# Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, homogeneous $G$-variety.

I recently learned that $X:=F_4/P_4$ is a hyperplane section of $Y:=E_6/P_6$. One obtains the Hasse diagramm of CH(X) by removing one dot in the Hasse diagramm of CH(Y) in degrees $8,12,16$, which is reflected by the fact that the Grothendieck-Chow motive of $Y$ has one more summand $R(8)$, namely a generalized Rost-Motive corresponding to $g_3 \in H^3(k,\mu_3)$ and thus splitting into

$\overline{R(8)} =$ $(\mathbb{Z}$ $\oplus$ $\mathbb{Z}(4)$ $\oplus$ $\mathbb{Z}(8))(8)$

Question: Are there other examples of algebraic groups $G,H$ with parabolic subgroups $P_i,P_j$ such that $G/P_i$ is a hyperplane section of $H/P_j$ known?

• A smooth hyperquadric in $\mathbb{P}^n$ (say, over an algebraically closed field) is a homogeneous space under $\mathrm{SO}(n+1)$; a smooth hyperplane section is again a hyperquadric, hence homogeneous under $\mathrm{SO}(n)$. Same with $\mathbb{P}^n$ (homogeneous under $\mathrm{PGL}(n+1)$).
– abx
May 28, 2015 at 9:09
• Very good, so far. Edit: Oh this introduces a new question for quadrics! But before i ask, is the same true for an anisotropic quadric (especially k not algebraically closed)?
– nxir
May 28, 2015 at 22:38
• Shouldn't the title be about principal homogeneous spaces, rather than groups? May 29, 2015 at 4:32
• Yes, i fixed that.
– nxir
May 29, 2015 at 9:45
• You might want to take a look at Ahiezer's classification of equivariant 2-Orbit completions in char. 0 (and Knop's amendment in char p). Nov 16, 2015 at 16:15