Variety of apolar schemes to powers of quadrics

Abstract

We study the variety ${\mathrm{VAPS}}_{\mathrm{G}}({\mathrm{q}}_2^3,10)$ (resp. ${\mathrm{VAPS}}_{\mathrm{G}}({\mathrm{q}}_3^2,10)$), 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 $\mathrm{VAP}{\mathrm{S}}_{\mathrm{G}}({\mathrm{q}}_2^3,10)$ is the tangent developable of a rational normal curve, while ${\mathrm{VAPS}}_{\mathrm{G}}({\mathrm{q}}_3^2,10)$ is reducible with three singular $5$-dimensional components.

53 pages, comments welcome