Dissertations / Theses on the topic 'Géométrie birationnelle des surfaces'
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Boitrel, Aurore. "Groupes d'automorphismes des surfaces del Pezzo sur un corps parfait." Electronic Thesis or Diss., université Paris-Saclay, 2025. http://www.theses.fr/2025UPASM002.
Full textDel Pezzo surfaces are algebraic surfaces with quite special properties, that play an importantpart in the classification of projective algebraic surfaces up to birational transformations.The classification of smooth rational del Pezzo surfaces of degree d over an arbitraryperfect field is classical for d = 7, 8, 9 and new for d = 6. The same is the case for thedescription of their groups of automorphisms. Their classification and the description of theirautomorphism groups is much more difficult for d ≤ 5, as one can see already if the groundfield is the field of real numbers, and the classification is open over a general perfect field.Partial classifications exist over finite fields. Accordingly, we do not know their automorphismgroups in general.The objective of the thesis is to classify the smooth rational del Pezzo surfaces of degreed = 5 and d = 4 over an arbitrary perfect field and describe their automorphism groups.Due to the difficulty of the project, the case d = 4 will only be studied over the field ofreal numbers
Benzerga, Mohamed. "Structures réelles sur les surfaces rationnelles." Thesis, Angers, 2016. http://www.theses.fr/2016ANGE0081.
Full textThe aim of this PhD thesis is to give a partial answer to the finiteness problem for R-isomorphism classes of real forms of any smooth projective complex rational surface X, i.e. for the isomorphism classes of R-schemes whose complexification is isomorphic to X. We study this problem in terms of real structures (or antiholomorphic involutions, which generalize complex conjugation) on X: the advantage of this approach is that it helps us rephrasing our problem with automorphism groups of rational surfaces, via Galois cohomology. Thanks to recent results on these automorphism groups, using hyperbolic geometry and, to a lesser extent, holomorphic dynamics and metric geometry, we prove several finiteness results which go further than Del Pezzo surfaces and can apply to some rational surfaces with large automorphism groups
Durighetto, Sara. "Géométrie birationnelle : classique et dérivée." Thesis, Toulouse 3, 2019. http://www.theses.fr/2019TOU30031.
Full textIn the field of algebraic geometry, the study of birational transformations and their properties plays a primary role. In this, there are two different approach: the classical one due to the Italian school who focuses on the Cremona group and a modern one which utilizes instruments like derived categories and semiorthogonal decompositions. About the Cremona group, that is the group of birational self- morphisms of Pn, we do not know much in general and we focus on the complex case. We know a set of generators only in dimension n = 2. Moreover, we do not have a classification of curves and linear systems in P2 up to Cremona transformations. Among the known results there are: irreducible curves and curves with two irreducible components. In this thesis we approach tha case of a configuration of lines in the projective plane. The last theorem lists the known contractible configurations. From a categorical point of view, the semiorthogonal decompositions of the derived category of a variety provide some useful invariants in the study of the variety. Following the work of Clemens-Griffiths about the complex cubic threefold, we want to characterize the obstructions to the rationality of a variety X of dimension n. The idea is to collect the component of a semiorthogonal decomposition which are not equivalent to the derived category of a variety of dimension at least n - 1. In this way we defined the so called Griffiths-Kuznetsov component of X. In this thesis we study the case of surfaces on an arbitrary field, we define that component and show that it is a birational invariant. It appears clearly that the Griffiths-Kuznetsov component vanishes only if the surface is rational
Beri, Pietro. "On birational transformations and automorphisms of some hyperkähler manifolds." Thesis, Poitiers, 2020. http://www.theses.fr/2020POIT2267.
