On spinor varieties and their secants

TL;DR

This paper investigates the algebraic equations defining the secant varieties of spinor and related Freudenthal varieties, revealing degree thresholds for cubic equations and similarities across these geometric objects.

Contribution

It establishes the existence of cubic equations for secant varieties of spinor varieties only when n≥9 and highlights the similar behaviors among Freudenthal varieties.

Findings

Cubic equations for secant varieties exist only when n≥9 in type D_n.

The ideal of these secant varieties is generated in degrees at least three and four.

Other Freudenthal varieties show similar algebraic behaviors.

Abstract

We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type $D_n$, cubic equations exist if and only if $n\ge 9$. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.