  We introduce a compactification of the space of simple positive divisors on a
Riemann surface, as well as a compactification of the universal family of
punctured surfaces above this space. These are real manifolds with corners. We
then study the space of constant curvature metrics on this Riemann surface with
prescribed conical singularities at these divisors. Our interest here is in the
local deformation for these metrics, and in particular the behavior as conic
points coalesce. We prove a sharp regularity theorem for this phenomenon in the
regime where these metrics are known to exist. This setting will be used in a
subsequent paper to study the space of spherical conic metrics with large cone
angles, where the existence theory is still incomplete.