Full textMy thesis work focuses on double EPW sextics, a family of hyperkähler manifolds which, in the general case, are equivalent by deformation to Hilbert's scheme of two points on a K3 surface. In particular I used the link that these manifolds have with Gushel-Mukai varieties, which are Fano varieties in a Grassmannian if their dimension is greater than two, K3 surfaces if their dimension is two.The first chapter contains some reminders of the theory of Pell's equations and lattices, which are fundamental for the study of hyperkähler manifolds. Then I recall the construction which associates a double covering to a sheaf on a normal variety.In the second chapter I discuss hyperkähler manifolds and describe their first properties; I also introduce the first case of hyperkähler manifold that has been studied, the K3 surfaces. This family of surfaces corresponds to the hyperkähler manifolds in dimension two.Furthermore, I briefly present some of the latest results in this field, in particular I define different module spaces of hyperkähler manifolds, and I describe the action of automorphism on the second cohomology group of a hyperkähler manifold.The tools introduced in the previous chapter do not provide a geometrical description of the action of automorphism on the manifold for the case of the Hilbert scheme of points on a general K3 surface. In the third chapter, I therefore introduce a geometrical description up to a certain deformation. This deformation takes into account the structure of Hilbert scheme. To do so, I introduce an isomorphism between a connected component of the module space of manifolds of type K3[n] with a polarization, and the module space of manifolds of the same type with an involution of which the rank of the invariant is one. This is a generalization of a result obtained by Boissière, An. Cattaneo, Markushevich and Sarti in dimension two. The first two parts of this chapter are a joint work with Alberto Cattaneo.In the fourth chapter, I define EPW sextics, using O'Grady's argument, which shows that a double covering of a EPW sextic in the general case is deformation equivalent to the Hilbert square of a K3 surface. Next, I present the Gushel-Mukai varieties, with emphasis on their connection with EPW sextics; this approach was introduced by O'Grady, continued by Iliev and Manivel and systematized by Kuznetsov and Debarre.In the fifth chapter, I use the tools introduced in the fourth chapter in the case where a K3 surface can be associated to a EPW sextic X. In this case I give explicit conditions on the Picard group of the surface for X to be a hyperkähler manifold. This allows to use Torelli's theorem for a K3 surface to demonstrate the existence of some automorphisms on X. I give some bounds on the structure of a subgroup of automorphisms of a sextic EPW under conditions of existence of a fixed point for the action of the group.Still in the case of the existence of a K3 surface associated with a EPW sextic X, I improve the bound obtained previously on the automorphisms of X, by giving an explicit link with the number of conics on the K3 surface. I show that the symplecticity of an automorphism on X depends on the symplecticity of a corresponding automorphism on the surface K3.The sixth chapter is a work in collaboration with Alberto Cattaneo. I study the group of birational automorphisms on Hilbert's scheme of points on a projective surface K3, in the generic case. This generalizes the result obtained in dimension two by Debarre and Macrì. Then I study the cases where there is a birational model where these automorphisms are regular. I describe in a geometrical way some involutions, whose existence has been proved before
Debin, Clément. "Géométrie des surfaces singulières." Thesis, Université Grenoble Alpes (ComUE), 2016. http://www.theses.fr/2016GREAM078/document.
Full textIf we look for a compactification of the space of Riemannian metrics with conical singularities on a surface, we are naturally led to study the "surfaces with Bounded Integral Curvature in the Alexandrov sense". It is a singular geometry, developed by A. Alexandrov and the Leningrad's school in the 70's, and whose main feature is to have a natural notion of curvature, which is a measure. This large geometric class contains any "reasonable" surface we may imagine.The main result of this thesis is a compactness theorem for Alexandrov metrics on a surface ; a straightforward corollary concerns Riemannian metrics with conical singularities. We describe here three hypothesis which pair with the Alexandrov surfaces, following Cheeger-Gromov's compactness theorem, which deals with Riemannian manifolds with bounded curvature, injectivity radius bounded by below and volume bounded by above. Among other things, we introduce the new notion of contractibility radius, which plays the role of the injectivity radius in this singular setting.We also study the (moduli) space of Alexandrov metrics on the sphere, with non-negative curvature along a closed curve. An interesting subset is the set of compact convex sets, glued along their boundaries. Following W. Thurston, C. Bavard and E. Ghys, who considered the moduli space of (convex) polyhedra and polygons with fixed angles, we show that the identification between a convex set and its support function give rise to an infinite dimensional hyperbolic geometry, for which we study the first properties
Philippe, Emmanuel. "Géométrie des surfaces hyperboliques." Toulouse 3, 2008. http://thesesups.ups-tlse.fr/270/.
Full textIn this report, we describe the beginning of the length spectra of the triangles groups associated with a hyperbolic triangle (r, p, q) with r, p, q integers were ordered in the increasing order. We show while the datum of the length spectra characterizes, except when r=3, the class of isometry of such a group among all the triangles groups
Zannad, Skander. "Surfaces branchées en géométrie de contact." Phd thesis, Université de Nantes, 2006. http://tel.archives-ouvertes.fr/tel-00103561.
Full textLe résultat principal est l'obtention d'une condition suffisante pour qu'une surface branchée B d'une variété V de dimension 3 porte pleinement une lamination. Il en découle une condition suffisante pour que le rappel de B dans le revêtement universel de V porte pleinement une lamination. Cette condition est nécessaire pour que cette lamination soit essentielle. Ce résultat apporte un élément de réponse à une question classique de Gabai.
On introduit ensuite une notion de structure de contact portée par une surface branchée qui généralise celle de Oertel-Swiatkowski. Enfin, on établit une condition sufisante pour que deux structures de contact soient, à isotopie près, portées par une même surface branchée.
Toubiana, Eric. "Géométrie des surfaces minimales de R³." Paris 7, 1988. http://www.theses.fr/1988PA077206.
Full textYassine, Zeina. "Géométrie systolique extrémale sur les surfaces." Thesis, Paris Est, 2016. http://www.theses.fr/2016PESC1074/document.
