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1
Wilson, Andrew. "Smooth exceptional del Pezzo surfaces." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4735.
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For a Fano variety V with at most Kawamata log terminal (klt) singularities and a finite group G acting bi-regularly on V , we say that V is G-exceptional (resp., G-weakly-exceptional) if the log pair (V,∆) is klt (resp., log canonical) for all G-invariant effective Q-divisors ∆ numerically equivalent to the anti-canonical divisor of V. Such G-exceptional klt Fano varieties V are conjectured to lie in finitely many families by Shokurov ([Sho00, Pro01]). The only cases for which the conjecture is known to hold true are when the dimension of V is one, two, or V is isomorphic to n-dimensional projective space for some n. For the latter, it can be shown that G must be primitive—which implies, in particular, that there exist only finitely many such G (up to conjugation) by a theorem of Jordan ([Pro00]). Smooth G-weakly-exceptional Fano varieties play an important role in non-rationality problems in birational geometry. From the work of Demailly (see [CS08, Appendix A]) it follows that Tian’s αG-invariant for such varieties is no smaller than one, and by a theorem of Tian such varieties admit G-invariant Kähler-Einstein metrics. Moreover, for a smooth G-exceptional Fano variety and given any G-invariant Kähler formin the first Chern class, the Kähler-Ricci iteration converges exponentially fast to the Kähler form associated to a Kähler- Einsteinmetric in the C∞(V)-topology. The termexceptional is inherited from singularity theory, to which this study enjoys strong links. We classify two-dimensional smooth G-exceptional Fano varieties (del Pezzo surfaces) and provide a partial list of all G-exceptional and G-weakly-exceptional pairs (S,G), where S is a smooth del Pezzo surface and G is a finite group of automorphisms of S. Our classification confirms many conjectures on two-dimensional smooth exceptional Fano varieties.
2
Loughran, Daniel Thomas. "Manin's conjecture for del Pezzo surfaces." Thesis, University of Bristol, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.544344.
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Kosta, Dimitra. "Del Pezzo surfaces with Du Val singularities." Thesis, University of Edinburgh, 2009. http://hdl.handle.net/1842/3934.
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A lot of attention has been drawn recently to global log canonical thresholds of Fano varieties, which are algebraic counterparts of the α-invariant of Tian for smooth Fano varieties. In particular, global log canonical thresholds are related to the existence of Kahler-Einstein metrics on Fano varieties. The purpose of this thesis is to apply techniques from singularity theory in order to compute the global log canonical thresholds of all Del Pezzo surfaces of degree 1 with Du Val singularities, as well as the global log canonical thresholds of all Del Pezzo surfaces of Picard rank 1 with Du Val singularities. As a consequence, it is proven that Del Pezzo surfaces of degree 1 with Du Val singularities admit a Kahler-Einstein metric if the singular locus consists of only A1, or A3, or A4 type Du Val singular points.
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Ueda, Kazushi. "Homological mirror symmetry for toric del Pezzo surfaces." 京都大学 (Kyoto University), 2006. http://hdl.handle.net/2433/144153.
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Kyoto University (京都大学)0048
新制・課程博士
博士(理学)
甲第12069号
理博第2963号
新制||理||1443(附属図書館)
23905
UT51-2006-J64
京都大学大学院理学研究科数学・数理解析専攻
(主査)助教授 河合 俊哉, 教授 齋藤 恭司, 教授 柏原 正樹
学位規則第4条第1項該当
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Manzaroli, Matilde. "Real algebraic curves in real del Pezzo surfaces." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLX017/document.
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L’étude topologique des variétés algébriques réelles remonte au moins aux travaux de Harnack, Klein, et Hilbert au 19éme siecle; en particulier, la classification des types d’isotopie réalisés par les courbes algébriques réelles d’un degré fixé dans RP2 est un sujet qui a connu un essor considérable jusqu'à aujourd'hui. En revanche, en dehors des études concernants les surfaces de Hirzebruch et les surfaces de degré au plus 3 dans RP3, à peu près rien n’est connu dans le cas de surfaces ambiantes plus générales. Cela est du en particulier au fait que les variétés construites en utilisant le "patchwork" sont des hypersurfaces de variétés toriques. Or, il existe de nombreuses autre surfaces algébriques réelles. Parmi celles-ci se trouvent les surfaces rationnelles réelles, et plus particulièrement les surfaces rèelles minimales. Dans cette thèse, on élargit l’étude des types d’isotopie réalisés par les courbes algébriques réelles aux surfaces réelles minimales de del Pezzo de degré 1 et 2. En outre, on termine la classification des types topologiques réalisés par les courbes algébriques réelles séparantes et non-séparantes de bidegré (5,5) sur la quadrique ellipsoideThe study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in RP2 is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in RP3, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the $mathbb{R}$-minimal surfaces. In this thesis, we extend the study of the topological types realized by real algebraic curves to the real minimal del Pezzo surfaces of degree 1 and 2. Furthermore, we end the classification of separating and non-separating real algebraic curves of bidegree $(5,5)$ in the quadric ellipsoid
6
Kleven, Stephanie. "Counting points of bounded height on del Pezzo surfaces." Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/2948.
