Limits of Stable Pairs

Let (X0, B0) be the canonical limit of a one-parameter family of stable pairs, provided by the log Minimal Model Program. We prove that X0 is S2 and that ⌊B0⌋ is S1, as an application of a general local statement: if (X,B+ ǫD) is log canonical and D is Q-Cartier then D is S2 and ⌊B⌋ ∩D is S1, i.e. has no embedded components. When B has coefficients < 1, examples due to Hacking and Hassett show that B0 may indeed have embedded primes. We resolve this problem by introducing a category of stable branchpairs. We prove that the corresponding moduli functor is proper for families with normal generic fiber.