Dimension of a meromorphic differentials space What is the dimension  $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ?
The question is asked in this paper, at the end of the page $(1)$, question $ii)$ "Counting parameters"
A simple example with, for instance $k=2, n= 3,4$ would be welcome too.
 A: First, a reminder of the definition of the residue:
$$
Res_0(\alpha z^{-k} +\beta z^{-k+1} +... )dz^k= \alpha.   
$$
I will do computations only for $k<0$, since otherwise you get only zero for ($k>0$) or constant ($k=0$) meromorphic differentials. 
Let $A=\{z_1,...,z_n\}$ be an $n$-element subset of $S^2$. I will regard $A$ as a divisor on $S^2$; I will use multiplicative notation for divisors. 
Recall that if $D=\prod_i P_i^{d_i}$ is a divisor on $S^2$ (we will need it only for $d_i>0$) then $O_D$ is the 
rank 1 sheaf on $S^2$ of meromorphic functions which are holomorphic away from $D$ and which have poles of order $\le d_i$ at the points $P_i$. 
The space $H^0(O_D)$ is the space of global sections of $O_D$.  To simplify the notation, I will denote this space simply by $L_D$. We will also need the sheaf $\Omega^k=\Omega^k(S^2)$ of holomorphic $k$-differentials on $S^2$; it is nothing but $K^k$, where $K$ is the canonical sheaf of $S^2$ (I will also use the notation $K$ for its divisor). 
We will, furthermore, need some degree computations: For $s>0$, $deg(A^s)=ns$ (here we have the divisor $A^s=\prod_{i=1}^n a_i^s$); and 
$deg(\Omega^k)=-2k$.   
Now, for $s>0$ consider the sheaf $O_{A^s}$ on $S^2$ corresponding to the divisor $A^s$. The space of meromorphic $k$-differentials on $S^2$ you are interested in, is $V_{k}=H^0(O_{A^k}\otimes \Omega^k(S^2))$, since you are allowing poles of order $\le k$ at the points $z_i$, $i=1,...,n$. 
I will use the notation $D_{s,k}$ for the divisor of the sheaf
$$
O_{A^s}\otimes \Omega^k(S^2)
$$
which we will need as well. Since the degree is additive for tensor products of line bundles (as divisors multiply), we obtain:
$$
deg (D_{s,k})= ns-2k. 
$$
By the Riemann-Roch formula, for $D=D_{s,k}$, we have
$$
dim L_{D}= deg (D_{s,k})+ 1 - dim L_{D^{-1} K}  = deg (D_{s,k})+ 1= ns-2k +1,
$$
since $deg(D^{-1} K)<0$ and, hence, the sheaf of this divisor has no nonzero (global) sections. 
At each $z_j$ we have the residue (linear function) 
$$
R_j: V_k\to {\mathbb C}, R_j(\omega)= Res_{z_j}(\omega).
$$ 
Set
$$
R=(R_1,...,R_n). 
$$
Your question then is to compute the dimensions of the affine spaces $R^{-1}({\alpha})$, which are spaces subspaces of $V_k$ consisting of 
meromorphic differentials with residues $\alpha_i$ at $z_i$. The kernel of the linear map $R$ is the space of meromorphic differentials which have poles of the order at most $k-1$ at each $a_i$. In other words, $Ker(R_k)=L_{D_{k-1,k}}$. Thus, the dimension formulae we have give us:
$$
dim V_k= dim L_{D_{k,k}}= kn-2k+1, dim L_{D_{k-1,k}}= (k-1)n-2k+1,
$$ 
which implies that $R: V_k\to {\mathbb C}^n$ is surjective. Therefore, we obtain:
$$
d_{k,n}=dim R^{-1}({\alpha})= dim L_{D_{k-1,k}}= (k-1)n-2k+1
$$
for every ${\alpha}\in {\mathbb C}^n$.  
