René Mboro - Remarks on the geometry of the variety of planes of a cubic fivefold

epiga:10806 - Épijournal de Géométrie Algébrique, October 25, 2023, Special volume in honour of Claire Voisin - https://doi.org/10.46298/epiga.2023.10806
Remarks on the geometry of the variety of planes of a cubic fivefoldArticle

Authors: René Mboro

    This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.


    Volume: Special volume in honour of Claire Voisin
    Published on: October 25, 2023
    Accepted on: June 27, 2023
    Submitted on: January 13, 2023
    Keywords: Mathematics - Algebraic Geometry,14J70, 14M15, 14C30 (primary), 14J42, 14J29 (secondary)

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