Identifiability and singular locus of secant varieties to spinor varieties

TL;DR

This paper investigates the structure and properties of secant varieties to Spinor varieties, focusing on orbit classification, identifiability issues, and the singular locus, providing new insights into their geometric and algebraic features.

Contribution

It determines the orbit structure of secant varieties to Spinor varieties and addresses problems of identifiability, tangential-identifiability, and the singular locus within this context.

Findings

Identified the poset of $Spin(V)$-orbits and their dimensions.

Solved identifiability and tangential-identifiability problems.

Characterized the singular locus and its relation to the tangential variety.

Abstract

In this work we analyze the $Spin(V)$-structure of the secant variety of lines $\sigma_{2}(\mathbb{S})$ to a Spinor variety $\mathbb{S}$ minimally embedded in its spin representation. In particular, we determine the poset of the $Spin(V)$-orbits and their dimensions. We use it for solving the problems of identifiability and tangential-identifiability in $\sigma_2(\mathbb S)$, and for determining the second Terracini locus of $\mathbb{S}$. Finally, we show that the singular locus $Sing(\sigma_{2}(\mathbb{S}))$ contains the two $Spin(V)$-orbits of lowest dimensions and it lies in the tangential variety $\tau(\mathbb{S})$: we also conjecture what it set-theoretically is.