We study static BPS black hole horizons in four dimensional
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0$ so that the spacetime metric is well defined. From~\eqref{ZSol} we see that $R_1$ can always be taken real since any phase is absorbed into $e^{i\psi}$. ]]>
0$ which gives a bound on the space of charges. Otherwise, if both $a_2\neq 0$ and $a_6\neq 0$ we have \be R_2^4=-\frac{a_2}{a_6} \label{R2Sola2a6} \ee and the constraint \be 0=a_0a_6^2-a_2a_4a_6+a_2^2 a_8\,. \label{chargeconstraint} \ee Again we have the bound $R_2^4>0$. In general there are two constraints:~\eqref{chargeconstraint} as well as~\eqref{PQ1}. It is only~\eqref{PQ1} where the genus of the horizon geometry (through $\kappa$) appears and the genus is thus only important when regularity of the solution is imposed. One can also use this explicit solution to obtain an explicit expression for $\cZ$~\eqref{ZSol} and thus $R_1$. It is of some interest to express this in a compact form in terms of the explicit invariants of the diagonal symplectic action on $(\cQ,\cP)$~\cite{NickWip}. \boldmath ]]>