Variety of apolar schemes to powers of quadrics

We study the variety ${\mathrm{V}\mathrm{A}\mathrm{P}\mathrm{S}}_G(q_2^3,10)$ (resp. ${\mathrm{V}\mathrm{A}\mathrm{P}\mathrm{S}}_G(q_3^2,10)$), a Grassmannian compactification of the variety of finite schemes of length $10$ apolar to $q_2^3$ (resp. $q_3^2$), where $\{q_n=0\}\subset {\mathbb{P}}^n$ is a smooth quadric hypersurface. In particular, we show that $\mathrm{V}\mathrm{A}\mathrm{P}{\mathrm{S}}_{\mathrm{G}}(q_2^3,10)$ is the tangent developable of a rational normal curve, while ${\mathrm{V}\mathrm{A}\mathrm{P}\mathrm{S}}_G(q_3^2,10)$ is reducible with three singular $5$-dimensional components.