Rationality properties of manifolds containing quasi-lines

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X' that contains a quasi-line, ie a smooth rational curve whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: There is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification X' which is strongly-rational We prove that strongly-rational varieties are stable under smooth, small deformations. Finally, we relate the previous results and formal geometry. This relies on \tilde{e}(X,Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion of X along Y. As applications we show various instances in which X is determined by this formal completion. We also formulate a basic question about the birational invariance of \tilde{e}(X,Y).Comment: 25 page

Mixed multiplier ideals and the irregularity of abelian coverings of the projective plane

A formula for the irregularity of abelian coverings of the projective plane is established and some applications are presented.Comment: 28 page

Jumping numbers of a unibranch curve on a smooth surface

A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.Comment: 20 pages; added reference

Numerical invariants of surfaces in P4{\mathbb P}^4 P 4 lying on small degree hypersurfaces

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Surfaces d'Enriques et une construction de surfaces de type général avecp g =0

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Variétés de Kummer généralisées

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