Spectral Geometry of Hurwitz Spaces with Arbitrary Ramification

We study the Friedrichs Laplacian associated with the pullback of the round metric on $\text{mb}{P}^1$ by a nonconstant meromorphic function $\phi :X\to \text{mb}{P}^1$ on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula $${\mathrm{Det}}_{\zeta }({\mathrm{\Delta }}_{[\phi ],\text{mc}F})=C\det \mathrm{Im}B\mathrm{\mid }{\tau }_B{\mathrm{\mid }}^2\prod _{k=1}^N\rho (z_k,\overline{z_k}{)}^{c_k}\mathrm{.}$$ The zero eigenvalue is omitted. Here $B$ is the period matrix, ${\tau }_B$ is the local Bergman tau-function, $z_k$ are the branch-value coordinates, $\rho (z,\bar{z})=4(1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{-2}$, and $c_k=\frac{1}{12}{\sum }_j(n_{kj}-n_{kj}^{-1})$, where $n_{kj}$ are the ramification indices over the $k$-th branch value. The constant $C>0$ is independent of the Hurwitz coordinates, and $\det \mathrm{Im}B$ is taken to be $1$ in genus zero. The proof combines smooth trivializations and trace-norm variation of resolvent powers with matrix comparison to spherical conic models. The Davies–Gaffney estimate provides the required uniform high-energy control, while the zero-energy terms are identified through the Schiffer bidifferential and Rauch variational formulas.