Full textIn 1949, C. Loewner proved in an unpublished work that the two-torus T satisfies an optimal systolic inequality relating the area of the torus to the square of its systole. By a systole here we mean the smallest length of a noncontractible loop in T. Furthermore, the equality is attained if and only if the torus is flat hexagonal. This result led to whatwas called later systolic geometry. In this thesis, we study several systolic-like inequalities. These inequalities involve the minimal length of various curves and not merely the systole.First we obtain three optimal conformal geometric inequalities on Riemannian Klein bottles relating the area to the product of the lengths of the shortest noncontractible loops in different free homotopy classes. We describe the extremal metrics in each conformal class.Then we prove optimal systolic inequalities on Finsler Mobius bands relating the systoleand the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimalsystolic inequality for Finsler Klein bottles with symmetries. We describe extremal metric families in both cases.Finally, we prove a critical systolic inequality on genus two surface. More precisely, it is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved Riemannian metrics. We show that this piecewise flat metric is also critical for slow metric variations, this time without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface
Guilbot, Robin. "Quelques aspects combinatoires et arithmétiques des variétés toriques complètes." Phd thesis, Université Paul Sabatier - Toulouse III, 2012. http://tel.archives-ouvertes.fr/tel-00832228.
Full textOu, Wenhao. "Géométrie des variétés rationnellement connexes." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAM060/document.
Full textIn this dissertation, we study several subjects on the geometry of rationally connected varieties. A complex variety is called rationally connected if for two general points, there is a rational curve passing through them. The first subject we study is the base of a Lagrangian fibration of a projective irreducible symplectic fourfold. We prove that there are at most two possibilities for the base. In the second part, we classify certain type of Fano varieties. In the end, we study the structures of singular rationally connected varieties which carry non-zero pluri-forms
Ortiz, Rodriguez Adriana. "Géométrie différentielle projective des surfaces algébriques réelles." Paris 7, 2002. http://www.theses.fr/2002PA077134.
Full textFloris, Enrica. "Deux aspects de la géométrie birationnelle des variétés algébriques : la formule du fibré canonique et la décomposition de Zariski." Phd thesis, Université de Strasbourg, 2013. http://tel.archives-ouvertes.fr/tel-00861470.
Full textRouyer, Joël. "En géométrie globale des surfaces : la notion d'antipode." Mulhouse, 2001. http://www.theses.fr/2001MULH0671.
Full textMoncet, Arnaud. "Géométrie et dynamique sur les surfaces algébriques réelles." Phd thesis, Rennes 1, 2012. https://ecm.univ-rennes1.fr/nuxeo/site/esupversions/6cd607e0-4a4e-4328-bf36-674a3bb9f4b8.
Full textThis thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex dynamics. This question is declined in two directions: the topological entropy and the Fatou set. For the first one, we introduce a purely geometric quantity depending only on the surface, and we call it concordance. Then we show that the ratio of real and complex entropies is linked to this quantity. The concordance is explicitely computed for many examples of surfaces, especially abelian surfaces which are broadly studied, as well assome K3 surfaces. In the second part, we are interested in the Fatou set, which corresponds to complex points for which the dynamics is simple. Thanks to previous results of Dinh and Sibony about closed positive currents, we prove that this set is hyperbolic in the sense of Kobayashi, after possibly deleting some curves which are fixed by (an iterate of) our transformation. From this property we deduce that, except for some exceptional cases in which the topology of the real locus is simple and the dynamics well understood, this real locus cannot be entirely contained in the Fatou set. Thus the complexity of the dynamics is observable on real points in most cases
Moncet, Arnaud. "Géométrie et dynamique sur les surfaces algébriques réelles." Phd thesis, Université Rennes 1, 2012. http://tel.archives-ouvertes.fr/tel-00724509.
Full textZang, Yiming. "Les surfaces de Ricci et les surfaces minimales dans les groupes de Lie métriques." Electronic Thesis or Diss., Université de Lorraine, 2022. https://docnum.univ-lorraine.fr/ulprive/DDOC_T_2022_0115_ZANG.pdf.
Full textIn this thesis, we will study some topics related to minimal surfaces in three-dimensional homogeneous manifolds. The first part is devoted to the study of non-positively curved Ricci surfaces with catenodial ends. The idea comes from a famous theorem of Huber. In the first place, we give a definition of catenoidal end for non-positively curved Ricci surfaces with finite total curvature. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data. By using this tool, we get some classification results and some non-existence results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. In the end of Chapter 2, we also prove an existence result for non-positively curved Ricci surfaces of arbitrary positive genus with finite many catenoidal ends.In the second part of this thesis, we concern about minimal surfaces in a three-dimensional metric Lie group widetilde{E(2)}, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. Firstly, a result of Patrangenaru describes the left-invariant metrics as a two-parameter family of metrics. Then we take advantage of a Weierstrass-type representation due to Meeks, Mira, Pérez and Ros to construct a one-parameter family of helicoidal minimal surfaces in widetilde{E(2)} as well as a one-parameter family of minimal annuli which are properly embedded in widetilde{E(2)}. In the end, by a discussion of the limit case of the second family of surfaces, we obtain a new proof of a half-space theorem for minimal surfaces in widetilde{E(2)}
Renaudineau, Arthur. "Constructions de surfaces algébriques réelles." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066249/document.
