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Local characterization of algebraic manifolds and characterization of components of the set S f

Zbigniew Jelonek (2000)

Annales Polonici Mathematici

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets U i which are isomorphic to closed smooth hypersurfaces in ℂ n + 1 . As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety X ⊂ ℂ m there is a generically-finite (even quasi-finite) polynomial mapping f : ℂ n → ℂ m such that X ⊂ S f . This gives (together with [3]) a full characterization of irreducible components of the set S f for generically-finite polynomial mappings f : ℂ n → ℂ m .

Non-zero constant Jacobian polynomial maps of ℂ ²

Nguyen Van Chau (1999)

Annales Polonici Mathematici

We study the behavior at infinity of non-zero constant Jacobian polynomial maps f = (P,Q) in ℂ² by analyzing the influence of the Jacobian condition on the structure of Newton-Puiseux expansions of branches at infinity of level sets of the components. One of the results obtained states that the Jacobian conjecture in ℂ² is true if the Jacobian condition ensures that the restriction of Q to the curve P = 0 has only one pole.

On quadrirational Yang-Baxter maps.

Papageorgiou, V.G., Suris, Yu.B., Tongas, A.G., Veselov, A.P. (2010)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

On the automorphisms of surfaces of general type in positive characteristic

Edoardo Ballico (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Here we give an explicit polynomial bound (in term of K X 2 and not depending on the prime p ) for the order of the automorphism group of a minimal surface X of general type defined over a field of characteristic p > 0 .

Quadro-quadric Cremona transformations in low dimensions via the  J C -correspondence

Luc Pirio, Francesco Russo (2014)

Annales de l’institut Fourier

It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “ J C -correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

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