Pour tout $t\in\mN$ nous définissons un certain entier positif $\N_t$ et nous conjecturons: si $H$ est un fibré de Gauss-Manin d'une fibration semi-stable alors la $t$-ème classe de Chern de $H$ est annulée par $\N_t$. Nous démontrons diverses conséquences de cette conjecture.
For any $t\in\mN$, we define a certain positive integer $\N_t$ and we conjecture: if $H$ is a Gauss-Manin bundle of a semi-stable fibration then the $t$-th Chern class of $H$ is kiled by $\N_t$. We prove various consequences of this conjecture.
Resume Soit SU_C(r) l'espace des modules des fibrés vectoriels semi-stables de déterminant trivial sur une courbe lisse $C$ de genre $g \geq 2$ sur $\mathbb{C}$. On étudie dans cet article, un exemple de fibré introduit par Raynaud dans ~\cite{R}, ne possédant pas de diviseur thêta. On construit ensuite des extensions stables de ce fibré ce qui conduit à une majoration de la codimension du lieu de base du fibré déterminant sur $\mathcal{SU}_{C}(r)$. Abstract Let $\mathcal{SU}_{C}(r)$ be the moduli space of semi stable bundles with trivial determinant on a smooth curve $C$ of genus $g\geq 2$ on…
We compute some cohomology spaces for the symmetric power of the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We compute the space of global sections for the symmetric power of the tautological bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We give a geometric Langlands type proof of the geometrization conjecture of the local Langlands correspondence introduced by the author for GL_1. For this we study an Abel-Jacobi morphism. We prove that this morphism is a pro-étale locally trivial fibration in simply connected diamonds in high degree. Those diamonds are absolute punctured Banach-Colmez spaces that we study in details.
En nous appuyant sur une construction due à Bedford et Taylor, et certains résultats récents de Demailly, nous présentons une extension (partielle) de la géométrie d'Arakelov aux fibrés en droites intégrables. (Ces derniers sont les fibrés en droites hermitiens sur une variété arithmétique $X$ pouvant se décomposer sous la forme $\ov{E} = \ov{E}_{1}\otimes (\ov{E}_{2})^{-1}$, où $\ov{E}_{1}$ et $\ov{E}_{2}$ sont des fibrés en droites munis à l'infini d'une métrique continue approchable uniformément sur $X(C)$ par des métriques positives $C^{\infty}$). Nous appliquons notre théorie aux fibrés e…
In this thesis we have proved a conjecture about the moduli space SU_X(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve X, due to I. Dolgachev. Given X a smooth projective curve of genus 2, and the embedding of the jacobian JX into |3Θ|, A. Coble proved, at the begining of XX century, that there exists a unique cubic hypersurface C in |3Θ|* \simeq P^8, JX[3]-invariant and singular along JX. On the other hand, we have a map of degree 2 from SU_X(3) over |3Θ|, ramified along a sextic hypersurface B. The Dolgachev's conjecture affirms that the sextic B is the du…
Let $X$ be a smooth projective toric variety over $Spec(Z)$. Let $\bar{\mathcal{O}(D)}$ be an equivariant hermitian line bundle on $X$, equipped with a positive metric and invariant under the action of the compact torus of $X(C)$, we provide a formula for the arithmetic volume in term of Fenchel-Legendre transform attached to the metric.
We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by t…
Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une variété propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des zé…
Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of $\sha(Jac(C))$, if the fibers over a generalised Hilbertian subset satisfy the Hasse principle (resp. weak approximation), then the Brauer-Manin obstruction coming from the base curve is the only obstruction to the Hasse principle (resp. to weak approximation) for zero-cycles of degree 1 on $X$.
Generalising work of La Bretèche, Browning and Peyre and of the author, we describe the geometry required by Manin's principle and Peyre's conjecture for a family of conic bundle surfaces containing Châtelet surfaces. These surfaces $S_{a,F}$ are the conic bundle surfaces obtained as smooth minimal proper model of $$ Y^2-aZ^2=F(X,1) $$ with $a \in \mathbf{Z}$ squarefree and $F \in \mathbf{Z}[x_1,x_2]$ a binary form of \textit{even} degree $n$ without repeated roots and whose irreducible factors over $\mathbf{Q}$ remain irreducible over $\mathbf{Q}\left( \sqrt{a} \right)$.
