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1
Barros, Ignacio. "K3 surfaces and moduli of holomorphic differentials." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19290.
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In dieser Arbeit behandeln wir die birationale Geometrie verschiedener Modulräume; die Modulräume von Kurven mit einem k-Differential mit vorgeschierbenen Nullen, besser bekannt als Strata von Differenzialen, Moduln von K3 Flächen mit markierten Punkten und Moduln von Kurven. Für bestimmte Geschlechter nennen wir Abschätzungen der Kodaira-Dimension, konstruieren unirationale Parametrisierungen, rationale deckende Kurven und unterschiedliche birationale Modelle.
In Kapitel 1 führen wir die zu untersuchenden Objekte ein und geben einen kurzen Überblick ihrer wichtigsten Eigenschaften und offenen Problemen. In Kapitel 2 konstruieren wir einen Hilfsmodulraum, der als Brücke zwischen bestimmten finiten Quotienten von Mgn für kleines g und den Moduln der polarisierten K3 Flächen vom Geschlecht 11 dient. Wir entwickeln die Deformationstheorie, die nötig ist, um die Eigenschaften und die oben genannten Modulräume zu erforschen.
In Kapitel 3 bedienen wir uns dieser Werkzeuge, um birationale Modelle für Moduln polarisierter K3 Flächen vom Geschlecht 11 mit markierten Punkten zu konstruieren. Diese nutzen wir, um Resultate über die Kodaira-Dimension herzuleiten. Wir beweisen, dass der Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten unirational ist, falls n<=6, und uniruled, falls n<=7. Wir beweisen auch, dass die Kodaira-Dimension von Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten nicht-negativ ist für n>= 9. Im letzten Kapitel gehen wir noch auf die fehlenden Fälle der Kodaira-Klassifizierung von Mgnbar ein.
Schliesslich behandeln wir in Kapitel 4 die birationale Geometrie mit Blick auf die Strata von holomorphen und quadratischen Differentialen. Wir zeigen, dass die Strata holomorpher und quadratischer Differentiale von niedrigem Geschlecht uniruled sind, indem wir rationale Kurven mit pencils auf K3 und del Pezzo Flächen konstruieren. Durch das Beschränken des Geschlechts 3<= g<=6 bilden wir projektive Bündel über rationale Varietäten, die die holomorphe Strata mit maximaler Länge g-1 dominieren. Also zeigen wir auch, dass diese Strata unirational sind.In this thesis we investigate the birational geometry of various moduli spaces; moduli spaces of curves together with a k-differential of prescribed vanishing, best known as strata of differentials, moduli spaces of K3 surfaces with marked points, and moduli spaces of curves. For particular genera, we give estimates for the Kodaira dimension, construct unirational parameterizations, rational covering curves, and different birational models. In Chapter 1 we introduce the objects of study and give a broad brush stroke about their most important known features and open problems. In Chapter 2 we construct an auxiliary moduli space that serves as a bridge between certain finite quotients of Mgn for small g and the moduli space of polarized K3 surfaces of genus eleven. We develop the deformation theory necessary to study properties of the mentioned moduli space. In Chapter 3 we use this machinery to construct birational models for the moduli spaces of polarized K3 surfaces of genus eleven with marked points and we use this to conclude results about the Kodaira dimension. We prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points is unirational when n<= 6 and uniruled when n<=7. We also prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points has non-negative Kodaira dimension for n>= 9. In the final section, we make a connection with some of the missing cases in the Kodaira classification of Mgnbar. Finally, in Chapter 4 we address the question concerning the birational geometry of strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3<= g<=6 we construct projective bundles over rational varieties that dominate the holomorphic strata with length at most g-1, hence showing in addition, these strata are unirational.
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Beri, Pietro. "On birational transformations and automorphisms of some hyperkähler manifolds." Thesis, Poitiers, 2020. http://www.theses.fr/2020POIT2267.
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Mon travail de thèse porte sur les doubles EPW sextiques, une famille de variétés hyperkähleriennes qui, dans le cas général, sont équivalentes par déformation au schéma de Hilbert de deux points sur une surface K3. Notamment j'ai utilisé le lien que ces variétés ont avec les variétés de Gushel-Mukai, qui sont des variétés de Fano dans une Grassmannienne si leur dimension est plus grande que deux, des surface K3 si la dimension est deux.Le premier chapitre contient quelques rappels de théorie des équations de Pell et des réseaux, qui sont fondamentals pour l’étude des variétés hyperkähleriennes. Ensuite je rappelle la construction qui associe un revêtement double à un faisceau sur une variété normale.Dans le deuxième chapitre j’aborde les variétés hyperkähleriennes et je décris leurs premières propriétés ; j’introduis aussi le premier cas de variété hyperkählerienne qui a été étudiée, les surfaces K3. Cette famille de surfaces correspond aux variétés hyperkähleriennes en dimension deux.Je présente ensuite brièvement certains des derniers résultats dans ce domaine, notamment je définis différents espaces de modules de variétés hyperkähleriennes et je décris l’action d’un automorphisme sur le deuxième groupe de cohomologie d’une variété hyperkähleriennes.Les outils introduits dans le chapitre précédent ne fournissent pas de description géométrique de l'action de l'automorphisme sur la variété, dans le cas où la variété est un schéma de Hilbert de points sur une surface K3. Dans le troisième chapitre, j’introduis donc une description géométrique à une certaine déformation près. Cette déformation prend en compte la structure du schéma de la variété de Hilbert. Pour ce faire, j'introduis un isomorphisme entre une composante connexe de l'espace de modules des variétés de type K3[n] avec une polarization, et l'espace de modules des variétés de même type avec une involution dont le rang de l'invariant est un. Il s’agit d’une généralisation d’un résultat obtenu par Boissière, An. Cattaneo, Markushevich et Sarti en dimension deux. Les deux premières parties de ce chapitre sont un travail en collaboration avec Alberto Cattaneo.Dans le quatrième chapitre, je définis les EPW sextiques, en présentant l'argument de O'Grady, qui montre qu'un double revêtement d'un EPW sextique dans le cas général est une variété de type K3[2]. Ensuite, je présente les variétés Gushel-Mukai, en mettant l'accent sur leur lien avec les EPW sextiques ; cette approche a été introduite par O'Grady, poursuivie par Iliev et Manivel et systématisée par Kuznetsov et Debarre.Dans le cinquième chapitre, j’utilise les outils introduits dans le quatrième chapitre dans le cas où on peut associer une surface K3 à une EPW sextique X. Dans ce cas je donne des conditions explicites sur le groupe de Picard de la surface pour que X soit une variété hyperkählerienne. Cela permet d'utiliser le théorème de Torelli pour une surface K3 pour démontrer l'existence de quelques automorphismes sur X. Je donne des bornes sur la structure d'un sous-groupe d'automorphismes d'une EPW sextique sous conditions d'existence d'un point fixe pour l'action du groupe.Toujours dans le cas d'existence d'une surface K3 associée à une EPW sextique X, j’améliore la borne obtenue précédemment sur les automorphismes de X, en donnant un lien explicite avec le nombre de coniques sur la surface K3. Je montre que la symplecticité d'un automorphisme sur X dépend de la symplecticité d'un automorphisme correspondant sur la surface K3.Le sixième chapitre est un travail en collaboration avec Alberto Cattaneo. J'étudie le groupe d'automorphismes birationels sur le schéma de Hilbert des points sur une surface projective K3, dans le cas générique. Cela généralise le résultat obtenu en dimension deux par Debarre et Macrì. Ensuite j’étudie les cas où il existe un modèle birationel où ces automorphismes sont réguliers. Je décris de façon géométrique quelques involutions dont on avait prouvé l'existence auparavantMy 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
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Fanelli, Andrea. "Two structural aspects in birational geometry : geography of Mori fibre spaces and Matsusaka's theorem for surfaces in positive characteristic." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/26285.
