  Branched covers between Riemann surfaces are associated with certain
combinatorial data, and Hurwitz existence problem asks whether given data
satisfying those combinatorial constraints can be realized by some branched
cover. We connect recent development in spherical conic metrics to this old
problem, and give a new method of finding exceptional (unrealizable) branching
data. As an application, we find new infinite sets of exceptional branched
cover data on the Riemann sphere.
