Asymptotic behaviour of rational curves

We investigate the asympotic behaviour of the moduli space of morphisms from the rational curve to a given variety when the degree becomes large. One of the crucial tools is the homogeneous coordinate ring of the variey. First we explain in details what happens in the toric case. Then we examine the general case.Comment: This is a revised and slightly expanded version of notes for a course delivered during the summer school on rational curves held in June 2010 at Institut Fourier, Grenobl

Exemples de comptage de courbes sur les surfaces

Let X be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. Under a simple assumption, we show that the geometric Manin's conjectures hold for some degrees lying in the dual of the effective cone of X (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin's conjecture

Comptage de courbes sur le plan projectif éclaté en trois points alignés

50 pages, in french, to appear in Annales de l'Institut FourierInternational audienceWe prove a version of Manin's conjecture for the projective plane blown up in three collinear points, the base field being a global field of positive characteristic

Produit eulérien motivique et courbes rationnelles sur les variétés toriques

International audienceWe study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of eulerian motivic product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula

La conjecture de Manin g\'eom\'etrique pour une famille de quadriques intrins\`eques

We prove a version of Manin's conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo surface

Stratosphere-troposphere exchange from the Lagrangian perspective: a case study and method sensitivities

International audienceAn important part of extra-tropical stratosphere-to-troposphere transport occurs in association with baroclinic wave breaking and cut-off decay at the tropopause. In the last decade many studies have attempted to estimate stratosphere-troposphere exchange (STE) in such synoptic events with various methods, and more recently efforts have been made to inter-compare these methods. These inter-comparisons show large variations between estimates from different methods. This large uncertainty points to a need to thoroughly evaluate such methods, assess the realism of the resulting STE estimates and determine the sensitivities to intrinsic parameters of the methods. The present study focuses on a trajectory-based Lagrangian method which has been applied in the past to climatological studies. This method is applied here to the quantification of STE in the context of a typical baroclinic wave breaking event. The analysis sheds light on (i) the complex three-dimensional temporal and spatial structures that are associated with the rapid inflow of stratospheric air into the troposphere, (ii) the variation of STE mass flux with the choice of the dynamical tropopause definition within 1.5 to 5 PVU, (iii) the sensitivity of the results to resolution, and in particular the minimum spatial resolution of 1°×1° required to reasonably capture STE fluxes in this wave breaking event, (iv) the effective removal of spurious exchange events using a threshold residence time larger than 8 h

Stratosphere-troposphere exchange from the Lagrangian perspective: a case study and method sensitivities

International audienceAn important part of extra-tropical stratosphere-to-troposphere transport occurs in association with baroclinic wave breaking and cut-off decay at the tropopause. In the last decade many studies have attempted to estimate stratosphere-troposphere exchange (STE) in such synoptic events with various methods, and more recently efforts have been put on inter-comparing these methods. However, large uncertainties remain on the sensitivities to methods intrinsic parameters, and on the best measure for STE with regard to end effects on chemistry. The goal of the present study is to address these two fundamental issues in the context of the application of a trajectory-based Lagrangian method, which has been applied in the past to climatological studies and has also been involved in inter-comparison studies, to a typical baroclinic wave breaking event. The analysis sheds light on (i) the fine mesoscale temporal and spatial structures that are associated with episodic, rapid inflows of stratospheric air into the troposphere; (ii) the spatial resolution of 1°×1° required to reasonably capture STE fluxes in such a wave breaking event; (iii) the effective removal of spurious exchange events using a threshold residence time; (iv) the relevance of residence time distributions for capturing the effective chemical forcing of STE; (v) the large differences in the temporal evolution and geographical distribution of STE fluxes across the 2 and the 4 potential vorticity unit iso-surface definitions of the tropopause

The Nash problem for torus actions of complexity one

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to construct examples of Nash valuations which are neither minimal nor terminal.Comment: minor correction