p elementary subgroups of the Cremona group
14 pages

p elementary subgroups of the Cremona group


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14 pages
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p-elementary subgroups of the Cremona group Arnaud BEAUVILLE Introduction Let k be an algebraically closed field. The Cremona group Crk is the group of birational transformations of P2k , or equivalently the group of k-automorphisms of the field k(x, y) . There is an extensive classical literature about this group, in particular about its finite subgroups – see the introduction of [dF] for a list of references. The classification of conjugacy classes of elements of prime order p in Crk has been given a modern treatment in [B-B] for p = 2 and in [dF] for p ≥ 3 (see also [B-Bl]). In this note we go one step further and classify p-elementary subgroups – that is, subgroups isomorphic to (Z/p)r for p prime. We will mostly describe such a subgroup as a group G of automorphisms of a rational surface S : we identify G to a subgroup of Crk by choosing a birational map ? : S 99K P 2 . Then the conjugacy class of G in Crk depends only on the data (G,S) . Theorem .? Let G be a subgroup of Crk of the form (Z/p) r with p prime 6= char(k) .

  • torus has

  • preserves gp

  • homographies z

  • then bav

  • let cp ?

  • identify ?

  • now let

  • quaternion algebra over

  • cremona group



Publié par
Nombre de lectures 9
Langue English


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