Full textIn this thesis, we focus on constructions of real algebraic surfaces. The main problem we focus on is to construct real algebraic surfaces with a big number of handles. This problem is related to Viro's conjecture. A couterexample to Viro's conjecture was constructed at the first time by I. Itenberg in 1993. The fundamental tool to our constructions is Viro's patchworking. Viro's patchworking can be reformulated in terms of tropical geometry. Using tropical geometry, and more precisely tropical modifications, we give a new construction of a family of real algebraic plane curves with asymptotically a maximal number of even ovals. This family was first constructed in 2006 by E. Brugallé. Using Viro's patchworking, we construct a real sextic with 45 handles, improving a result of F. Bihan obtained in 2001. At least, we focus on the study of real algebraic surfaces in P1xP1xP1. More precisely, we construct a family of real algebraic surfaces of tridegree (2k,2l,2) in P1xP1xP1 with asymptotically a maximal first Betti number. This construction uses a more general version of Viro's patchworking due to E. Shustin in 1998
Ancona, Michele. "Moments en géométrie algébrique réelle." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSE1274.
Full textIt is well known that the number of real roots of a real degree d polynomial is at most d. In the 90s, E. Kostlan proved that the average number of real roots equals the square root of d, once we equip the space of polynomials with some natural Gaussian measure. This result has a geometric interpretation, in which the real polynomials are sections of a line bundle over the Riemann sphere. We can extend this study in a more general case of a real Riemann surface equipped with ample line bundle and study the expected value of the number of real zeros of a random section. In this thesis, we compute all the central moments of these random variables. As an application, we prove that the measure of the space of real sections whose number of real zeros deviates from the expected one goes to zeros, as the degree of the line bundle goes to infinity.In a second part, we present analogues results in real Hurwitz theory, in which we study the real critical points of a random branched covering of the Riemann sphere. We compute the expected value of this number and also all the central moments.The techniques we use are of analytique nature (Bergman kernel, L^2 estimates) and gometric one (Olver multispaces, coarea formula)
Jeannin, Pierre. "Contrôle des courbes et surfaces rationnelles par vecteurs massiques." Lille 1, 1988. http://www.theses.fr/1988LIL10111.
Full textTrin, Marie. "Application de courants géodésiques à la géométrie des surfaces." Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. http://www.theses.fr/2024URENS025.
Full textLet Z be a finite surface with negative Euler characteristic. A geodesic current on Z is a Radon measure on the set of bi-infinite unoriented geodesics of the universal cover of Z which is stable by the action of the fundamental group. This notion has been introduced by F.Bonahon in 1986 and has since proven to be a fertile concept in the study of geometry of surfaces. In this thesis we are interested in two main applications of geodesic currents: the compactification of Teichmüller space and counting problems in surfaces. The first chapter is dedicated to the necessary definitions and properties. Chapter 2 deals with Thurston compactification of Teichmüller space. Especially, we will prove that the method developed by Bonahon using geodesic currents can be extended for non-compact surfaces of finite area. This chapter also contains some results about sequences of random geodesics. The last two chapters focus on counting problems. In chapter 3 we prove that arcs in surfaces with boundary can be counted thanks to sequences of measures on geodesic currents. Hence, chapter 4 is dedicated to the different perspectives associated to this manuscipt. The main one being to count elements in the orbits for the action of subgroups of mapping class groups on curves
Toussaint, Antoine. "Real Structures of Phase Tropical Surfaces." Electronic Thesis or Diss., Sorbonne université, 2023. http://www.theses.fr/2023SORUS246.
Full textIntroduced in the late 1970's, Viro's patchworking is one of the main approaches for constructing real algebraic varieties with prescribed topological properties. Viro's method is closely related to tropical geometry. In particular, the simplest case, known as primitive combinatorial patchworking, is based on data partially dual to a non-singular tropical variety. For plane curves, the real structures arising in a primitive combinatorial patchworking were described by B. Haas in 1997, enabling a combinatorial criterion to be established for the maximality (in the Smith-Thom sense) of the plane curves under construction. This description was reinterpreted in 2017 by B. Bertrand, E. Brugallé and A. Renaudineau, paving the way for a generalization to higher dimensions. We generalize their approach to the case of surfaces and give a description of a certain class of real structures on a phase tropical surface that form an affine space whose direction is given in terms of the cohomology of the wave space of the underlying tropical surface. We then give a filtration of the homology of the phase tropical surface by lifting tropical cycles of the tropical surface. Finally, we establish a necessary combinatorial criterion for the maximality of a phase tropical surface endowed with a real structure, as well as a necessary and sufficient criterion for real phase tropical surfaces to be of type I
Kerautret, Bertrand. "Reconstruction et lissage de surfaces discrètes." Bordeaux 1, 2004. http://www.theses.fr/2004BOR12938.