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del Pezzo surfaces are isomorphic to either P1 x P1 or P2 blown up a times, where a ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P2 blown up a times with a ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.
7
Martinez, Garcia Jesus. "Dynamic alpha-invariants of del Pezzo surfaces with boundary." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/8090.
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The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.
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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
9
Festi, D. "Topics in the arithmetic of Del Pezzo and K3 surfaces." Doctoral thesis, Università degli Studi di Milano, 2016. http://hdl.handle.net/2434/411137.
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In this thesis we study the arithmetic of certain del Pezzo surfaces and K3 surfaces.We prove that all the del Pezzo surfaces of degree 2 over a finite field are unirational. We compute the Picard lattice of the members of a family of K3 surfaces given by double covers of the projective plane. Finally, we provide an explicit example of a K3 surface over the field of rational numbers with a particular Picard lattice of rank 2.
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Testa, Damiano. "The Severi problem for rational curves on del Pezzo surfaces." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/30356.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 141-142).
Let X be a smooth projective surface and choose a curve C on X. Let VC be the set of all irreducible divisors on X linearly equivalent to C whose normalization is a rational curve. The Severi problem for rational curves on X with divisor class [C] consists of studying the irreducibility of the spaces VC as C varies among all curves on X. In this thesis, we prove that all the spaces VC are irreducible in the case where X is a del Pezzo surface of degree at least two. If the degree of X is one, then we prove the same result only for a general X, with the exception of V-KX, where KX is the canonical divisor of X. It is well known that for general del Pezzo surface of degree one, V-KX consists of twelve points, and thus cannot be irreducible.
by Damiano Testa.
Ph.D.
11
Le, Boudec Pierre. "Répartition des points rationnels sur certaines surfaces de del Pezzo." Paris 7, 2012. http://www.theses.fr/2012PA077138.
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Dans cette thèse, nous nous intéressons à des problèmes de comptage de points rationnels sur certaines variétés algébriques. Une conjecture de Manin prévoit avec précision le comportement asymptotique du nombre de points rationnels de hauteur bornée sur les variétés de Fano. Notre but principal est de prouver la conjecture de Manin pour certains exemples de surfaces de del Pezzo définies sur Q. Pour cela, nous avons recours à des torseurs universels pour paramétrer les points rationnels et nous utilisons ensuite divers résultats de théorie analytique des nombres, tels que par exemple l'équidistribution des valeurs de certaines fonctions diviseur dans les progressions arithmétiques. Tout d'abord, nous traitons dans une première partie les cas de trois surfaces de del Pezzo de degré quatre, dont les types de singularité sont respectivement 3A1, A1+A2 et A3. Ensuite, nous traitons dans une seconde partie les cas de deux surfaces cubiques, dont les types de singularité sont respectivement 2A2+A1 et D4. Cette première est seulement le troisième exemple de surface cubique non-torique pour laquelle la conjecture de Manin est prouvée. Notons par ailleurs que le travail concernant cette dernière améliore un résultat de Browning et répond à un problème initialement posé par Tschinkel. Enfin, dans une annexe, comme autre application des résultats d'équidistribution mentionnés ci-dessus, nous établissons une formule asymptotique pour le nombre de valeurs sans puissance k ème du polynôme en r variables t1⋯tr−1In this thesis, we are interested in counting rational points on certain algebraic varieties. A conjecture of Manin predicts precisely the asymptotic behaviour of the number of rational points of bounded height on Fano varieties. Our main goal is to prove Manin's conjecture for some examples of del Pezzo surfaces defined over Q. For this, we resort to universal torsors to parametrize the rational points and then we make use of various analytic number theory results, such as for instance the equidistribution of the values of certain divisor functions in arithmetic progressions. To begin with, we deal in a first part with the cases of three quartic del Pezzo surfaces, whose singularity types are respectively 3A1, A1+A2 and A3. Afterwards, we deal in a second part with the cases of two cubic surfaces, whose singularity types are respectively 2A2+A1 and D4. The former is only the third example of non-toric cubic surface for which Manin's conjecture is proved. Note in addition that the work about the latter improves on a result of Browning and answers a problem initially posed by Tschinkel. Finally, in an appendix, as another application of the equidistribution results mentioned above, we establish an asymptotic formula for the number of power-free values of the r variables polynomial t1⋯tr−1
12
Blunk, Mark Alan. "Del Pezzo surfaces of degree 6 over an arbitrary field." Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1835130531&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Desjardins, Julie. "Densité des points rationnels sur les surfaces elliptiques et les surfaces de Del Pezzo de degré 1." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCC229/document.
