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.

我们提出了稳定双（家庭的替代定义X，  d）在任意的情况下（可能非还原的）基，其中d减小时，通过用（X，  d与适当orbifold对）\（（\ mathcal {X}，\ mathcal {D}）\）。这个稳定族的定义最终与之前的定义相同，但具有更适合变形理论工具的优点。红利为\（（\ mathcal {X}，\ mathcal {d}）\）保持在鼻子上; 没有来自不同的修正项。这导致了稳定曲面族的函子胶合态射和来自 的函子态射的存在\((n + 1)\)维稳定对到n维极化轨道空间。作为一个应用，我们研究了一些表面对的变形理论。