Alternative moduli of twisted curves

Abstract

We define new Artin stack compactifications of the moduli of smooth curves, which parameterize "relatively polarized curves" consisting of stacky semistable curves equipped with line bundles that are trivialized away from the rational bridges. Using techniques from birational geometry, we prove that stacks of relatively polarized curves admit the classical coarse space $\overline{M}_g$ of stable curves as a good moduli space. One of our main motivations is a companion paper where we construct a proper moduli space of $\mathrm{SL}_N$-bundles on relatively polarized curves. This suggests the hypothesis that relatively polarized curves are suitable sources for defining Gromov-Witten theory for Artin stack targets.