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: Soit E→P1 une surface elliptique sur Q de base P1 non triviale. On s’intéresse à la Zariski-densité des points rationnels de E . Il est conjecturé que le signe de l’équation fonctionnelle d’une courbe elliptique est relié à la parité du rang de celle-ci. Modulo cette conjecture, il est suffisant de démontrer que le signe des fibres de E varie pour démontrer la Zariski-densité de E (Q). Un théorème conditionnel de Helfgott garantit que le signe moyen d’une surface non isotriviale est strictement compris entre -1 et 1. Dans le cas où E possède une place générique de réduction multiplicative, le signe moyen serait nul. Ce travail est conditionnel à deux conjectures de théorie analytique des nombres : la conjecture sans facteur carré et la conjecture de Chowla. L’objectif principal de cette thèse est d’éviter les conjectures utilisées par Helfgott pour démontrer la variation du signe sur les surfaces elliptiques non triviales. On réussit à se passer de la conjecture sans facteur carré sous certaines hypothèses techniques. On démontre ainsi (sous l’hypothèse de la conjecture de parité) la densité des points rationnels sur certaines surfaces elliptiques dont les coefficients sont des polynômes de degré arbitraire. Une surface de Del Pezzo de degré 1 est reliée par l’éclatement d’un point canonique à une surface elliptique rationnelle. On démontre inconditionnellement la densité des points rationnels dans plusieurs cas par des arguments géométriques. On étudie aussi la variation du signe de l’équation fonctionnelle pour des surfaces elliptiques rationnelles isotriviales et on cerne des conditions pour que le signe soit fixé. Dans le cas où le signe est +1, on en déduit des exemples de surfaces elliptiques non triviales dont les points rationnels pourraient ne pas être densesLet E→P1 be an non-trivial elliptic surface over Q with base P1. We are interested in the Zariski density of the rational points of E. It is conjectured that the root number of an elliptic curve E has the same parity as its rank. Assuming this conjecture, it is enough to show that the root number of the fibre of E varies to prove the Zariski density of E(Q). A conditional theorem of Helfgott garanties that the average root number of a non-isotrivial elliptic surface is strictly between -1 et 1. In the case where E has a generic place of multiplicative reduction, the average root number should be zero. This work is conditional to two analytic number theory conjectures : the squarefree conjecture and the Chowla conjecture. The main aim of this Ph.D thesis is to avoid the conjectures used by Helfgott when proving the variation of the root number on non-trivial elliptic surfaces. We manage to drop the squarefree conjecture assumption under some technical hypothesis. We show thus (under the parity conjecture) the density of the rational points on some elliptic surfaces whose coefficients have arbitrary large degree. Blowing up the anticanonical point on a del Pezzo surface of degree 1, one obtains a rational elliptic surface. We show unconditionally the density of the rational points in many cases by means of geometric arguments. We also study the variation of the root number on some isotrivial rational elliptic surfaces and we state the conditions under which it is constant. When it is +1, we deduce examples of non trivial elliptic surfaces whose rational points might not be dense
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Jost, Jan Niklas [Verfasser], and Thomas [Akademischer Betreuer] Reichelt. "Mirror Symmetry for Del Pezzo Surfaces / Jan Niklas Jost ; Betreuer: Thomas Reichelt." Heidelberg : Universitätsbibliothek Heidelberg, 2021. http://d-nb.info/1227711492/34.
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Wittenberg, Olivier. "Principe de Hasse pour les surfaces de del Pezzo de degré 4." Paris 11, 2005. http://www.theses.fr/2005PA112277.
