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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 Jacobian Conjecture in case of "non-negative coefficients"

Ludwik M. Drużkowski (1997)

Annales Polonici Mathematici

It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form F ( x , . . . , x n ) = x - H ( x ) : = ( x - H ( x , . . . , x n ) , . . . , x n - H n ( x , . . . , x n ) ) , where H j are homogeneous polynomials of degree 3 with real coefficients (or H j = 0 ), j = 1,...,n and H’(x) is a nilpotent matrix for each x = ( x , . . . , x n ) n . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case d e g F - 1 ( d e g F ) i n d F - 1 , where i n d F : = m a x i n d H ' ( x ) : x n . Note that the above inequality is not true when the coefficients of...

The local lifting problem for actions of finite groups on curves

Ted Chinburg, Robert Guralnick, David Harbater (2011)

Annales scientifiques de l'École Normale Supérieure

Let k be an algebraically closed field of characteristic p > 0 . We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k [ [ t ] ] . To each such  φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k . We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic  0 in such a way that X / H and Y / H have the same genus for all subgroups H G . We determine for which G the KGB obstruction...

Topological types of symmetries of elliptic-hyperelliptic Riemann surfaces and an application to moduli spaces.

José A. Bujalance, Antonio F. Costa, Ana M. Porto (2003)

RACSAM

Sea X una superficie de Riemann de género g. Diremos que la superficie X es elíptica-hiperelíptica si admite una involución conforme h de modo que X/〈h〉 tenga género uno. La involución h se llama entonces involución elíptica-hiperelíptica. Si g > 5 entonces la involución h es única, ver [1]. Llamamos simetría a toda involución anticonforme de X. Sea Aut±(X) el grupo de automorfismos conformes y anticonformes de X y σ, τ dos simetrías de X con puntos fijos y tales que {σ, hσ} y {τ, hτ} no...

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