2024/11/25 by Di Lorenzo, Andrea, Inchiostro, Giovanni · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.16141
We compactify the moduli stack of maps from curves to certain quotient stacks X=[W/G] with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational transformation for algebraic stacks, the extended weighted blow-up, to prove that any algebraic stack with a properly stable point can be enlarged so that it contains an open substack which is proper and Deligne-Mumford. As a first application, we use our main theorem to construct a compact moduli stack for certain fibered log-Calabi-Yau pairs. We further apply our result to construct a compactification of the space of maps to X when X is respectively: a quotient by a torus of a proper Deligne-Mumford stack; a GIT compactification of the stack of binary forms of degree 2n; a GIT compactification of the stack of 2n-marked smooth rational curves, and a GIT compactification of the stack of smooth plane cubics. In the appendix, we give a criterion for when a morphism of algebraic stacks is an extended weighted blow-up, and we use it in order to give a modular proof of a conjecture of Hassett on weighted pointed rational curves.