Counting conics in complete intersections
We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.
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arXiv subjects
Laurent Bonavero
Publications and source records attributed to Laurent Bonavero.
Algebraic foliations defined by quasi-lines
Let X be a projective manifold containing a quasi-line l. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety.
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Quasi-lines and their degenerations
In this paper we study the structure of manifolds that contain a quasi-line and give some evidence towards the fact that the irreducible components of degenerations of the quasi-line should determine the Mori cone. We show that the minimality with respect to a quasi-line yields strong restrictions on fibre space structures of the manifold.
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Pseudo-index of Fano manifolds and smooth blow-ups
Suppose $π: X \to Y$ is a smooth blow-up along a submanifold $Z$ of $Y$ between complex Fano manifolds $X$ and $Y$ of pseudo-indices $i_X$ and $i_Y$ respectively (recall that $i_X$ is defined by $i_X := \min \{-K_X \cdot C | C {is a rational curve of} X\}$). We prove that $i_X \leq i_Y$ if $2\dim (Z)< \dim(Y) + i_Y -1$ and show that this result is optimal by classifying the ``boundary'' cases. As expected, these results are obtained by studying rational curves on $X$ and $Y$.
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Toric varieties whose blow-up at a point is Fano
We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a $T$-fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to $\PP^n$ or to the blow-up of $\PP^n$ along a $\PP^{n-2}$. As expected, such results are proved using toric Mori theory due to Reid.
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Factorisation faible des applications birationnelles [d'après Abramovich, Karu, Matsuki, Włodarczyk et Morelli]
Dans cet exposé, nous expliquons le résultat suivant démontré récemment par Abramovich, Karu, Matsuki et Włodarczyk : toute application birationnelle entre deux variétés algébriques complètes et lisses sur un corps algébriquement clos de caractéristique nulle se factorise en une suite finie d'éclatements et de contractions de centres lisses. Ce résultat n'aurait sans doute pas vu le jour sans la démonstration de son analogue dans le cadre de la géométrie torique par Morelli et Włodarczyk au milieu des années 90. La démonstration donnée par Abramovich, Karu, Matsuki et Włodarczyk est une réduction au cas torique, nous en donnons les grandes étapes.
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Sur des variétés toriques non projectives
With the help of Mori theory for projective toric manifolds due to M. Reid, we study non projective toric manifolds which become projective after a single blow up along an invariant curve.
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Inégalités de Morse et variétés de Moishezon
The central topic of this thesis is the study of some properties of a class of complex compact manifolds~: Moishezon manifolds. In the first part, we generalize J.-P. Demailly's holomorphic Morse inequalities to the case of a line bundle equipped with a metric with analytic singularities on an arbitrary compact complex manifold. Our inequalities give an estimate of the cohomology groups with values in the line bundle tensor powers twisted by the corresponding sequence of multiplier ideal sheaves introduced by Nadel. As a consequence, we obtain a necessary and sufficient analytic condition, invariant by bimeromorphism, for a manifold to be Moishezon. In the second part, we use Mori theory to analyze the structure of Moishezon manifolds with infinite cyclic Picard group, with big canonical bundle, and which become projective after one single blow-up with smooth center. We study the dimension and the structure of the center of the blow-up. In dimension four, we show that this locus is always a surface, and when the canonical bundle
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