Configurations of points on degenerate varieties and properness of moduli spaces

Abramovich, Dan; Fantechi, Barbara

Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 1-17.

We generalize Kim and Sato's construction of the relative configuration space $X_D^{\left[n\right]}$ to the case where $X$ is an algebraic stack; and construct an analogous projective moduli space $W_{\pi }^{\left[n\right]}$ for a degeneration $\pi :W\to B$. We construct $X_D^{\left[n\right]}$ and $W_{\pi }^{\left[n\right]}$ 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.