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Soient k un corps de nombres et X une intersection lisse de deux quadriques dans P^n. On dit que X satisfait au principe de Hasse si l'existence d'un k_v-point de X pour toute place v de k suffit à assurer l'existence d'un k-point de X. Il est conjecturé que (i) X satisfait au principe de Hasse si n>=5; (ii) X satisfait au principe de Hasse si n=4 et si Br(X)/Br(k)=0. Le but de là thèse est de démontrer la conjecture (i), ainsi qu'une grande partie de la conjecture (ii), en admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres. Les deux premiers chapitres contiennent d'autre part des résultats d'intérêt indépendant sur l'arithmétique des surfaces munies d'un pinceau de courtes de genre 1Let k be a number field and X be a smooth intersection of two quadrics in P^n. The variety X is said to satisfy the Hasse principle if the existence of a k_v-point of X for each place v of k implies the existence of a k-point of X. It is conjectured that (i) X satisfies the Hasse principle if n>=5; (ii) X satisfies the Hasse principle if n=4 and Br(X)/Br(k)=0. The aim of this thesis is to establish conjecture (i) as well as a good deal of conjecture (ii), assuming Schinzels hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields
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Jost, Jan [Verfasser], and Thomas [Akademischer Betreuer] Reichelt. "Mirror Symmetry for Del Pezzo Surfaces / Jan Niklas Jost ; Betreuer: Thomas Reichelt." Heidelberg : Universitätsbibliothek Heidelberg, 2021. http://d-nb.info/1227711492/34.
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Kaplan, Nathan. "Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory." Thesis, Harvard University, 2013. http://dissertations.umi.com/gsas.harvard:10896.
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The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation between the weight enumerator of a linear code and the weight enumerator of its dual code. For certain codes C coming from families of varieties where it is not known how to determine the distribution of point counts directly, we analyze low-weight codewords of the dual code and apply the MacWilliams theorem and its generalizations to gain information about the weight enumerator of C. These low-weight dual codes can be described in terms of point sets that fail to impose independent conditions on this family of varieties. Our main results concern rational point count distributions for del Pezzo surfaces of degree 2, and for certain families of genus 1 curves. These weight enumerators have interesting geometric and coding theoretic applications for small q.Mathematics
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Heuberger, Liana. "Deux points de vue sur les variétés de Fano : géométrie du diviseur anticanonique et classification des surfaces à singularités 1/3(1,1)." Thesis, Paris 6, 2016. http://www.theses.fr/2016PA066129/document.
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Cette thèse concerne l'étude des variétés de Fano, qui sont des objets centraux de la classification des variétés algébriques. La première question abordée concerne les variétés de Fano lisses de dimension quatre. On cherche a étudier les potentielles singularités d'un diviseur anticanonique de sorte qu'on puisse les écrire sous une forme locale explicite. En tant qu'étape intermédiaire, on démontre aussi que ces points sont au plus des singularités terminales, c'est-à-dire les singularités les plus proches du cas lisse du point de vue de la géométrie birationnelle. On montre ensuite que ce dernier résultat se généralise en dimension arbitraire en admettant une conjecture de non-annulation de Kawamata.De façon complémentaire, on s¿intéresse à des variétés de Fano de dimension plus petite, mais admettant des singularités. Il s¿agit des surfaces de del Pezzo ayant des singularités de type 1/3(1,1). Ceci est l'exemple le plus simple de singularité rigide, c'est-à-dire qui reste inchangée à une déformation Q-Gorenstein près. On classifie entièrement ces objets en trouvant 29 familles. On obtient ainsi un tableau contenant des modèles de ces surfaces, qui pour la plupart sont des intersections complètes dans des variétés toriques. Ce travail s'inscrit dans un contexte plus large, qui a pour cible de calculer leur cohomologie quantique pour ensuite vérifier si deux conjectures en symmetrie miroirThis thesis concerns Fano varieties, which are central objects within the classification of algebraic varieties.The first problem we discuss involves smooth Fano varieties of dimension four. We study the potential singularities of an anticanonical divisor and determine their explicit local expression. As an intermediate step, we show that they are terminal points, that is the singularities which are closest to the smooth case from the point of view of birational geometry. We then show that the latter result generalizes in arbitrary dimension if we suppose that a nonvanishing conjecture of Kawamata holds.The second approach is to examine Fano varieties of smaller dimensions which admit singularities. The objects we consider are log del Pezzo surfaces with 1/3(1,1) points. This is the simplest example of a rigid singularity, that is it remains unchanged under Q-Gorenstein deformations. We give a complete classification of these surfaces, finding 29 families. We also provide a table describing almost all of them as complete intersections in toric varieties. This work belongs to an overarching project that aims at studying mirror symmetry for del Pezzo surfaces with cyclic quotient singularities
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Nguyen, Dong Quan Ngoc. "Nonexistence of Rational Points on Certain Varieties." Diss., The University of Arizona, 2012. http://hdl.handle.net/10150/238653.
