Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let $\mathrm{HolEmb}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the subspace of those holomorphic maps which are also embeddings, so there is an inclusion
$$i : \mathrm{HolEmb}^d(\Sigma, \mathbb{C} \mathbb{P}^n) \longrightarrow \mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n).$$


If we were discussing smooth maps, instead of holomorphic, Whitney's embedding theorem would say that this map is approximately $(\frac{n}{2}-2)$-connected. Is there a connectivity range for this map of spaces of holomorphic maps?



Approximating holomorphic maps by holomorphic embeddings
I will assume that *d* is the degree of the pull-back of $\mathcal{O}(1)$ to $\Sigma$ and that it is sufficiently large with respect to the genus *g* of $\Sigma$. In this case, the dimension of the space of holomorphic maps of $\Sigma$ in $\mathbb{P}^n$ is 
$$
D\_n := (n+1)d + n(1-g) ,
$$
while the dimension of the space of holomorphic maps that are not isomorphisms onto their image is 
$$
(n+1)d + n(1-g) -n+2 = D\_n -(n-2) .
$$
In particular, since the inclusion that you are interested in has complement of (complex) codimension *n-2*, it follows that it is "quite connected", roughly *(2n-1)*-connected?


Unless I made some mistakes in my computations, the estimates for the dimensions above are only valid if *d* is sufficiently large, otherwise they should simply be lower bounds on the actual dimensions of the spaces. In the case of *n=2* you obviously must allow singularities in the image, but you can also prove that the locus where the morphism is not a local embedding (i.e. when the derivative is not injective somewhere) has codimension one.


