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A boundedness theorem for morphisms between threefolds

Ekatarina Amerik, Marat Rovinsky, Antonius Van de Ven (1999)

Annales de l'institut Fourier

The main result of this paper is as follows: let X , Y be smooth projective threefolds (over a field of characteristic zero) such that b 2 ( X ) = b 2 ( Y ) = 1 . If Y is not a projective space, then the degree of a morphism f : X Y is bounded in terms of discrete invariants of X and Y . Moreover, suppose that X and Y are smooth projective n -dimensional with cyclic Néron-Severi groups. If c 1 ( Y ) = 0 , then the degree of f is bounded iff Y is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional...

A classification theorem on Fano bundles

Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)

Annales de l’institut Fourier

In this paper we classify rank two Fano bundles on Fano manifolds satisfying H 2 ( X , ) H 4 ( X , ) . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization ( ) , that allows us to obtain the cohomological invariants of X and . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

A computation of invariants of a rational self-map

Ekaterina Amerik (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

I prove the algebraic stability and compute the dynamical degrees of C. Voisin’s rational self-map of the variety of lines on a cubic fourfold.

A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski (1998)

Colloquium Mathematicae

Let F=X-H: k n k n be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of G i of degree 2d+1 can be expressed as G i ( d ) = T α ( T ) - 1 σ i ( T ) , where T varies over rooted trees with d vertices, α(T)=CardAut(T) and σ i ( T ) is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, F is an automorphism or, equivalently, G i ( d ) is zero for sufficiently large d....

A group law on smooth real quartics having at least 3 real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

Let C be a smooth real quartic curve in 2 . Suppose that C has at least 3 real branches B 1 , B 2 , B 3 . Let B = B 1 × B 2 × B 3 and let O B . Let τ O be the map from B into the neutral component Jac ( C ) ( ) 0 of the set of real points of the jacobian of C , defined by letting τ O ( P ) be the divisor class of the divisor P i - O i . Then, τ O is a bijection. We show that this allows an explicit geometric description of the group law on Jac ( C ) ( ) 0 . It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...

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