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Any group of type F4 is obtained as the automorphism group of an Albert algebra. We prove that such a group is R‐trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras. [ABSTRACT FROM AUTHOR]

Let G be a group definable in an o‐minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a split semisimple algebraic group over an arbitrary field F, let E be a G‐torsor over F, and let P be a parabolic subgroup of G. The quotient variety X : = E / P, known as a flag variety, is generically split, if the parabolic subgroup P is special. It is generic, provided that the G‐torsor E over F is a standard generic G k‐torsor for a subfield k ⊂ F and a split semisimple algebraic group G k over k with ( G k ) F = G. For any generically split generic flag variety X, we show that the Chow ring CH X is generated by Chern classes (of vector bundles over X). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup. We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a del Pezzo surface of degree 5 defined over a field F. A theorem of Manin and Swinnerton‐Dyer asserts that every Del Pezzo surface of degree 5 is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree 5 over a field F are precisely the F‐forms of the moduli space M ¯ 0 , 5 of stable curves of genus 0 with five marked points. Suppose n ⩾ 5 is an integer, and F is an infinite field of characteristic ≠2. It is easy to see that every twisted F‐form of M ¯ 0 , n is unirational over F. We show that (a)if n is odd, then every twisted F‐form of M ¯ 0 , n is rational over F; (b)if n is even, there exists a field extension F / k and a twisted F‐form X of M ¯ 0 , n such that X is not retract rational over F. [ABSTRACT FROM AUTHOR]

Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G-torsor. We start a systematic study of the conjecture, raised in in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of [ABSTRACT FROM AUTHOR]

In this article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety of full flags of a split semisimple algebraic group [ABSTRACT FROM AUTHOR]

Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so-called mixed groups of type [ABSTRACT FROM AUTHOR]