On linear sections of the spinor tenfold, I

TL;DR

This paper explores the geometry of linear sections of the spinor tenfold, revealing their Chow motives, classifying small codimension sections, and introducing a quadratic line complex linked to these sections.

Contribution

It provides a detailed classification of smooth linear sections of the spinor tenfold and establishes their Chow motives as Lefschetz type for sections of dimension at least 5.

Findings

Chow motives of large linear sections are of Lefschetz type.

Unique isomorphism class for smooth hyperplane sections.

Two isomorphism classes for smooth codimension 2 sections.

Abstract

We discuss the geometry of transverse linear sections of the spinor tenfold $X$, the connected component of the orthogonal Grassmannian of 5-dimensional isotropic subspaces in a 10-dimensional vector space equipped with a non-degenerate quadratic form. In particular, we show that as soon as the dimension of a linear section of $X$ is at least 5, its integral Chow motive is of Lefschetz type. We discuss classification of smooth linear sections of $X$ of small codimension; in particular we check that there is a unique isomorphism class of smooth hyperplane sections and exactly two isomorphism classes of smooth linear sections of codimension 2. Using this, we define a natural quadratic line complex associated with a linear section of $X$. We also discuss the Hilbert schemes of linear spaces and quadrics on $X$ and its linear sections.