In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singulari…
Dans cette note, nous donnons une autre démonstration du résultat démontré par D. Barlet dans http://aif.cedram.org/aif-bin/fitem?id=AIF_2010__60_1_319_0 (arXiv:0809.4981), en nous appuyant sur la transformation de Mellin et des propriétés classiques des fonctions de Bessel (au lieu de la convolution). Le théoréme principal de cette note est un peu plus précis que l'énoncé donné dans loc. cit. [English: In this article, we give another proof of the result shown by D. Barlet in http://aif.cedram.org/aif-bin/fitem?id=AIF_2010__60_1_319_0 (arXiv:0809.4981), relying on the Mellin transform and on …
Nous calculons le groupe de Picard de l'espace des modules $Θ_{\planp}(d)$ des thêta-caractéristiques des courbes planes (non forcément lisses) de degré $d$. Nous montrons que pour $d\ge 6$, le groupe de Picard de la composante paire est engendré par le fibré déterminant ${\cal{D}}$ et l'image réciproque ${\cal{L}}$, sous le morphisme support schématique, de ${\cal{O}}_{\proj}(1)$ où $\proj$ est l'espace des courbes de degré $d$ sur $\planp$. Pour $1\le d \le 4$, ce groupe est engendré uniquement par ${\cal{L}}$, le fibré déterminant étant trivial dans ces cas. Dans le groupe $\Cl$ des diviseu…
Let $X$ be a complete symmetric variety i.e. the wonderful compactification of a symmetric $G-$homogeneous space (where $G$ is a simply-connected semi-simple linear algebraic group). If $L$ is a line bundle over $X$ and if $C$ is a Bialynicki-Birula cell of codimension $c$ in $X$, then the Lie algebra $\mathfrak g$ of $G$ operates naturally on the cohomology group with support : $H^c_C(L)$. One gives here a necessary condition on the cell $C$ for that $\mathfrak g-$module have a finite dimensional simple subquotient. As applications one calculates the Euler-Poincaré characteristic of $L$ over …
Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ impl…
This paper is devoted to the estimation of the number of points of bounded height on fibrations in toric varieties over algebraic varieties, generalizing previous work by Strauch and the second author. Under reasonable hypotheses on ``Arakelov L-functions'' of the base, we show how to deduce a good estimate for the open subset of the total space of the unerlying fibration in torus. In passing, we improve drastically the error term for toric varieties themselves, generalizing a theorem by de la Breteche over any number field.
In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of P^2 ramified along six lines in general position.
We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
In this paper we give the Weierstrass equations for Jacobian fibrations on the K3 surface that is the minimal resolution of the double covering of projective plane ramified along generic six lines.
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surface…
We describe the scheme of jumping lines of logarithmic vector bundles on the projective plane. This result is already proved by Dolgachev and Kapranov when the first Chern class is even, it is new when the first Chern class is odd.
This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over the rationals. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. This is related to work of J. Top and N. Yui which exhibited the same procedure for the se…
We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.
We study equations in four variables (x,y,z,t) of the shape q(x,y,z)=P(t), where q(x,y,z) is an indefinite ternary quadratic form over the integers and P(t) is a polynomial in one variable with integral coefficients. If P(t) is not the product of a constant and the square of a polynomial, strong approximation holds for integral solutions (x,y,z,t). In the general case, we show that the integral Brauer-Manin conditions are the only obstructions to strong approximation. We actually study the analogous situation over an arbitrary number field. --- Nous étudions les équations à quatre variables (x…
We study in this paper a new family of stable algebraic vector bundles of rank $ n-1 $ on the complex projective space $\mathbb{P}^{n}$ whose weighted Tango bundles of Cascini \cite{ca} belongs to. We show that these bundles are invariant under a miniversal deformation.
We provide supporting examples to Le Potier's Strange duality conjecture, in the case of the moduli space M of rank 2 semi-stable sheaves on the projective plane, with even first Chern class, and second Chern class less or equal to 19. We compute in this case the dimension of the space of global sections of the determinant bundle on M.
Soit $X$ une hypersurface lisse de degré $δ\geq 4$ dans $\pp ^3$ telle que $Pic(X)=Z$. On désigne par $M_X(c_2)$ l'espace de modules des fibrés vectoriels sur $X$ de classes de Chern $c_1 = 0$ et $c_2$, semi-stables par rapport au diviseur hyperplan. Nous contribuons ici à la recherche de différentes composantes irréductibles pour $c_2$ petit. On prouve que pour tout entier $c_2 \geq δ^3/4 - δ^2/2$ l'espace des modules $M_X(c_2)$ contient une composante irréductible réduite de dimension attendue. En utilisant un résultat de O'Grady, on en déduit que pour tout entier $c_2$ tel que $$ {1/4}(δ^3-…
We prove the existence (in characteristic 0) on every polarized (smooth, projective and connected) surface of stable bundles of rank $r\geq 2$, arbitrary first Chern class and large enough $c_2$.
We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that the arithmetic hypotheses are supposed only on the fibres over a generalized Hilbertian subset.
The Borel-Weil-Bott theorem describes the cohomology of line bundles over flag varieties. Here, one generalizes this theorem to a wider class of projective varieties : the wonderful varieties of minimal rank.
An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.
Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of discriminant $-12$. We prove here that the list is complete with a $53$th fibration, thanks to a remark of Elkies and Schütt. We characterize the fibration both theoretically and with a Weierstrass model.
We show that Lang's hyperbolic and function version conjectures hold for surfaces $S$ of general type having a fibration of general type onto a curve $C$. The notion of multiplicity used is natural, but not classical, which leds to orbifold versions of Mordell's conjectures not reducible by ramified covering tricks to the already solved cases. These are solved here in the two above case (hyperbolic and function field cases). But the arithmetic case (which follows from the abc conjecture) is left open. We also give an example of a simply connected surface of general type having a fibration of g…
