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 of these families 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. Of independent interest is how setting up the analysis
on these compactified configuration spaces provides a good framework for
analyzing “confluent families” of regular singular, ie conic, elliptic differential
operators.
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