Rendiconti del Seminario Matematico della Università di Padova


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Volume 137, 2017, pp. 1–17
DOI: 10.4171/RSMUP/137-1

Published online: 2017-05-12

Configurations of points on degenerate varieties and properness of moduli spaces

Dan Abramovich[1] and Barbara Fantechi[2]

(1) Brown University, Providence, USA
(2) SISSA, Trieste, Italy

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

Keywords: Moduli spaces, algebraic stacks, Gromov–Witten invariants, configuration spaces

Abramovich Dan, Fantechi Barbara: Configurations of points on degenerate varieties and properness of moduli spaces. Rend. Sem. Mat. Univ. Padova 137 (2017), 1-17. doi: 10.4171/RSMUP/137-1