New PDF release: Some properties of the Cremona group

Nonfiction 6

By Julie Déserti

Summary. We bear in mind a few homes, regrettably now not all, of the Cremona
We first commence by way of proposing a pleasant evidence of the amalgamated product
structure of the well known subgroup of the Cremona workforce made of the
polynomial automorphisms of C2. Then we care for the type of
birational maps and a few purposes (Tits substitute, non-simplicity...)
Since any birational map might be written as a composition of quadratic
birational maps as much as an automorphism of the complicated projective plane,
we spend time on those targeted maps. a few questions of staff idea are
evoked: the class of the finite subgroups of the Cremona crew and
related difficulties, the outline of the automorphisms of the Cremona
group and the representations of a few lattices within the Cremona group.
The description of the centralizers of discrete dynamical platforms is an
important challenge in genuine and complicated dynamic, we describe the kingdom of
the artwork for this challenge within the Cremona group.
Let S be a compact advanced floor which consists of an automorphism f
of confident topological entropy. both the Kodaira size of S is zero
and f is conjugate to an automorphism at the special minimum version of
S that is both a torus, or a K3 floor, or an Enriques floor, or S is
a non-minimal rational floor and f is conjugate to a birational map of
the advanced projective aircraft. We take care of effects bought during this last
case: building of such automorphisms, dynamical homes (rotation

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P2ν−1 . The corresponding Cremona transformation is a de Jonqui`eres transformation. 2 Here a rational fibration is a rational application from P2 (C) into P1 (C) whose fibers are rational curves. 3 The de Jonqui` eres group is birationally isomorphic to the subgroup of Bir(P1 (C) × P1 (C)) which preserves the first projection p : P1 (C) × P1 (C) → P1 (C). 38 Julie D´eserti Indeed let Γ be an element of Λf . Let Ξ be the pencil of curves of Λf that have in common with Γ a point m distinct from p1 , .

The length of the element given by Danilov is 26. ) is invariant by inner conjugacy, we can thus assume that f has minimal length in its conjugacy class. 8 ([94]). Let f be an element of Aut(C2 ). Assume that det jac f = 1 and that f has minimal length in its conjugacy class. • If f is non trivial and if ℓ(f ) ≤ 8, the normal subgroup generated by f coincides with the group of polynomial automorphisms f of C2 with det jac f = 1; • if f is generic6 and if ℓ(f ) ≥ 14, the normal subgroup generated by f is strictly contained in the subgroup f ∈ Aut(C2 ) det jac f = 1 of Aut(C2 ).

Let us denote by Eclat(S) the union of the surfaces endowed with a birational morphism π : S′ → S modulo the following equivalence relation: S ∋ p1 ∼ p2 ∈ S if and only if ε−1 2 ε1 sends p1 onto p2 and is a local isomorphism between a neighborhood of p1 and a neighborhood of p2 . A point of Eclat(S) corresponds either to a point of S, or to a point on an exceptional divisor of a blow-up of S etc. Any surface S′ which dominates S embeds into Eclat(S). Let us consider the free abelian group Ec(S) generated by the points of Eclat(S); we have a scalar product on Ec(S) (p, p)E = −1, (p, q) = 0 if p = q.

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Some properties of the Cremona group by Julie Déserti

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