We generalize Kim and Sato's construction of the relative configuration space to the case where is an algebraic stack; and construct an analogous projective moduli space for a degeneration . We construct and and prove their properness using a universal construction of expanded degenerations and pairs introduced in our paper with Cadman and Wise. We use these spaces to re-prove the properness of Jun Li’s moduli of relative and degenerate stable maps, and generalize this properness result to the case of target stacks.
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Dan Abramovich, Barbara Fantechi, Configurations of points on degenerate varieties and properness of moduli spaces. Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 1–17DOI 10.4171/RSMUP/137-1