Geodesic
On linear sections of the spinor tenfold.~I
Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 694-751

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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 endowed with a non-degenerate quadratic form. In particular, we show that if the dimension of a linear section of $X$ is at least 5, then its integral Chow motive is of Lefschetz type. We discuss the 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 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.
Keywords: spinor variety, linear sections, Chow motives, birational transformations, classification of algebraic varieties, Hilbert schemes. 
@article{IM2_2018_82_4_a2,
     author = {A. G. Kuznetsov},
     title = {On linear sections of the spinor {tenfold.~I}},
     journal = {Izvestiya. Mathematics },
     pages = {694--751},
     publisher = {mathdoc},
     volume = {82},
     number = {4},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a2/}
}
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A. G. Kuznetsov. On linear sections of the spinor tenfold.~I. Izvestiya. Mathematics , Tome 82 (2018) no. 4, pp. 694-751. http://geodesic.mathdoc.fr/item/IM2_2018_82_4_a2/