Variety of apolar schemes to powers of quadrics

TL;DR

This paper investigates the geometric structure of varieties of schemes apolar to specific powers of quadrics, revealing that one forms a tangent developable while the other is reducible with singular components.

Contribution

It characterizes the structure of apolar scheme varieties for powers of quadrics, identifying one as a tangent developable and another as reducible with singular components.

Findings

VAPS_G(q_2^3, 10) is the tangent developable of a rational normal curve.

VAPS_G(q_3^2, 10) is reducible with three singular 5-dimensional components.

Provides geometric descriptions of apolar scheme varieties for specific quadric powers.

Abstract

We study the variety $\rm{VAPS}_G(q_2^3, 10)$ (resp. $\rm{VAPS}_G(q_3^2, 10)$), 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 $\rm{VAPS_G}(q_2^3, 10)$ is the tangent developable of a rational normal curve, while $\rm{VAPS}_G(q_3^2, 10)$ is reducible with three singular $5$-dimensional components.