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A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen (1995)

Annales Polonici Mathematici

Let F = X + H be a cubic homogeneous polynomial automorphism from n to n . Let p be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that d e g F - 1 3 p - 1 . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

A geometric approach to the Jacobian Conjecture in ℂ²

Ludwik M. Drużkowski (1991)

Annales Polonici Mathematici

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set g - 1 ( 0 ) (resp. f - 1 ( 0 ) ), then (f,g) is bijective.

Basis of homology adapted to the trigonal automorphism of a Riemann surface.

Helena B. Campos (2007)


A closed (compact without boundary) Riemann surface S of genus g is said to be trigonal if there is a three sheeted covering (a trigonal morphism) from S to the Riemann sphere, ƒ : S →Ĉ. If there is an automorphism of period three, φ, on S permuting the sheets of the covering, we shall call S cyclic trigonal and will be called trigonal automorphism. In this paper we determine the intersection matrix on the first homology group of a cyclic trigonal Riemann surface on an adapted basis B to the trigonal...

Chen–Ruan Cohomology of 1 , n and ¯ 1 , n

Nicola Pagani (2013)

Annales de l’institut Fourier

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable n -pointed curves of genus 1 . In the first part of the paper we study and describe stack theoretically the twisted sectors of 1 , n and ¯ 1 , n . In the second part, we study the orbifold intersection theory of ¯ 1 , n . We suggest a definition for an orbifold tautological ring in genus 1 , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Classification of degree 2 polynomial automorphisms of C3.

John Erik Fornaess, He Wu (1998)

Publicacions Matemàtiques

For the family of degree at most 2 polynomial self-maps of C3 with nowhere vanishing Jacobian determinant, we give the following classification: for any such map f, it is affinely conjugate to one of the following maps:(i) An affine automorphism;(ii) An elementary polynomial autormorphismE(x, y, z) = (P(y, z) + ax, Q(z) + by, cz + d),where P and Q are polynomials with max{deg(P), deg(Q)} = 2 and abc ≠ 0.(iii)⎧ H1(x, y, z) = (P(x, z) + ay, Q(z) + x, cz + d)⎪ H2(x, y, z) = (P(y, z) + ax, Q(y)...

Conformal actions with prescribed periods on Riemann surfaces

G. Gromadzki, W. Marzantowicz (2011)

Fundamenta Mathematicae

It is a natural question what is the set of minimal periods of a holomorphic maps on a Riemann surface of negative Euler characteristic. Sierakowski studied ordinary holomorphic periods on classical Riemann surfaces. Here we study orientation reversing automorphisms acting on classical Riemann surfaces, and also automorphisms of non-orientable unbordered Klein surfaces to which, following Singerman, we shall refer to as non-orientable Riemann surfaces. We get a complete set of conditions for the...

Curves in P2(C) with 1-dimensional symmetry.

A. A. du Plessis, Charles Terence Clegg Wall (1999)

Revista Matemática Complutense

In a previous paper we showed that the existence of a 1-parameter symmetry group of a hypersurface X in projective space was equivalent to failure of versality of a certain unfolding. Here we study in detail (reduced) plane curves of degree d ≥ 3, excluding the trivial case of cones. We enumerate all possible group actions -these have to be either semisimple or unipotent- for any degree d. A 2-parameter group can only occur if d = 3. Explicit lists of singularities of the corresponding curves are...

Field of moduli versus field of definition for cyclic covers of the projective line

Aristides Kontogeorgis (2009)

Journal de Théorie des Nombres de Bordeaux

We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli that can not be defined over is also given.

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