Stable pairs with a twist and gluing morphisms for moduli of surfaces

Abstract

We propose an alternative definition for families of stable pairs (X, D) over an arbitrary (possibly non-reduced) base in the case in which D is reduced, by replacing (X, D) with an appropriate orbifold pair \((\mathcal {X},\mathcal {D})\). This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Adjunction for \((\mathcal {X},\mathcal {D})\) holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial morphisms from \((n + 1)\) dimensional stable pairs to n dimensional polarized orbispaces. As an application, we study the deformation theory of some surface pairs.