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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 4, Pages 53–114 (Mi izv8756)  

This article is cited in 5 scientific papers (total in 6 papers)

On linear sections of the spinor tenfold. I

A. G. Kuznetsov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

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 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.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.4213/im8756

Full text: PDF file (1075 kB)
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English version:
Izvestiya: Mathematics, 2018, 82:4, 694–751

Bibliographic databases:

UDC: 512.7
MSC: 14J40, 14J60, 14M17, 14C05, 14C15
Received: 29.12.2017

Citation: A. G. Kuznetsov, “On linear sections of the spinor tenfold. I”, Izv. RAN. Ser. Mat., 82:4 (2018), 53–114; Izv. Math., 82:4 (2018), 694–751

Citation in format AMSBIB
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    Erratum

    This publication is cited in the following articles:
    1. Watanabe K., “Fano Manifolds of Coindex Three Admitting Nef Tangent Bundle”, Geod. Dedic.  crossref  mathscinet  isi
    2. A. G. Kuznetsov, “Pismo v redaktsiyu”, Izv. RAN. Ser. matem., 82:5 (2018), 227–228  mathnet  crossref  mathscinet  elib
    3. Kuznetsov A.G., Prokhorov Yu.G., Shramov C.A., “Hilbert Schemes of Lines and Conics and Automorphism Groups of Fano Threefolds”, Jap. J. Math., 13:1 (2018), 109–185  crossref  mathscinet  zmath  isi  scopus
    4. Ch. Bai, B. Fu, L. Manivel, “On fano complete intersections in rational homogeneous varieties”, Math. Z., 295:1-2 (2020), 289–308  crossref  mathscinet  zmath  isi
    5. V. Przyjalkowski, C. Shramov, “Hodge level for weighted complete intersections”, Collect. Math., 71:3 (2020), 549–574  crossref  mathscinet  zmath  isi
    6. L. Manivel, “Topics on the geometry of rational homogeneous spaces”, Acta. Math. Sin.-English Ser., 36:8 (2020), 851–872  crossref  mathscinet  zmath  isi
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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