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The aim of this thesis is to investigate two questions which naturally arise in the context of the classification of algebraic varieties. The first project concerns the structure of Mori fibre spaces: these objects naturally appear in the birational classification of higher dimensional varieties and the minimal model program. We ask which Fano varieties can appear as a fibre of a Mori fibre space and introduce the notion of fibre-likeness to study this property. This turns out to be a rather restrictive condition: in order to detect this property, we obtain two criteria (one sufficient and one necessary), which turn into a characterisation in the rigid case. Many applications are discussed and the basis for the classification of fibre-like Fano varieties is presented. In the second part of the thesis, an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic is provided: this gives an effective bound on the multiple which makes an ample line bundle D very ample. A careful study of pathological surfaces is presented here in order to bypass the classical cohomological approach. As a consequence, we obtain a Kawamata-Viehweg-type vanishing theorem for arbitrary smooth algebraic surfaces in positive characteristic.
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Benzerga, Mohamed. "Structures réelles sur les surfaces rationnelles." Thesis, Angers, 2016. http://www.theses.fr/2016ANGE0081.
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Le but de cette thèse est d’apporter des éléments de réponse au problème de la finitude du nombre de classes de R-isomorphisme de formes réelles d’une surface rationnelle projective complexe lisse X quelconque, i.e. du nombre de classes d’isomorphisme de R-schémas dont le complexifié est isomorphe à X. Nous étudions ce problème en termes de structures réelles (ou involutions antiholomorphes, généralisant la conjugaison complexe) sur X : l’intérêt de cette approche est qu’elle permet une réécriture du problème faisant intervenir les groupes d’automorphismes de surfaces rationnelles, à travers la cohomologie galoisienne. Grâce à des résultats récents concernant ces groupes et en nous appuyant sur de la géométrie hyperbolique et aussi dans une moindre mesure sur de la dynamique holomorphe et de la géométrie métrique, nous prouvons plusieurs résultats généraux de finitude qui dépassent largement le seul cadre des surfaces de Del Pezzo et peuvent s’appliquer à certaines surfaces rationnelles à grands groupes d’automorphismesThe 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
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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.
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Les surfaces del Pezzo sont des surfaces algébriques dotées de propriétés particulières, et qui jouent un rôle important dans la classification des surfaces algébriques projectives à transformations birationnelles près.La classification des surfaces del Pezzo rationnelles et lisses de degré d sur un corps parfait arbitraire est classique pour d = 7, 8, 9 et nouvelle pour d = 6. Il en va de même pour ladescription de leurs groupes d'automorphismes. Leur classification et la description de leursgroupes d'automorphismes sont beaucoup plus difficiles pour d ≤ 5, comme on peut déjà le voir si le corps de base est le corps des nombres réels, et la classification est ouverte sur un corps parfait général. Des classifications partielles existent sur des corps finis. Par conséquent, nous ne connaissons pas leurs groupes d'automorphismes en général.L'objectif de la thèse est de classifier les surfaces del Pezzo rationnelles lisses de degréd = 5 et d = 4 sur un corps parfait arbitraire et de décrire leurs groupes d'automorphismes.En raison de la difficulté du projet, le cas d = 4 ne sera étudié que sur le corps des nombres réelsDel 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
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Krylov, Igor. "Birational geometry of Fano fibrations." Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/28857.
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An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is variety admitting Mori fiber spaces over a rationally connected base. In this thesis we study the birational geometry of a particular class of rationally connected Mori fiber spaces: Fano fibrations over the projective line. We construct examples of Fano fibrations with a unique Mori fiber space in their birational classes. We prove that these examples are not birational to varieties of Fano type, thus answering the question of Cascini and Gongyo. That is we prove that the classes of rationally connected varieties and varieties of Fano type are not birationally equivalent. To construct the examples we use the techniques of birational rigidity. A Mori fiber space is called birationally rigid if there is a unique Mori fiber space structure in its birational class. The birational rigidity of smooth varieties admitting a del Pezzo fibration of degrees 1 and 2 is a well studied question. Unfortunately it is not enough to study smooth del Pezzo fibrations as there are fibrations which do not have smooth or even smoothable minimal models. In the case of fibrations of degree 2 we know that there is a minimal model with 2-Gorenstein singularities. These singularities are degenerations of the simplest terminal quotient singularity: singular points of the type 1/2(1,1,1). We give first examples of birationally rigid del Pezzo fibrations with 2-Gorenstein singularities. We then apply this result to study finite subgroups of the Cremona group of rank three. We then study the birational geometry of Fano fibrations from a different side. Using the reduction to characteristic 2 method we prove that double covers of Pn-bundles over Pm branched over a divisor of sufficiently high degree are not stably rational. For a del Pezzo fibration Y→P1 of degree 2 such that X is smooth there is a double cover Y→ X, where X is a P2-bundle over P1. In this case a stronger result holds: a very general Y with Pic(Y)≅Z⊕Z is not stably rational. We discuss the proof of this statement.
