  The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact
Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace
operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold
$(X,|df|^2)$. We define a regularized determinant for $\Delta^{|df|^2}$ and
study it as a functional on the Hurwitz space. We prove that this functional is
related to a system of PDE which admits explicit integration. This leads to an
explicit expression for the determinant of the Laplace operator in terms of the
basic objects on the underlying Riemann surface (the prime form,
theta-functions, the canonical meromorphic bidifferential) and the divisor of
the meromorphic differential $df$. The proof has several parts that can be of
independent interest. As an important intermediate result we prove a
decomposition formula of the type of Burghelea-Friedlander-Kappeler for the
determinant of the Laplace operator on flat surfaces with conical singularities
and Euclidean or conical ends. We introduce and study the $S$-matrix,
$S(\lambda)$, of a surface with conical singularities as a function of the
spectral parameter $\lambda$ and relate its behavior at $\lambda=0$ with the
Schiffer projective connection on the Riemann surface $X$. We also prove
variational formulas for eigenvalues of the Laplace operator of a compact
surface with conical singularities when the latter move.
