on $$\mathbb{CP}^1$$, the surfaces which cover the sphere with those ramifications localize in the moduli space of surfaces.

What, if anything, can we say about maps $$f:S\to\mathbb{CP}^3$$ (and, in general, for $$\mathbb{CP}^r$$ with $$r\geq 1$$)? What would be the analog of the ramification $$e(p)$$ at a point $$p\in S$$? Are there similar localization principles for 'branched' maps to $$\mathbb{CP}^3$$? Is there an equivalent of the Riemann-Hurwitz formula for maps into higher dimensional projective space?