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

TL;DR

This paper characterizes the incidence variety compactification of strata of d-differentials in genus 0 as a blow-up of the moduli space, providing explicit divisor representatives and setting the stage for volume calculations of flat metrics.

Contribution

It explicitly describes the incidence variety compactification as a blow-up along a specific ideal sheaf and provides a divisor representative for the tautological line bundle.

Findings

Incidence variety compactification is isomorphic to a blow-up of the moduli space.

Explicit divisor representative of the tautological line bundle is obtained.

Construction facilitates volume computations of flat metrics with fixed angles.

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 [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.