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Stable pairs with a twist Bejleri, Dori


The moduli of stable log varieties or stable pairs \((X,D)\) are the higher dimensional analogue of the compactified moduli of stable pointed curves. The existence of a proper moduli space has been established thanks to the last several decades of advancements in the minimal model program. However, the notion of a family of stable pairs remains quite subtle, and in particular a deformation-obstruction theory for these moduli is not known. When the boundary divisor \(D\) is empty, Abramovich and Hassett gave an approach to stable varieties that replaces \(X\) with an associated orbifold. They show in this setting that the quite subtle notion of family of stable varieties becomes simply a flat family of the associated orbifolds. We extend this approach to the case where there is a nonempty but reduced boundary divisor \(D\) with the hopes of producing a deformation-obstruction theory for these moduli spaces. As an application we show that this approach leads to functorial gluing morphisms on the moduli spaces, generalizing the clutching and gluing morphisms that describe the boundary strata of the moduli of curves. This is joint work with G. Inchiostro.

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