The Incidence Variety Compactification of Strata of d-Differentials in Genus 0

Abstract

Abstract Given $d\in \mathbb {Z}_{\geq 2}$, for every $\kappa =(k_{1},\dots ,k_{n}) \in \mathbb {Z}^{n}$ such that $k_{i}\geq 1-d$ and $k_{1}+\dots +k_{n}=-2d$, denote by $\Omega ^{d}\mathcal {M}_{0,n}(\kappa )$ and $\mathbb {P}\Omega ^{d}\mathcal {M}_{0,n}(\kappa )$ the corresponding stratum of $d$-differentials in genus $0$ and its projectivization, respectively. We specify an ideal sheaf of the structure sheaf of $\overline {\mathcal {M}}_{0,n}$ and show that the incidence variety compactification $\mathbb {P}\overline {\Omega }^{d}\mathcal {M}_{0,n}(\kappa )$ of $\mathbb {P}\Omega ^{d}\mathcal {M}_{0,n}(\kappa )$ is isomorphic to the blow-up of $\overline {\mathcal {M}}_{0,n}$ along this sheaf of ideals. We also obtain an explicit divisor representative of the tautological line bundle on the incidence variety. In an accompanying work [22], the construction of $\mathbb {P}\overline {\Omega }^{d}\mathcal {M}_{0,n}(\kappa )$ in this paper will be used to prove a recursive formula computing the volumes of the spaces of flat metric with fixed conical angles on the sphere.