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DURIGHETTO, Sara. "Classical and Derived Birational Geometry." Doctoral thesis, Università degli studi di Ferrara, 2019. http://hdl.handle.net/11392/2488324.
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In the field of algebraic geometry, the study of birational transforma-
tions and their properties plays a primary role. In this, there are two different
approaches: 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 P^n, 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 classication of
curves and linear systems in P^2 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 conguration 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.Nell'ambito della geometria algebrica, lo studio delle trasformazioni birazionali e delle loro proprietà riveste un ruolo di importanza primaria. In questo, si affiancano l'approccio classico della scuola italiana che si concentra sul gruppo di Cremona e quello più moderno che utilizza strumenti come categorie derivate e decomposizioni semiortogonali. Del gruppo di Cremona Cr_n, cioé il gruppo degli automorfismi birazionali di P^n, in generale non si conosce molto e ci si concentra sul caso complesso. Si conosce un insieme di generatori solo nel caso di dimensione 2. Inoltre non é ancora nota una classicazione tramite trasformazioni di Cremona delle curve e dei sistemi lineari di P^2. Tra i casi noti ci sono: le curve irriducibili e quelle formate da due componenti irriducibili. In questa tesi ci si approccia al caso di una configurazione di d rette nel piano proiettivo. Il teorema finale fornisce condizioni necessarie o sufficienti alla contraibilità. Da un punto di vista categoriale invece, le decomposizioni semiortogonali della cat- egoria derivata di una varietà ci forniscono degli invarianti utili nello studio della varietà. Seguendo l'approccio di Clemens-Griffiths riguardante la cubica complessa di dimensione 3, si vuole caratterizzare le ostruzioni alla razionalità di una varietà X di dimensione n. L'idea è di raccogliere le componenti di una decomposizione ortog- onale che non sono equivalenti a categorie derivate di varietà di dimensione almeno n-1 e in questo modo definire quella che chiamiamo componente di Griffiths- Kuznetsov di X. In questa tesi si studia il caso delle superci geometricamante razionali su un campo arbitrario, si definisce tale componente e si mostra che essa è un invariante birazionale. Si vede anche che la componente di Griffiths-Kuznetsov è nulla solo se la supercie è razionale.
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Zong, Hong R. "Topics in birational geometry of algebraic varieties." Thesis, Princeton University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3665359.
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Various questions related to birational properties of algebraic varieties are concerned.
Rationally connected varieties are recognized as the buildings blocks of all varieties by the Minimal Model theory. We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. As a consequence, we solve a question of Professor Burt Totaro on integral Hodge classes on rationally connected 3-folds. And by a result of Professor Claire Voisin, the general case will be a consequence of the Tate conjecture for surfaces over finite fields.
Using the same philosophy looking for degenerated rational components through forgetful maps between moduli spaces of curves, we prove Weak Approximation conjecture to Prof. Hassett and Prof. Tschinkel for isotrivial families of rationally connected varieties. Theory of Twisted Stable maps is essentially used, with an alternative proof where some notion from Derived Algebraic Geometry is applied. It is remarkable that technics and ideas developed in this part, shed light upon and essentially led to the final solution to weak approximation of Cubic Surfaces, which is a problem concerned by Number Theorists for many years, and this is currently the best known result in this subject.
Then we turn to Minimal Model theory in both zero and positive characteristics. Firstly, projective globally F-regular threefolds of characteristic p ≥ 11, are shown to be rationally chain connected, and back to characteristic zero, we use hard-core technics of Minimal Model program, esp. finite generate of canonical rings due to Professor Hacon, Professor McKernan et al. to characterize Toric varieties and geometric rational varieties as log canonical log-Calabi Yau varieties with "large" boundary, where the specific meanings of "large" are originated from some notion of "charges" from String theory, and hence is related to Mirror Symmetry. This part of works also answered a Conjecture due to Prof. Shokurov.
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Rulla, William Frederick. "The birational geometry of M₃ and M₂, ₁ /." Full text (PDF) from UMI/Dissertation Abstracts International, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3008434.
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Massarenti, Alex. "Biregular and Birational Geometry of Algebraic Varieties." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4679.
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Every area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory.
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Johnstone, E. "On the birational geometry of singular Fano varieties." Thesis, University of Liverpool, 2017. http://livrepository.liverpool.ac.uk/3008126/.
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This thesis investigates the birational geometry of a class of higher dimensional Fano varieties of index 1 with quadratic hypersurface singularities. The main investigating question is, what structures of a rationally connected fibre space can these varieties have? Two cases are considered: double covers over a hypersurface of degree two, known as double quadrics and double covers over a hypersurface of degree three, known as double cubics. This thesis extends the study of double quadrics and cubics, first studied in the non-singular case by Iskovskikh and Pukhlikov, by showing that these varieties have the property of birational superrigidity, under certain conditions on the singularities of the branch divisor. This implies, amongst other things, that these varieties admit no non-trivial structures of a rationally connected fibre space and are thus non-rational. Additionally, the group of birational automorphisms coincides with the group of regular automorphisms. This is shown using the ``Method of maximal singularities" of Iskovskikh and Manin, expanded upon by Pukhlikov and others, in conjunction with the connectedness principal of Shokurov and Kollar. These results are then used to give a lower bound on the codimension of the set of all double quadrics (and double cubics) which are either not factorial or not birationally superrigid, in the style of the joint work of Pukhlikov and Eckl on Fano hypersurfaces. Such a result has applications to the study of varieties which admit a fibration into double quadrics or cubics.
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Tyler, Michael Peter. "On the birational section conjecture over function fields." Thesis, University of Exeter, 2017. http://hdl.handle.net/10871/31600.
