Stable pairs with a twist

It is well known that, for pointed nodal curves, considering flat and proper families of pairs (X,D) leads to a proper moduli space. Still, while the notion of stable pairs is a higher dimensional analogue of pointed nodal curves, the right definition of a family of stable pairs is far from obvious. In this work, building on an idea of Kollár and the work of Abramovich and Hassett, we give an alternative definition of a family of stable pairs, in the case where the divisor D is reduced. This definition is more amenable to the tools of deformation theory. As an application we produce functorial gluing morphisms on the moduli spaces of surfaces, generalizing the clutching and gluing morphisms that describe the boundary strata of the moduli of curves. This is joint work with D. Bejleri.