“…Quadric threefolds: We consider first X ⊂ P 4 given by 5 j=1 x 2 j = 0, with a natural action of the Weyl group W (D 5 ). This group has 197 conjugacy classes of subgroups, examined in [23,Section 5] in connection with the analysis of possible Galois actions (or automorphisms) on Picard groups of Del Pezzo surfaces of degree 4; the goal there was to identify potentially rational surfaces over nonclosed fields (see also [31]). There are 112 (conjugacy classes of) subgroups G ⊂ W (D 5 ) which give rise to fixed-point free actions.…”
“…Quadric threefolds: We consider first X ⊂ P 4 given by 5 j=1 x 2 j = 0, with a natural action of the Weyl group W (D 5 ). This group has 197 conjugacy classes of subgroups, examined in [23,Section 5] in connection with the analysis of possible Galois actions (or automorphisms) on Picard groups of Del Pezzo surfaces of degree 4; the goal there was to identify potentially rational surfaces over nonclosed fields (see also [31]). There are 112 (conjugacy classes of) subgroups G ⊂ W (D 5 ) which give rise to fixed-point free actions.…”
“…Such actions have been classified for degree 4 del Pezzo surfaces in [19, p. 15] and for degrees 3, 2, and 1 in [31]. When the surface admits a conic bundle structure W → P 1 over k then the Galois action can be described as follows:…”
Section: Recollections On Stably Rational Non-rational Surfacesmentioning
We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
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