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The birational variant of Grothendieck's section conjecture proposes a characterisation of the rational points of a curve over a finitely generated field over Q in terms of the sections of the absolute Galois group of its function field. While the p-adic version of the birational section conjecture has been proven by Jochen Koenigsmann, and improved upon by Florian Pop, the conjecture in its original form remains very much open. One hopes to deduce the birational section conjecture over number fields from the p-adic version by invoking a local-global principle, but if this is achieved the problem remains to deduce from this that the conjecture holds over all finitely generated fields over Q. This is the problem that we address in this thesis, using an approach which is inspired by a similar result by Mohamed Saïdi concerning the section conjecture for étale fundamental groups. We prove a conditional result which says that, under the condition of finiteness of certain Shafarevich-Tate groups, the birational section conjecture holds over finitely generated fields over Q if it holds over number fields.
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Daigle, Daniel. "Birational endomorphisms of the affine plane." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=75337.
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Birational morphisms f: X $ to$ Y of nonsingular surfaces are studied first. Properties of the surfaces X and Y are shown to be related to certain numerical data extracted from the configuration of "missing curves" of f, that is, the curves in Y whose generic point is not in f (X). These results are then applied to the problem of decomposing birational endomorphisms of the plane into a succession of irreducible ones.A graph-theoretic machinery is developed to keep track of the desingularization of the divisors at infinity of the plane. That machinery is then used to investigate the problem of classifying all birational endomorphisms of the plane, and a complete classification is given in the case of two fundamental points.
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Farkas, Gavril Marius. "The birational geometry of the moduli space of curves." [S.l. : Amsterdam : s.n.] ; Universiteit van Amsterdam [Host], 2000. http://dare.uva.nl/document/84192.
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Perry, Alexander Richard. "Derived categories and birational geometry of Gushel-Mukai varieties." Thesis, Harvard University, 2016. http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493330.
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We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group.
We study the K3 category associated to a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.Mathematics
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Krashen, Daniel Reuben. "Birational isomorphisms between Severi-Brauer varieties." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3034558.
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Kaloghiros, Anne-Sophie. "The topology of terminal quartic 3-folds." Thesis, University of Cambridge, 2007. https://www.repository.cam.ac.uk/handle/1810/214794.
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Let Y be a quartic hypersurface in P⁴ with terminal singularities. The Grothendieck-Lefschetz theorem states that any Cartier divisor on Y is the restriction of a Cartier divisor on P⁴. However, no such result holds for the group of Weil divisors. More generally, let Y be a terminal Gorenstein Fano 3-fold with Picard rank 1. Denote by s(Y )=h_4 (Y )-h² (Y ) = h_4 (Y )-1 the defect of Y. A variety is Q-factorial when every Weil divisor is Q-Cartier. The defect of Y is non-zero precisely when the Fano 3-fold Y is not Q-factorial. Very little is known about the topology of non Q-factorial terminal Gorenstein Fano 3-folds. Q-factoriality is a subtle topological property: it depends both on the analytic type and on the position of the singularities of Y . In this thesis, I endeavour to answer some basic questions related to this global topolgical property. First, I determine a bound on the defect of terminal quartic 3-folds and on the defect of terminal Gorenstein Fano 3-folds that do not contain a plane. Then, I state a geometric motivation of Q-factoriality. More precisely, given a non Q-factorial quartic 3-fold Y , Y contains a special surface, that is a Weil non-Cartier divisor on Y . I show that the degree of this special surface is bounded, and give a precise list of the possible surfaces. This question has traditionally been studied in the context of Mixed Hodge Theory. I have tackled it from the point of view of Mori theory. I use birational geometric methods to obtain these results.
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Venkatram, Kartik (Kartik Swaminathan). "Birational geometry of the space of rational curves in homogeneous varieties." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/68484.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 55-56).
In this thesis, we investigate the birational geometry of the space of rational curves in various homogeneous spaces, with a focus on the quasi-map compactification induced by the Quot and Hyperquot functors. We first study the birational geometry of the Quot scheme of sheaves on P1 via techniques from the Mori program, explicitly describing its associated cones of ample and effective divisors as well as the various Mori chambers within the latter. We compute the base loci of all effective divisors, and give a conjectural description of the induced birational models. We then partially extend our results to the Hyperquot scheme of sheaves on P', which gives the analogous compactification for rational curves in flag varieties. We fully describe the cone of ample divisors in all cases and the cone of effective divisors in certain ones, but only claim a partial description of the latter in general.
by Kartik Venkatram.
Ph.D.
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Duran, James Joseph. "Differential geometry of surfaces and minimal surfaces." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1542.
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Aghasi, Mansour. "Geometry of arithmetic surfaces." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5270/.
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In this thesis my emphasis is on the resolution of the singularities of fibre products of Arithmetic Surfaces. In chapter one as an introduction to my thesis some elementary concepts related to regular and singular points are reviewed and the concept of tangent cone is defined for schemes over a discrete valuation ring. The concept of arithmetic surfaces is introduced briefly in the end of this chapter. In chapter 2 my new procedures namely the procedure of Mojgan(_1) and the procedure of Mahtab(_2) and a new operator called Moje are introduced. Also the concept of tangent space is defined for schemes over a discrete valuation ring. In chapter 3 the singularities of schemes which are the fibre products of some surfaces with ordinary double points are resolved. It is done in two different methods. The results from both methods are consistent. In chapter 4, I have tried to resolve the singularities of a special class of arithmetic three-folds, namely those which are the fibre product of two arithmetic surfaces, which were very helpful to achieve my final results about the resolution of singularities of fibre products of the minimal regular models of Tate. Chapter 5 includes my final results which are about the resolution of singularities of the fibre product of two minimal regular models of Tate.
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Chaparro, Maria Guadalupe. "Minimal surfaces." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3118.
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The focus of this project consists of investigating when a ruled surface is a minimal surface. A minimal surface is a surface with zero mean curvature. In this project the basic terminology of differential geometry will be discussed including examples where the terminology will be applied to the different subjects of differential geometry. In addition the focus will be on a classical theorem of minimal surfaces referred to as the Plateau's Problem.
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Mboro, René. "Birational invariants : cohomology, algebraic cycles and Hodge theory cohomologie." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLX049/document.
