On linear sections of the spinor tenfold. I

Abstract

We discuss the geometry of transverse linear sections of the spinor tenfold , the connected component of the orthogonal Grassmannian of 5-dimensional isotropic subspaces in a 10-dimensional vector space endowed with a non-degenerate quadratic form. In particular, we show that if the dimension of a linear section of is at least 5, then its integral Chow motive is of Lefschetz type. We discuss the classification of smooth linear sections of 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 sections of codimension 2. Using this, we define a natural quadratic line complex associated with a linear section of . We also discuss the Hilbert schemes of linear spaces and quadrics on and its linear sections.