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The set of points at which a polynomial map is not proper

Zbigniew Jelonek (1993)

Annales Polonici Mathematici

We describe the set of points over which a dominant polynomial map f = ( f 1 , . . . , f n ) : n n is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by ( i = 1 n d e g f i - μ ( f ) ) / ( m i n i = 1 , . . . , n d e g f i ) .

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.

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