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Dans cette thèse, nous étudions certains invariants birationnels des variétés projectives lisses, en lien avec les questions de rationalité de ces variétés. Elle se compose de trois chapitres qui peuvent être lus indépendamment.Dans le premier chapitre, nous étudions, pour certaines familles de variétés, certains invariants birationnels stables, nuls pour l'espace projectif, apparaissant naturellement avec les formules de Manin. D'une part, nous montrons que l'invariant birationnel qu'est le groupe des cycles de torsion de codimension 3 contenus dans le noyau de l'application classe de cycle de Deligne est pour, les hypersurfaces cubiques complexes de dimension 5, contrôlé par l'invariant birationnel de sa variété des droites donné par le groupe des 1-cycles de torsion contenus dans le noyau de l'application classe de cycle de Deligne. D'autre part on établit la nullité du groupe de Griffiths des 1-cycles pour la variété des droites d'une hypersurface de l'espace projectif sur un corps algébriquement clos de caractérsitique 0, lorsque celle-ci est lisses et de Fano d'indice au moins 3.Les deux derniers chapitres se concentrent sur des aspects différents d'une propriété invariante par équivalence birationnelle stable introduite récemment par Voisin: l'existence d'une décomposition de Chow de la diagonale. Dans le second chapitre, nous étendons à la caractéristique positive p > 2 une partie des résultats obtenus par Voisin sur la décomposition de Chow de la diagonale des hypersurfaces cubiques complexes de dimension 3.Dans le dernier chapitre, on étudie la notion de dimension CH0 essentielle introduite par Voisin et reliée à l’existence d’une décomposition de Chow de la diagonale en ce que dire d’une variété qu’elle est de dimension CH0 essentielle nulle équivaut à affirmer l’existence d’une décomposition de Chow de sa diagonale. Nous présentons des conditions suffisantes (et nécessaires) pour assurer qu’une variété complexe dont le groupe des 0-cycle est trivial et dont la dimension CH_0 essentielle est au plus 2 est de dimension CH_0 essentielle nulleIn this thesis, we study some birational invariants of smooth projective varieties, in view of rationality questions for these varieties. It consists of three parts, that can be read independently.In the first chapter, we study, for some families of varieties, some stable birational invariants, that vanish for projective space and that appear naturally with Manin formulas. On one hand, we show for complex cubic 5-folds that the birational invariant given by the group of torsion codimension 3 cycles annihilated by the Deligne cycle map is controlled by the group of torsion 1-cycles of its variety of lines annihilated by the Deligne cycle map. We also prove that the Griffiths group of 1-cycles for the variety of lines of a hypersurface of the projective space over an algebraically closed field of characteristic 0, is trivial when the variety is smooth and Fano of index at least 3.The two last chapters focus on different aspects of the Chow-theoretic decomposition of the diagonal, a property which is invariant under stable birational equivalence, recently introduced by Voisin. In the second chapter, we adapt in characteristic greater than 2, part of the results, obtained by Voisin over the complex numbers, on the decomposition of the diagonal of cubic threefolds.In the last chapter, we study the concept of essential CH_0-dimension introduced by Voisin and related to the decomposition of the diagonal in that having essential CH_0-dimension 0 is equivalent to admitting a Chow-theoretic decomposition of the diagonal. We give sufficient (and necessary) conditions, for a complex variety with trivial group of 0-cycles, having essential CH_0-dimension non greater than 2 to admit a Chow-theoretic decomposition of the diagonal
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Syzdek, Wioletta. "Seshadri constants and geometry of surfaces." [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975820532.
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Mitsui, Kentaro. "Bimeromorphic geometry of rigid analytic surfaces." 京都大学 (Kyoto University), 2011. http://hdl.handle.net/2433/142438.
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He, Zhuang. "On Moduli Spaces of Weighted Pointed Stable Curves and Applications." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1437187765.
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Bråmå, Erik. "Strain Energy of Bézier Surfaces." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145645.
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Bézier curves and surfaces are used to great success in computer-aided design and finite element modelling, among other things, due to their tendency of being mathematically convenient to use. This thesis explores the different properties that make Bézier surfaces the strong tool that it is. This requires a closer look at Bernstein polynomials and the de Castiljau algorithm. To illustrate some of these properties, the strain energy of a Bézier surface is calculated. This demands an understanding of what a surface is, which is why this thesis also covers some elementary theory regarding parametrized curves and surface geometry, including the first and second fundamental forms.
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O'Neill, Edward Finbar. "Geometry based constructions for curves and surfaces." Thesis, University of Birmingham, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.251132.
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Dangskul, Supreedee. "Construction of Seifert surfaces by differential geometry." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20382.
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A Seifert surface for a knot in ℝ³ is a compact orientable surface whose boundary is the knot. Seifert surfaces are not unique. In 1934 Herbert Seifert provided a construction of such a surface known as the Seifert Algorithm, using the combinatorics of a projection of the knot onto a plane. This thesis presents another construction of a Seifert surface, using differential geometry and a projection of the knot onto a sphere. Given a knot K : S¹⊂ R³, we construct canonical maps F : ΛdiffS² → ℝ=4πZ and G : ℝ³ - K(S¹) → ΛdiffS² where ΛdiffS² is the space of smooth loops in S². The composite FG : ℝ³ - K(S¹) → ℝ=4πZ is a smooth map defined for each u∈2 ℝ³ - K(S¹) by integration of a 2- form over an extension D² → S² of G(u) : S1 → S². The composite FG is a surjection which is a canonical representative of the generator 1∈H¹(ℝ³- K(S¹)) = Z. FG can be defined geometrically using the solid angle. Given u ∈ ℝ³ - K(S¹), choose a Seifert surface Σu for K with u ∉ Σu. It is shown that FG(u) is equal to the signed area of the shadow of Σu on the unit sphere centred at u. With this, FG(u) can be written as a line integral over the knot. By Sard's Theorem, FG has a regular value t ∈ ℝ=4πZ. The behaviour of FG near the knot is investigated in order to show that FG is a locally trivial fibration near the knot, using detailed differential analysis. Our main result is that (FG)-¹(t)⊂ ℝ³ can be closed to a Seifert surface by adding the knot.
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Kaba, Mustafa Devrim. "On The Arithmetic Of Fibered Surfaces." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613674/index.pdf.
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In the first three chapters of this thesis we study two conjectures relating arithmetic with geometry, namely Tate and Lang&rsquos conjectures, for a certain class of algebraic surfaces. The surfaces we are interested in are assumed to be defined over a number field, have irregularity two and admit a genus two fibration over an elliptic curve. In the final chapter of the thesis we prove the isomorphism of the Picard motives of an arbitrary variety and its Albanese variety.
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Liu, Yang. "Optimization and differential geometry for geometric modeling." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40988077.
