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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?

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For your 2nd and 4th questions: yes. The reference is Proposition 1.1 in D. Eisenbud and J. Harris "Divisors on general curves and cuspidal rational curves".

(I'm afraid I don't understand your 'localization' principle. Edit: ok, maybe I can guess what it wants to say, though it's not stated clearly.)

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