On linear sections of the spinor tenfold. I

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.

GLSMs for exotic Grassmannians

“…The Z 2 subgroup arises as a subgroup of det U (n) that acts trivially on the q ab , and so, from decomposition [19,[45][46][47][48][49], we expect that the Landau-Ginzburg geometry is a disjoint union of two spaces, just as the r 0 geometry. We shall not pursue the geometry of this Landau-Ginzburg phase further in this paper, but it would be interesting to do so, especially to compare to the predictions for this phase from homological projective duality [50][51][52][53].…”

The Cayley Grassmannian

“…There are very nice examples of linear sections (of codimension two and three) of the ten dimensional spinor variety S 10 , which are defined by the generic point of a representation with an open orbit, and turn our for this reason to be locally rigid. However, they are not globally rigid because the generic points of some smaller orbits still define smooth sections, but of a different type [13,4]. In our case, what does happen if we replace the general point z of ∆ by a general point of its invariant octic divisor?…”

“…Can we deduce that of DG? 13 The Bialynicki-Birula decomposition of the wonderful compactification has been studied in [7]. Can one extract a Pieri formula, and push it down to DG?…”

On the automorphisms of Mukai varieties

“…This happens for g = 7, k = 2, 3, and for g = 8, k = 2 (see [BFM18] for the connection with exceptional Lie groups). The case g = 7, k = 2 was studied in detail in [Kuz18].…”