Full textAbuaf, Roland. "Dualité homologique projective et résolutions catégoriques des singularités." Thesis, Grenoble, 2013. http://www.theses.fr/2013GRENM057/document.
Full textLet $X$ be an algebraic variety with Gorenstein rational singularities. A crepant resolution of $X$ is often considered to be a minimal resolution of singularities for $X$. Unfortunately, crepant resolution of singularities are very rare. For instance, determinantal varieties of skew-symmetric matrices never admit crepant resolution of singularities. In this thesis, we discuss various notions of categorical crepant resolution of singularities as defined by Alexander Kuznetsov. Conjecturally, these resolutions are minimal from the categorical point of view. We introduce the notion of wonderful resolution of singularities and we prove that a variety endowed with such a resolution admits a weakly crepant resolution of singularities. As a corollary, we prove that all determinantal varieties (square, as well as symmetric and skew-symmetric) admit weakly crepant resolution of singularities. Finally, we study some quartics hypersurfaces which come from the Tits-Freudenthal magic square. Though they do no admit any wonderful resolution of singularities, we are still able to prove that they have a weakly crepant resolution of singularities. This last result should be of interest in order to construct homological projective duals for some symplectic Grassmannians over the composition algebras
Borot, Gaetan. "Quelques problèmes de géométrie énumérative, de matrices aléatoires, d'intégrabilité, étudiés via la géométrie des surfaces de Riemann." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00625776.
Full textBorot, Gaëtan. "Quelques problèmes de géométrie énumérative, de matrices aléatoires, d'intégrabilité, étudiés via la géométrie des surfaces de Riemann." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112092/document.
Full textComplex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory, … All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently : it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications.First, we consider all order large N asymptotics in some N-dimensional integrals coming from random matrix theory, or more generally from "log gases" problems. We shall explain how to use loop equations to establish those asymptotics in beta matrix models within a one cut regime. This can be applied in the study of large fluctuations of the maximum eigenvalue in beta matrix models, and lead us to heuristic predictions about the asymptotics of Tracy-Widom beta law to all order, and for all positive beta. Second, we study the interplay between integrability and loop equations. As a corollary, we are able to prove the previous prediction about the asymptotics to all order of Tracy-Widom law for hermitian matrices.We move on with the solution of some combinatorial problems in all topologies. In topological string theory, a conjecture from Bouchard, Klemm, Mariño and Pasquetti states that certain generating series of Gromov-Witten invariants in toric Calabi-Yau threefolds, are solutions of loop equations. We have proved this conjecture in the simplest case, where those invariants coincide with the "simple Hurwitz numbers". We also explain recent progress towards the general conjecture, in relation with our work. In statistical physics on the random lattice, we have solved the trivalent O(n) model introduced by Kostov, and we explain the method to solve more general statistical models.Throughout the thesis, the computation of some "generalized matrices integrals" appears to be increasingly important for future applications, and this appeals for a general theory of loop equations
Laurain, Paul. "Comportement asymptotique des surfaces à courbure moyenne constante." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2010. http://tel.archives-ouvertes.fr/tel-00559640.
Full textCaissard, Thomas. "Opérateur de Laplace–Beltrami discret sur les surfaces digitales." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSE1326/document.
Full textThe central issue of this thesis is the development of a discrete Laplace--Beltrami operator on digital surfaces. These surfaces come from the theory of discrete geometry, i.e. geometry that focuses on subsets of relative integers. We place ourselves here in a theoretical framework where digital surfaces are the result of an approximation, or discretization process, of an underlying smooth surface. This method makes it possible both to prove theorems of convergence of discrete quantities towards continuous quantities, but also, through numerical analyses, to experimentally confirm these results. For the discretization of the operator, we face two problems: on the one hand, our surface is only an approximation of the underlying continuous surface, and on the other hand, the trivial estimation of geometric quantities on the digital surface does not generally give us a good estimate of this quantity. We already have answers to the second problem: in recent years, many articles have focused on developing methods to approximate certain geometric quantities on digital surfaces (such as normals or curvature), methods that we will describe in this thesis. These new approximation techniques allow us to inject measurement information into the elements of our surface. We therefore use the estimation of normals to answer the first problem, which in fact allows us to accurately approximate the tangent plane at a point on the surface and, through an integration method, to overcome topological problems related to the discrete surface. We present a theoretical convergence result of the discretized new operator, then we illustrate its properties using a numerical analysis of it. We carry out a detailed comparison of the new operator with those in the literature adapted on digital surfaces, which allows, at least for convergence, to show that only our operator has this property. We also illustrate the operator via some of these applications such as its spectral decomposition or the mean curvature flow
Mondal, Sugata. "Small eigenvalues of hyperbolic surfaces." Toulouse 3, 2013. http://thesesups.ups-tlse.fr/2233/.