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Berardini, Elena. "Algebraic geometry codes from surfaces over finite fields." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.
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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principaleIn this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization
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Corman, Etienne. "Functional representation of deformable surfaces for geometry processing." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX075/document.
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La création et la compréhension des déformations de surfaces sont des thèmes récurrent pour le traitement de géométrie 3D. Comme les surfaces lisses peuvent être représentées de multiples façon allant du nuage de points aux maillages polygonales, un enjeu important est de pouvoir comparer ou déformer des formes discrètes indépendamment de leur représentation. Une réponse possible est de choisir une représentation flexible des surfaces déformables qui peut facilement être transportées d'une structure de données à une autre.Dans ce but, les "functional map" proposent de représenter des applications entre les surfaces et, par extension, des déformations comme des opérateurs agissant sur des fonctions. Cette approche a été introduite récemment pour le traitement de modèle 3D, mais a été largement utilisé dans d'autres domaines tels que la géométrie différentielle, la théorie des opérateurs et les systèmes dynamiques, pour n'en citer que quelques-uns. Le principal avantage de ce point de vue est de détourner les problèmes encore non-résolus, tels que la correspondance forme et le transfert de déformations, vers l'analyse fonctionnelle dont l'étude et la discrétisation sont souvent mieux connues. Cette thèse approfondit l'analyse et fournit de nouvelles applications à ce cadre d'étude. Deux questions principales sont discutées.Premièrement, étant donné deux surfaces, nous analysons les déformations sous-jacentes. Une façon de procéder est de trouver des correspondances qui minimisent la distorsion globale. Pour compléter l'analyse, nous identifions les parties les moins fiables du difféomorphisme grâce une méthode d'apprentissage. Une fois repérés, les défauts peuvent être éliminés de façon différentiable à l'aide d'une représentation adéquate des champs de vecteurs tangents.Le deuxième développement concerne le problème inverse : étant donné une déformation représentée comme un opérateur, comment déformer une surface en conséquence ? Dans une première approche, nous analysons un encodage de la structure intrinsèque et extrinsèque d'une forme en tant qu'opérateur fonctionnel. Dans ce cadre, l'objet déformé peut être obtenu, à rotations et translations près, en résolvant une série de problèmes d'optimisation convexe. Deuxièmement, nous considérons une version linéarisée de la méthode précédente qui nous permet d'appréhender les champs de déformation comme agissant sur la métrique induite. En conséquence la résolution de problèmes difficiles, tel que le transfert de déformation, sont effectués à l'aide de simple systèmes linéaires d'équationsCreating and understanding deformations of surfaces is a recurring theme in geometry processing. As smooth surfaces can be represented in many ways from point clouds to triangle meshes, one of the challenges is being able to compare or deform consistently discrete shapes independently of their representation. A possible answer is choosing a flexible representation of deformable surfaces that can easily be transported from one structure to another.Toward this goal, the functional map framework proposes to represent maps between surfaces and, to further extents, deformation of surfaces as operators acting on functions. This approach has been recently introduced in geometry processing but has been extensively used in other fields such as differential geometry, operator theory and dynamical systems, to name just a few. The major advantage of such point of view is to deflect challenging problems, such as shape matching and deformation transfer, toward functional analysis whose discretization has been well studied in various cases. This thesis investigates further analysis and novel applications in this framework. Two aspects of the functional representation framework are discussed.First, given two surfaces, we analyze the underlying deformation. One way to do so is by finding correspondences that minimize the global distortion. To complete the analysis we identify the least and most reliable parts of the mapping by a learning procedure. Once spotted, the flaws in the map can be repaired in a smooth way using a consistent representation of tangent vector fields.The second development concerns the reverse problem: given a deformation represented as an operator how to deform a surface accordingly? In a first approach, we analyse a coordinate-free encoding of the intrinsic and extrinsic structure of a surface as functional operator. In this framework a deformed shape can be recovered up to rigid motion by solving a set of convex optimization problems. Second, we consider a linearized version of the previous method enabling us to understand deformation fields as acting on the underlying metric. This allows us to solve challenging problems such as deformation transfer are solved using simple linear systems of equations
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Cox, Anna Lee. "A categorization of piecewise-linear surfaces." Virtual Press, 1994. http://liblink.bsu.edu/uhtbin/catkey/902464.
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Any Piecewise-Linear (PL) surface can be formed from a regular polygon (including the interior) with an even number of edges, where the edges are identified in pairs to form a two-dimensional manifold. The resulting surfaces can be distinguished by algebraic means. An analysis of the construction algorithm can also be used to determine the resulting surface. Knowledge of the polygon used can also yield information about the surfaces formed.In this thesis, an algorithm is developed that will analyze all possible edge pairings for an arbitrary regular polygon. The combination of this data, along with known techniques from geometric topology, will categorize the constructions of these PL surfaces. A procedure using matrices is developed that will determine the Euler number and establish which algebraic words are equivalent.This topic extends to two-dimensional manifolds a classical method of analysis for three-dimensional manifolds. It therefore provides a more geometrical approach than has traditionally been used for two dimensional surfaces.Department of Mathematical Sciences
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Deopurkar, Anand. "Alternate Compactifications of Hurwitz Spaces." Thesis, Harvard University, 2012. http://dissertations.umi.com/gsas.harvard:10308.
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We construct several modular compactifications of the Hurwitz space \(H^d_{g/h}\) of genus g curves expressed as d-sheeted, simply branched covers of genus h curves. They are obtained by allowing the branch points of the cover to collide to a variable extent, generalizing the spaces of twisted admissible covers of Abramovich, Corti, and Vistoli. The resulting spaces are very well-behaved if d is small or if relatively few collisions are allowed. In particular, for d = 2 and 3, they are always well-behaved. For d = 2, we recover the spaces of hyperelliptic curves of Fedorchuk. For d = 3, we obtain new birational models of the space of triple covers. We describe in detail the birational geometry of the spaces of triple covers of \(P^1\) with a marked fiber. In this case, we obtain a sequence of birational models that begins with the space of marked (twisted) admissible covers and proceeds through the following transformations: (1) sequential contractions of the boundary divisors, (2) contraction of the hyperelliptic divisor, (3) sequential flips of the higher Maroni loci, (4) contraction of the Maroni divisor (for even g). The sequence culminates in a Fano variety in the case of even g, which we describe explicitly, and a variety fibered over \(P^1\) with Fano fibers in the case of odd g.Mathematics
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Rockwood, A. P. "Blending surfaces in solid geometric modelling." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234923.
