Stable pairs with a twist and gluing morphisms for moduli of surfaces
Stable pairs \((X, D)\), was first introduced in dimension 2 by \textit{J. Kollár} and \textit{N. I. Shepherd-Barron} [Invent. Math. 91, No. 2, 299--338 (1988; Zbl 0642.14008)], give a natural generalization of stable curves with marked points to higher dimensions. To extend the notion of stable pairs to higher dimensions, main subtle aspect occurs the notion of a flat family of stable pairs over an arbitrary base. One difficulty is that the natural polarization \(K_X + D\) on a stable pair is only a \(\mathbb Q\)-divisor. The second difficulty is that the family of divisors \(D\) is not flat when the coefficients are small. Based on significant advances in the minimal model program and on the notion of \(K\)-flatness of relative Mumford divisors introduced by Kollár, a moduli theory of stable pairs over arbitrary base scheme in characteristic 0 is obtained and projectivity of the coarse moduli space has been proven by \textit{J. Kollár} [``Families of divisors'', Preprint, \url{arXiv:1910.00937}]. However, infinitesimal deformations and obstruction theories for these moduli problems are not well understood in general since the moduli functor does not simply parametrize flat families of pairs \((X, D)\to B\) which are fiberwise stable. In this paper, building upon the work of \textit{D. Abramovich} and \textit{B. Hassett} [in: Classification of algebraic varieties. Based on the conference on classification of varieties, Schiermonnikoog, Netherlands, May 2009. Zürich: European Mathematical Society (EMS). 1--38 (2011; Zbl 1223.14039)], the authors propose an alternative solution for pairs \((X, D)\) over an arbitrary base in the case in which \(D\) is reduced, by replacing \((X, D)\) with an associated twisted stable pair (an appropriate orbifold pair) \((\mathcal X , \mathcal D)\), which is more amenable to the tools of deformation and obstruction theory. Admissible families of stable pairs become just flat families of twisted stable pairs with no extra condition. Adjunction for \((\mathcal X , \mathcal D)\) holds without 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, they study the deformation theory of some surface pairs.
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