Full textA hyperbolic surface S is a complete two dimensional manifold of sectional curvature -1. In this thesis we consider the Laplace operator associated to this metric (acting on functions). Any eigenvalue below 1/4 is called a small eigenvalue. The general theme of our research is to bound the number of small eigenvalues of S in terms of the topology of S when S has finite area. A theorem of Otal-Rosas says that the number of small eigenvalues of a closed hyperbolic surface of genus g is not more than 2g -2, confirming a conjecture of P. Buser. We prove a quantitative version of this result by giving the lower bound for the (2g- 2)-th eigenvalue : {\lambda_{2g-2}}(S) > 1/4 +{\epsilon_0}(S) where {\epsilon_0}(S) > 0 is an explicit function that depends only on the geometry of S. Our proof uses geometric inequalities of Faber-Krahn and of Cheeger. For a hyperbolic surface of finite area and type (g, n) it is a conjecture that the number of small cuspidal eigenvalues is <= 2g- 3. We show that on a non-empty open unbounded subset of the moduli space Mg;n, this number of eigenvalues is <= 2g -2. The proof is based on a theorem, motivated by results of Lizhen Ji and Scott Wolpert, that describes the behavior of small cuspidal eigenfunctions of surfaces Sm when the sequence (Sm) tends to the boundary of the moduli space. We use this theorem to give a new and elementary proof of a result of D. Hejhal also. In the last chapter, we study the maximum of {\lambda_1} viewed as a function on Mg. More precisely, we ask if the maximum is more than 1/4. Using topological arguments, we prove that in the case for genus two : there exist surfaces in Mg for which {\lambda_1} > 1/4
Devernay, Frédéric. "Vision stéréoscopique et propriétés différentielles des surfaces." Phd thesis, Ecole Polytechnique X, 1997. http://tel.archives-ouvertes.fr/tel-00005629.
Full textOudot, Steve. "Echantillonnage et maillage de surfaces avec garanties." Palaiseau, Ecole polytechnique, 2005. http://www.theses.fr/2005EPXX0060.
Full textGuilbot, Robin. "Quelques aspects combinatoires et arithmétiques des variétés toriques complètes." Phd thesis, Toulouse 3, 2012. http://thesesups.ups-tlse.fr/1905/.
Full textIn this thesis we study two distinct aspects of toric varieties, one purely geometric, over C, and the other of arithmetic nature, over quasi algebraically closed fields (C1 fields). Extremal curves, which generate the Mori cone of a projective toric variety, are primitive curves (V. Batyrev). In 2009, D. Cox and C. Von Renesse conjectured that the classes of primitive curves generate the Mori cone of any toric variety whose fan has full dimensional convex support. We present a family of counterexamples to this conjecture and propose a new formulation based on the notion of local contractibility, generalizing the contractibility defined by C. Casagrande. Using the corridors, a combinatorial tool that we introduce, we show how to write any given 1-cycle class as a linear combination with integer coefficients of toric curve classes. Corridors enable us to give an explicit decomposition of any class that is not contractible (straights corridors) as well as contractible classes in some particular cases (circular corridors). C1 fields are those over which the existence of rational points on a variety Y is ensured by any small degree embedding of Y in a projective space (by definition) or in a weighted projective space (according to an easy theorem of Kollar). For an ample divisor in a toric variety whose fan is simplicial and complete, we show that there is also a notion of small degree which ensures the existence of rational points. This way, we show the existence of rational points on a large class of rationally connected varieties
Uribe, Vargas Eduardo Ricardo. "Singularités symplectiques et de contact en géométrie différentielle des courbes et des surfaces." Paris 7, 2001. http://www.theses.fr/2001PA077154.
Full textThibaut, Muriel. "Géométrie des surfaces de faille et dépliage 3D : (méthodes et applications)." Phd thesis, Université Joseph Fourier (Grenoble), 1994. http://tel.archives-ouvertes.fr/tel-00745509.
Full textLevallois, Jérémy. "Estimateurs différentiels en géométrie discrète : Applications à l'analyse de surfaces digitales." Thesis, Lyon, INSA, 2015. http://www.theses.fr/2015ISAL0103.
