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We introduce a compactification of the space of simple positive divisors on aRiemann surface, as well as a compactification of the universal family ofpunctured surfaces above this space. These are real manifolds with corners. Wethen study the space of constant curvature metrics on this Riemann surface withprescribed conical singularities at these divisors. Our interest here is in thelocal deformation for these metrics, and in particular the behavior as conicpoints coalesce. We prove a sharp regularity theorem for this phenomenon in theregime where these metrics are known to exist. This setting will be used in asubsequent paper to study the space of spherical conic metrics with large coneangles, where the existence theory is still incomplete.

Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity