Characterization of spectral curves of hyperbolic monopoles

Let $S$ be a curve of bidegree $(k,k)$ in ${\mathbb P}_1\times {\mathbb P}_1$. Let $L^r(a,b)$ denote the line bundle used in the hyperbolic-monopole spectral-curve construction, restricted to $S$, and let a real structure mean the corresponding antiholomorphic involution on the line bundle. Spectral-curve characterization. $S$ is the spectral curve of a hyperbolic monopole of charge $k$ and mass $p\in {\frac{1}{2}}{\mathbb Z}_{>0}$ if and only if all of the following hold: (o) $S$ does not intersect the anti-diagonal; (i) $S$ has no multiple components; (ii) $S$ is real and $L^{2p+k}_{|S}$ is holomorphically trivial; (iii) $L^{p+1/2}(k-1,0)_{|S}$ has a real structure; and (iv) $H^0(S,L^r(k-2,0))=0$ for $r=1,2,\ldots,2p+1$.