Full text3D image acquisition devices are now ubiquitous in many domains of science, including biomedical imaging, material science, or manufacturing. Most of these devices (MRI, scanner X, micro-tomography, confocal microscopy, PET scans) produce a set of data organized on a regular grid, which we call digital data, commonly called pixels in 2D images and voxels in 3D images. Properly processed, these data approach the geometry of imaged shapes, like organs in biomedical imagery or objects in engineering. In this thesis, we are interested in extracting the geometry of such digital data, and, more precisely, we focus on approaching geometrical differential quantities such as the curvature of these objects. These quantities are the critical ingredients of several applications like surface reconstruction or object recognition, matching or comparison. We focus on the proof of multigrid convergence of these estimators, which in turn guarantees the quality of estimations. More precisely, when the resolution of the acquisition device is increased, our geometric estimates are more accurate. Our method is based on integral invariants and on digital approximation of volumetric integrals. Finally, we present a surface classification method, which analyzes digital data in a multiscale framework and classifies surface elements into three categories: smooth part, planar part, and singular part (tangent discontinuity). Such feature detection is used in several geometry pipelines, like mesh compression or object recognition. The stability to parameters and the robustness to noise are evaluated with respect to state-of-the-art methods. All our tools for analyzing digital data are applied to 3D X-ray tomography of snow microstructures and their relevance is evaluated and discussed
Otal, Jean-Pierre. "Courants géodésiques et surfaces." Paris 11, 1989. http://www.theses.fr/1989PA112051.
Full textThe first part of this work is concerned with some geometric questions about the 3-dimensional manifolds called "compression bodies". In the first chapter, one defines a space associated to a compression body N : it is the quotient of an open subset of the space of measured laminations on the compressible component S of ∂N by the action of a certain subgroup of the modular group of S. This space carries a natural map to the space of geodesic currents C(N) of the group π1(N). The main result is that this map is an homeomorphism on its image L(N). The second chapter introduces some technics to understand the frontier of L(N) in C(N). One considers there the problem oh characterizing the conjugacy classes of the free group G on g generators which can be represented by an embedded loop on the boundary of an handlebody with fundamental group G. One studies therefore some equivalence relations on the space of ends of the free group G. The second part is concerned with the problem of reconstructing Riemannian metric on a surface from some spectral data. One shows in the third chapter that two negatively curved metrics on a closed surface S which give the same length to each homotopy class π1(S) are isotopic. In the fourth chapter, one shows that two negatively curved metrics on a compact disc D² which induce the same distance function on ∂D² are isotopic
Gresser, Laurent. "Modification des propriétés adhésives de surfaces de polycarbonate par décharge couronne en géométrie fil-plan." Pau, 1998. http://www.theses.fr/1998PAUU3014.
Full textSabourau, Stéphane. "Sur quelques problèmes de la géométrie des systoles." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2001. http://tel.archives-ouvertes.fr/tel-00001175.
Full textTout d'abord, nous étudions les métriques extrémales pour le problème isosystolique sur les surfaces. Nous établissons un critère à l'extrémalité des métriques sur les surfaces orientables et examinons le cas de genre deux.
Ensuite, nous montrons que la longueur de la plus courte trajectoire non triviale parmi les géodésiques fermées simples d'indice un et les géodésiques en huit d'indice nul minore l'aire et le diamètre des sphères riemanniennes.
Nous discutons aussi de la rigidité et de la souplesse du rayon de remplissage par rapport aux longueurs de courtes géodésiques provenant de la théorie de Morse sur l'espace des 1-cycles.
Finalement, nous minorons le volume et le diamètre des variétés riemanniennes complètes à l'aide de la longueur du plus court lacet géodésique non trivial. De plus, nous obtenons une minoration de la croissance du volume des boules de ``petit'' rayon, ainsi qu'un résultat de finitude homotopique.
Sanner, Michel. "Sur la modélisation des surfaces moléculaires." Mulhouse, 1992. http://www.theses.fr/1992MULH0245.
Full textCoeurjolly, David. "Algorithmique et géométrie discrète pour la caractérisation des courbes et des surfaces." Phd thesis, Université Lumière - Lyon II, 2002. http://tel.archives-ouvertes.fr/tel-00167370.
Full textBiswas, KIngshook. "Sur la géométrie des hérissons, et des tube-log surfaces de Riemann." Paris 13, 2005. http://www.theses.fr/2005PA132017.
Full textMontero, Silva Pedro Pablo. "Géométrie des variétés de Fano singulières et des fibrés projectifs sur une courbe." Thesis, Université Grenoble Alpes (ComUE), 2017. http://www.theses.fr/2017GREAM050/document.
Full textThis thesis is devoted to the geometry of Fano varieties and projective vector bundles over a smooth projective curve.In the first part we study the geometry of mildly singular Fano varieties on which there is a prime divisor of Picard number 1. By studying the contractions associated to extremal rays in the Mori cone of these varieties, we provide a structure theorem in dimension 3 for varieties with maximal Picard number. Afterwards, we address the case of toric varieties and we extend the structure theorem to toric varieties of dimension greater than 3 and with maximal Picard number. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.In the second part we study Newton-Okounkov bodies on projective vector bundles over a smooth projective curve. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags and we study how these bodies depend on the Schubert cell decomposition with respect to linear flags which are compatible with the Harder-Narasimhan filtration of the bundle. Moreover, we characterize semi-stable vector bundles over smooth projective curves via Newton-Okounkov bodies
Dehlinger, Christophe. "Spécifications et preuves en Coq pour les surfaces combinatoires et leur classification." Université Louis Pasteur (Strasbourg) (1971-2008), 2003. http://www.theses.fr/2003STR13236.
