  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.