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Mechanical CAD/CAM (computer aided design/manufacturing) as a field research concerns itself with the algorithms and the mathematics necessary to simulate mechanical parts of the computer, that is to produce a computer model. Solid modelling is a subdiscipline in which the computer model accurately simulates volumetric, i.e. 'solid', properties of mechanical parts. This dissertation deals with a particular type of free-form surface, the blending surface, which is particularly well-suited for solid modelling. A blending surface is one which replaces creases and kinks in the original model with smooth surfaces. A fillet surface is a simple example. We introduce an intuitive paradigm for devising different types of blending forms. Using the paradigm, three forms are derived: the circular, the rolling-ball, and the super-elliptic forms. Important mathematical properties are investigated for the blending surfaces, e.g. continuity, smoothness, containment etc. Blending on blends is introduced as a notion which both extends the flexibility of blending surfaces and allows the blending of multiple surfaces. Blending on blends requires one to think about the way in which the defining functions act as a distance measure from a point in space to a surface. The function defining the super-elliptic blend is offered as an example or a poor distance measure. The zero surface of this function is then embedded within a function which provides an improved distance measure. Mathematical properties are derived for the new function. A weakness in the continuity properties of above blending form is rectified by defining another method to embed the super elliptic blend into a function with better distance properties. This is the displacement form. The concern with this form is its computational reliability which is, therefore, considered in more depth. In the process of integrating the blending surface geometry into a solid modelling environment so it was usable, it was discovered that three other formidable problems needed some type of resolution. These were the topological, the intersection and the display problems. We report on the problems, and solutions which we developed.
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McKinnon, David N. R. "The multiple view geometry of implicit curves and surfaces /." [St. Lucia, Qld.], 2006. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe19677.pdf.
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Björklund, Johan. "Knots and Surfaces in Real Algebraic and Contact Geometry." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-156908.
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This thesis consists of a summary and three articles. The thesis is devoted to the study of knots and surfaces with additional geometric structures compared to the classical smooth structure. In Paper I, real algebraic rational knots in real projective space are studied up to rigid isotopy and we show that two real rational algebraic knots of degree at most 5 are rigidly isotopic if, and only if, their degree and encomplexed writhe are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree at most 6. Furthermore, an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented. In Paper II, we construct an invariant of parametrized generic real algebraic surfaces in real projective space which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: an intersection of two real sheets, an intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend their definition of this self linking number to the case of parametrized generic real algebraic surfaces. In Paper III, we give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in the product of a punctured Riemann surface with the real line. As an application we show that for any nonzero homology class h, and for any integer k there exist k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.
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Randecker, Anja [Verfasser]. "Geometry and topology of wild translation surfaces / Anja Randecker." Karlsruhe : KIT-Bibliothek, 2015. http://d-nb.info/1084112442/34.
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Yilmaz, Oguzhan. "Repair of complex geometry components and free-form surfaces." Thesis, University of Nottingham, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.437089.
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Ruffoni, Lorenzo <1989>. "The Geometry of Branched Complex Projective Structures on Surfaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amsdottorato.unibo.it/7860/1/ruffoni_lorenzo_tesi.pdf.
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We study the geometry of deformations of structures locally modelled on the Riemann sphere, up to branched covers, focusing on structures with quasi-Fuchsian holonomy and on structures which admit holomorphically trivial deformations. Applications to Riemann-Hilbert problems are discussed.
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Liu, Yang, and 劉洋. "Optimization and differential geometry for geometric modeling." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40988077.
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Chiek, Veasna. "Geodesic on surfaces of constant Gaussian curvature." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3045.
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The goal of the thesis is to study geodesics on surfaces of constant Gaussian curvature. The first three sections of the thesis is dedicated to the definitions and theorems necessary to study surfaces of constant Gaussian curvature. The fourth section contains examples of geodesics on these types of surfaces and discusses their properties. The thesis incorporates the use of Maple, a mathematics software package, in some of its calculations and graphs. The thesis' conclusion is that the Gaussian curvature is a surface invariant and the geodesics of these surfaces will be the so-called best paths.
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Kotschick, Dieter. "On the geometry of certain 4 - manifolds." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236179.
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Turowski, Gudrun. "Nichtparametrische Minimalflächen vom Typ des Kreisrings und ihr Verhalten längs Kanten der Stützfläche." Bonn : [s.n.], 1998. http://catalog.hathitrust.org/api/volumes/oclc/41464677.html.
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Huang, Hui. "Efficient reconstruction of 2D images and 3D surfaces." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/2821.
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The goal of this thesis is to gain a deep understanding of inverse problems arising from 2D image and 3D surface reconstruction, and to design effective techniques for solving them. Both computational and theoretical issues are studied and efficient numerical algorithms are proposed.
The first part of this thesis is concerned with the recovery of 2D images, e.g., de-noising and de-blurring. We first consider implicit methods that involve solving linear systems at each iteration. An adaptive Huber regularization functional is used to select the most reasonable model and a global convergence result for lagged diffusivity is proved. Two mechanisms---multilevel continuation and multigrid preconditioning---are proposed to improve efficiency for large-scale problems. Next, explicit methods involving the construction of an artificial time-dependent differential equation model followed by forward Euler discretization are analyzed. A rapid, adaptive scheme is then proposed, and additional hybrid algorithms are designed to improve the quality of such processes. We also devise methods for more challenging cases, such as recapturing texture from a noisy input and de-blurring an image in the presence of significant noise.
It is well-known that extending image processing methods to 3D triangular surface meshes is far from trivial or automatic. In the second part of this thesis we discuss techniques for faithfully reconstructing such surface models with different features. Some models contain a lot of small yet visually meaningful details, and typically require very fine meshes to represent them well; others consist of large flat regions, long sharp edges (creases) and distinct corners, and the meshes required for their representation can often be much coarser. All of these models may be sampled very irregularly. For models of the first class, we methodically develop a fast multiscale anisotropic Laplacian (MSAL) smoothing algorithm. To reconstruct a piecewise smooth CAD-like model in the second class, we design an efficient hybrid algorithm based on specific vertex classification, which combines K-means clustering and geometric a priori information. Hence, we have a set of algorithms that efficiently handle smoothing and regularization of meshes large and small in a variety of situations.