Full textFavreau, Jean-Marie. "Outils pour le pavage de surfaces." Phd thesis, Université Blaise Pascal - Clermont-Ferrand II, 2009. http://tel.archives-ouvertes.fr/tel-00440730.
Full textBiard, Luc. "Modélisation Géométrique et Reconstruction de Surfaces." Habilitation à diriger des recherches, Université de Grenoble, 2009. http://tel.archives-ouvertes.fr/tel-00994789.
Full textRoux, Jean-Christophe. "Méthodes d'approximation et de géométrie algorithmique pour la reconstruction de courbes et surfaces." Phd thesis, Grenoble 1, 1994. http://tel.archives-ouvertes.fr/tel-00344528.
Full textPouget, Marc. "Géométrie des surfaces :de l'estimation des quantités différentielles localesà l'extraction robuste d'éléments caractéristiquesglobaux." Phd thesis, Université de Nice Sophia-Antipolis, 2005. http://tel.archives-ouvertes.fr/tel-00102998.
Full textIl est fortement motivé par des applications telles que la conception assistée par ordinateur,
l'imagerie médicale, le calcul scientifique et la simulation ou encore la réalité virtuelle et
le multimédia. Plus précisément, cette thèse propose une analyse de la géométrie des surfaces
tant d'un point de vue local que global.
Tout d'abord, étant donnée une surface lisse connue via un échantillonnage, nous étudions le
problème de l'estimation des quantités différentielles locales: normale, courbures et quantités
d'ordre supérieur. Une méthode d'estimation utilisant un ajustement polynomial est développée:
les propriétés de convergence sont établies et un algorithme est proposé et implémenté.
D'un point de vue global, nous analysons les lignes d'extrême de courbure sur une surface,
appelées ridges. Pour le cas d'une surface discrétisée par un maillage, des conditions
précises d'échantillonnage sont données, et sous ces hypothèses, un algorithme produisant une
approximation topologiquement certifiée des ridges est développé. Dans le cas d'une surface
paramétrée, nous établissons que les ridges ont une structure implicite globale, et étudions les
singularités de la courbe associée dans le domaine de paramétrage en termes de systèmes zerodimensionnels.
Pour une paramétrisation polynomiale, ces équations sont aussi polynomiales
et des méthodes spécifiques de calcul formel sont développées pour calculer la topologie de la
courbe singulière des ridges.
Hajli, Mounir. "Théorie spectrale pour certaines métriques singulières et géométrie d'Arakelov." Paris 6, 2012. http://www.theses.fr/2012PA066507.
Full textIn this thesis we are interested in the holomorphic analytictorsion and its extension (in the setting of Arakelov geometry)to integrable line bundles on a compact Riemann surface. We propose two different approaches: The first approach isan approximation process which uses Bismut, Gillet , Soulé anomaly formula. The second one introduces the notion of singular Laplacian which extend the classical one. We apply both approaches to line bundles on \mathbb{P}^1 endowedwith their canonical metric. By direct computations, we establish that both approaches define the same notion of analytic torsion in the case of canonical metrics. We propose a general spectral theory which take into account this kind of metrics and generalizes the classical theory for line bundles on compact Riemann surfaces equipped with \mathcal{C}^\infty. As a consequence, we provide a new proof or the previous results, obtained by direct computations
Devernay, Frédéric. "Vision stereoscopique et proprietes differentielles de surfaces." Palaiseau, Ecole polytechnique, 1997. http://www.theses.fr/1997EPXX0052.
Full textDedieu, Thomas. "Auto-transformations et géométrie des variétés de Calabi-Yau." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2008. http://tel.archives-ouvertes.fr/tel-00358735.
Full textDans la première, je démontre que si certaines variétés de Severi universelles, qui paramètrent les courbes nodales de degré et de genre fixés existant sur une surface K3, sont irréductibles, alors une surface K3 projective générique ne possède pas d'endomorphisme rationnel de degré >1. J'établis également un certain nombre de contraintes numériques satisfaites par ces endomorphismes.
Voisin a modifié la pseudo-forme volume de Kobayashi en introduisant les K-correspondances holomorphes. Dans la seconde partie, j'étudie une version logarithmique de cette pseudo-forme volume. J'associe une pseudo-forme volume logarithmique intrinsèque à toute paire (X,D) constituée d'une variété complexe et d'un diviseur à croisements normaux et partie positive réduite. Je démontre qu'elle est génériquement non dégénérée si X est projective et K_X+D est ample. Je démontre d'autre part qu'elle s'annule pour une grande classe de paires à fibré canonique logarithmique trivial.