Stringy Chow rings and weighted blow ups
Qiangru Kuang, Yeqin Liu, Rachel Webb, Weihong Xu
TL;DR
The paper extends the computation of the stringy Chow ring $A^*_{st}$ to tame DM stacks of the form $\mathcal{X}=[\widetilde{X}/G]$ with diagonalizable $G$ over general base fields, identifying the cyclotomic inertia, obstruction sheaf, and a base-field–independence framework. It then specializes to weighted blow-ups, providing explicit presentations of $A^*_{st}(\mathscr{B}l_YX)$ via Rees algebra data, and proving finite-generation results under regularity assumptions on the Rees algebra. A central technical achievement is the identification $\mathcal{K}(\mathcal{X})\cong\mathrm{II}_\mu(\mathcal{X})$ with a sectorwise universal family, enabling concrete product formulas in $A^*_{st}(\mathcal{X})$ through the obstruction class and sector geometry. The results generalize complex-analytic computations to arbitrary base fields, yield computable presentations for stringy Chow rings of weighted blow-ups, and offer tools for orbifold intersection theory and birational geometry in a broad setting.
Abstract
We compute the stringy chow ring of a general Deligne-Mumford stack of the form [X/G] for a smooth variety X and diagonalizable group scheme G, working over a base field that is not necessarily algebraically closed. We then specialize to the stringy chow ring of the weighted blow up of a smooth variety along a smooth center. We explore finite generation properties of this ring.