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Rycroft, Jeanette Erica. "A geometrical investigation into the projections of surfaces and space curves." Thesis, University of Liverpool, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.322492.
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Ginebre, Emmanuel. "Geometry-dependence of the adhesive strength of biomimetic, micropatterned surfaces." Thesis, Linköpings universitet, Mekanik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-81067.
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Pressure sensitive adhesive surfaces are often inspired by nature. Miming the toe-surface of gecko, engineered surfaces made of thousands of micro-pillars show promising adhesive properties. This surfaces, covered with cylindrical pillars arranged into a pattern have adhesive properties greatly dependent on the geometrical characteristics. In this thesis, have been studied successively two models of micro-patterned surfaces, one two-dimensional, the other in three-dimensional using a FEM tool. Varying geometry parameters, has been determined optimal geometries to improve adhesive strength on these biomimetic, micropatterned surfaces. This study concludes to the non-adaptability of one-level scale micropatterned surface to large area of adhesion, to the strong advantage from the point of adhesion per contact area for high aspect ratio at each level of the geometry and study the opportunity of hierarchical structures. Some further suggestions of improvements to adhesion properties are discussed in the final chapter.
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Sánchez, Luis Florial Espinoza. "Surfaces in 4-space from the affine differential geometry viewpoint." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-23032015-142340/.
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In this thesis, we study locally strictly convex surfaces from the affine differential viewpoint and generalize some tools for locally strictly submanifolds of codimension 2. We introduce a family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally atrictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal to the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes. More generally, by using the metric of the transversal vector field on M we introduce the affine normal plane and the families of the affine distance and height functions on M. We show that the singularities of the family of the affine height functions appear at directions on the affine normal plane and the singularities of the family of the affine distance functions appear at points on the affine normal plane and the affine focal points correspond as degenerate singularities of the family of affine distance functions. Moreover we show that if M is immersed in a locally strictly convex hypersurface then the affine normal plane contains the affine normal vector to the hypersurface. Finally, we conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.Nesta tese estudamos as superfícies localmente estritamente convexas desde o ponto de vista da geometria diferencial afim e generalizamos algumas ferramentas para subvariedades localmente estritamente convexas de codimensão 2. Introduzimos uma família de métricas afins sobre uma superfície localmente estritamente convexa M no 4-espaço afim. Então, definimos os planos equiafins simétrico e antissimétrico associados com alguma métrica. Mostramos que se M é imersa em uma hiperquádrica localmente estritamente convexa, então os planos simétrico e assimétrico são iguais e contêm o campo vetorial normal afim à hiperquádrica. Em particular, qualquer superfície imersa em uma hiperquádrica localmente estritamente convexa é semiumbílica afim com relação ao plano equiafim simétrico ou antissimétrico. Mais geralmente, usando a métrica do campo transversal sobre M introduzimos o plano normal afim e as famílias de funções distância e altura afim sobre M. Provamos que as singularidades da família de funções altura afim aparecem como direções do plano normal afim e as singularidades da família de funções distância afim aparecem como pontos sobre o plano normal afim e os pontos focais correspondem às singularidades degeneradas da família de funções distância afim. Também provamos que se M é uma superfície imersa em uma hipersuperfície localmente estritamente convexa, então o plano normal afim contém o vetor normal afim à hipersuperfície. Finalmente, concluímos que qualquer superfície imersa em uma hiperesfera localmente estritamente convexa é semiumbílica afim.
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Alagal, Wafa Abdullah. "Application of Bridgeland stability to the geometry of abelian surfaces." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20440.
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A key property of projective varieties is the very ampleness of line bundles as this provides embeddings into projective space and allows us to express the variety in equational terms. In this thesis we study the general version of this property which is k- very ampleness of line bundles. We introduce the notation of critical k-very ampleness and compute it for abelian surfaces. The property of k-very ampleness is usually discussed using tools from divisor theory but we take a different approach and use methods from derived algebraic geometry as part of program to use properties of the derived category of a variety to access the geometry of the variety. In particular, we use the Fourier-Mukai transform, moduli spaces of sheaves and properties of Bridgeland stability. We compute walls for certain Bridgeland stable spaces and certain Chern characters and to complete the picture we study the moduli spaces of torsion sheaves with minimal first Chern class and we go on to compute the walls for these as well building on tools developed earlier in the thesis.
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Bozhkov, Yuri Dimitrov. "Specific complex geometry of certain complex surfaces and three-folds." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/110781/.
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One of the most important consequences of Yau's proof of the Calabi’s conjecture is the existence of a non-trivial Ricci-flat metric on K3 surfaces. For its explicit construction would be of great interest. Since it is not available yet the qualitative description of this metric would also have certain significance. In Chapter 1 we propose an approximation of the K3 Kahler-Einstein-Calabi-Yau metric for Kummer surfaces. It is obtained by gluing 16 pieces of the Eguchi-Hanson metric and 16 pieces of the Euclidean metric. Two estimates on its curvature are proved. Then we discuss the possibility of application of C.Taubes’s iteration scheme for solving anti-self-duality equations. The reason is that the curvature of the metric in question is concentrated in small thin regions and it is almost anti-self-dual. It can be also used later to deduce stability of Kummer surfaces’ tangent bundle. In Chapter 2 we consider a special case of compact 3-folds M which are diffeo- morphic to the connected sum of n copies of S3 x S3. If n > 103, there is a complex structure of C1 = 0 on M, which is a non-Kahler manifold. We prove that there are no non-trivial fine bundles on M and hence we deduce that its tangent bundle is stable with respect to any Gauduchon metric. By a theorem of Li and Yau we conclude that there is an Hermitian-Einstein metric on M. Our basic hypothesis is that the Hermitian-Einstein metric and the Gauduchon metric coincide. This is similar to the previous situation on K3. Then we consider the deformations of this metric, keeping the volume and the complex structure fixed. We seek the place of M in the classification of almost Hermitian manifolds by Gray and Hervella and explore some sorts of conditions which can be imposed on M and which can substitute the Kiihler one. We also show that on Hermitian non-Kaliler manifolds with h2'0 = 0 there are no non-zero anti-symmetric deformations of the complex structure.