The Incidence Variety Compactification of strata of d-differentials in genus 0

Abstract

Given $d\in {\mathbb{Z}}_{\ge 2}$, for every $\kappa =(k_1,\dots ,k_n)\in {\mathbb{Z}}^n$ such that $k_i\ge 1-d$ and $k_1+\cdots +k_n=-2d$, 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 [29], 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.

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