2023/10/23 by Arena, Veronica, Di Lorenzo, Andrea, Inchiostro, Giovanni +3 · 1 citation
#14D23 #14E05 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.15076
We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack X and a Cartier divisor E ⊂ X such that (1) E is a weighted projective bundle over a smooth Deligne-Mumford stack Y and (2) for every y\inY we have OX(E)|Ey≃ OEy(-1), then there exists a contraction X\toZ to a smooth Deligne-Mumford stack Z. Moreover, the stack X can be recovered as a weighted blow-up along Y⊂ Z with exceptional divisor E, and Z is a pushout in the category of algebraic stacks. As an application, we show that the moduli stack \mathscrM1,n of stable n-pointed genus one curves is a weighted blow-up of the stack of pseudo-stable curves. Along the way we also prove a reconstruction result for smooth Deligne-Mumford stacks that is of independent interest.