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In the paper

Ilev, Manivel - The Chow ring of the Cayley plane

we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of an 8-dimensional quadric [Q]. We consider $CH(-)$ mod rational equivalence.

Now I would like to know more about these quadrics.

$\bullet$ What is known about them? (How abstract "are" we ?)

$\bullet$ Is there any chance for example to rule out possible splitting patterns of the respective quadratic form (of dimension 10)?

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    $\begingroup$ Iliev and Manivel work over $\mathbb{C}$, so the quadratic form is the hyperbolic one. Of course, the construction can be considered over an arbitrary field, but then one would use the split form of $E_6$ and I think in this case, the quadratic form also has to be the hyperbolic one. $\endgroup$
    – Matthias Wendt
    Commented Dec 19, 2016 at 19:18
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    $\begingroup$ The quadrics can be described as follows. One can consider $Y := E_6/P_5$ (so that $Y$ is another Cayley plane) and $Z := E_6/P_{1,5}$. Then there are natural maps $p:Z \to X$ and $q:Z \to Y$. The quadrics are $Q_y = p(q^{-1}(y))$, where $y \in Y$. $\endgroup$
    – Sasha
    Commented Dec 19, 2016 at 19:36
  • $\begingroup$ I studied the mod 2 motivic decompositions of $E_6$ varieties (mod 3 case is known) and noticed, that in the isotropic case, where $E_6$ has kernel $D_4$, there are Tate motives of $M(X)$ in dimensions $0,8,16$, with the dimension $8$ one beeing realized "onto" the cycle $Z = [6,5,..]$ (Bruhat order), which should be the quadric. So i thought that the quadric may be isotropic iff $X$ is, such that one might be able to construct an invariant to some $H^n(k,\mu_2)$ by knowing more about these quadrics. But i guess there is no chance to extract such delicate information out of a cylce class. $\endgroup$
    – nxir
    Commented Dec 19, 2016 at 20:45
  • $\begingroup$ @nxir that sounds interesting. My point was just that the description of the Chow ring by Iliev and Manivel is for the split case and in this case the quadric is also split. It is not clear to me what the Chow ring would look like for twisted forms of the Cayley plane (and in particular if the class of the quadric is visible and has a quadric representative). $\endgroup$
    – Matthias Wendt
    Commented Dec 20, 2016 at 10:37
  • $\begingroup$ Ok it seems that my mistake was to assume that the class of the quadric IS visible, even when $X$ is anisotropic. Maybe i should study the isotropic, but non split case. $\endgroup$
    – nxir
    Commented Dec 20, 2016 at 17:28

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