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The automorphism group of M ¯ 0 , n

Andrea Bruno, Massimiliano Mella (2013)

Journal of the European Mathematical Society

The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of M ¯ 0 , n is the permutation group on n elements as soon as n 5 .

The automorphism groups of Zariski open affine subsets of the affine plane

Zbigniew Jelonek (1994)

Annales Polonici Mathematici

We study some properties of the affine plane. First we describe the set of fixed points of a polynomial automorphism of ℂ². Next we classify completely so-called identity sets for polynomial automorphisms of ℂ². Finally, we show that a sufficiently general Zariski open affine subset of the affine plane has a finite group of automorphisms.

The number of conjugacy classes of elements of the Cremona group of some given finite order

Jérémy Blanc (2007)

Bulletin de la Société Mathématique de France

This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n = 3 or n = 5 , and that it is equal to 3 (respectively 9 ) if n = 9 (respectively if n = 15 ) and to 1 for all remaining odd orders. Some precise representative elements of the classes are given.

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

We study the group Tame ( SL 2 ) of tame automorphisms of a smooth affine 3 -dimensional quadric, which we can view as the underlying variety of SL 2 ( ) . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT ( 0 ) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame ( SL 2 ) is linearizable, and that Tame ( SL 2 ) satisfies the Tits alternative.

Towards the classification of weak Fano threefolds with ρ = 2

Joseph Cutrone, Nicholas Marshburn (2013)

Open Mathematics

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases...

Transformations birationnelles quadratiques de l'espace projectif complexe à trois dimensions

Ivan Pan, Felice Ronga, Thierry Vust (2001)

Annales de l’institut Fourier

Nous classifions les transformations birationnelles quadratiques de l'espace projectif complexe de dimension trois, à des isomorphismes linéaires près. Elles sont de trois sortes, selon que le degré de leur inverse est 2, 3 ou 4. Il y a en tout 30 types différents; en 1871, L. Cremona en avait déjà décrit 23.

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