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In this thesis, we study the Hasse principle for curves and K3 surfaces. We give several sufficient conditions under which the Brauer-Manin obstruction is the only obstruction to the Hasse principle for curves and K3 surfaces. Using these sufficient conditions, we construct several infinite families of curves and K3 surfaces such that these families are counterexamples to the Hasse principle that are explained by the Brauer-Manin obstruction.
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Henry-Labordère, Pierre. "Symétries en théorie M." Paris 7, 2003. http://www.theses.fr/2003PA077234.
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Koshelev, Dmitrii. "Nouvelles applications des surfaces rationnelles et surfaces de Kummer généralisées sur des corps finis à la cryptographie à base de couplages et à la théorie des codes BCH." Thesis, université Paris-Saclay, 2021. http://www.theses.fr/2021UPASM001.
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Il y a une théorie bien développée de ce qu'on appelle codes toriques, c'est-à-dire des codes de géométrie algébrique sur des variétés toriques sur un corps fini. A côté des tores et variétés toriques ordinaires (c'est-à-dire déployés), il y a non-déployés. La thèse est donc dédiée à l'étude des codes de géométrie algébrique sur les derniersThere is well developed theory of so-called toric codes, i.e., algebraic geometry codes on toric varieties over a finite field. Besides ordinary (i.e., split) tori and toric varieties there are non-split ones. Therefore the thesis is dedicated to the study of algebraic geometry codes on the latter
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Derenthal, Ulrich. "Geometry of universal torsors." Doctoral thesis, [S.l.] : [s.n.], 2006. http://webdoc.sub.gwdg.de/diss/2006/derenthal.
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Çelik, Türkü Özlüm. "Propriétés géométriques et arithmétiques explicites des courbes." Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S032/document.
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Les courbes algébriques sont des objets centraux de la géométrie algébrique. Dans cette thèse, nous étudions ces objets sous différents angles de la géométrie algébrique tels que la géométrie algébrique effective et la géométrie arithmétique. Dans le premier chapitre, nous étudions les courbes non-hyperelliptiques de genre g et leurs jacobiennes liées par l’intermédiaire de diviseurs thêta caractéristiques. Ces derniers contiennent des propriétés géométriques extrinsèques qui permettent de calculer les constantes thêta. Dans le deuxième chapitre, nous nous concentrons sur les courbes hyperelliptiques de genre 2 et leur surface de Kummer associée avec une motivation cryptographique. Dans le troisième et dernier chapitre, nous étudions les revêtements doubles non-ramifiés des courbes non-hyperelliptiques de genre g pour obtenir des informations sur le p-rangAlgebraic curves are central objects in algebraic geometry. In this thesis, we consider these objects from different angles of algebraic geometry such as computational algebraic geometry and arithmetic geometry. In the first chapter, we study non-hyperelliptic curves of genus g and their Jacobians linked via theta characteristic divisors. Such divisors provide extrinsic geometric properties which allow us to compute theta constants. In the second chapter, we focus on hyperelliptic curves of genus 2 and the associated Kummer surface with a cryptographic motivation. In the third and final chapter, we examine unramified double covers of non-hyperelliptic curves of genus g to obtain information about p-rank
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Paulot, Louis. "Théorie M et dualités." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2003. http://tel.archives-ouvertes.fr/tel-00005254.
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Dans leur recherche d'une théorie unifiée des interactions fondamentales, contenant en particulier un modèle quantique de la gravitation, les physiciens ont imaginé des théories de supercordes, dans lesquelles, en plus des cordes, on trouve des objets étendus de diverses dimensions, reliés par le groupe de U-dualité. De plus, on conjecture l'existence d'une théorie mère, la théorie M, dont la limite de basse énergie serait la supergravité à onze dimensions. Dans ce travail, nous montrons qu'en partant des surfaces de del Pezzo, on peut construire des superalgèbres de Kac-Moody généralisées qui contiennent les groupes de U-dualité et donnent le contenu en champs bosoniques (doublé) de la théorie M et de ses réductions dimensionnelles. On retrouve alors les équations du mouvement comme une condition d'auto-dualité, associée à une symétrie du réseau de Picard de la surface de del Pezzo correspondante. Cela permet d'expliquer la symétrie du «triangle magique» de Cremmer, Julia, Lü etPope.
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Lin, Chin-Yi, and 林金毅. "On Geometry of Del Pezzo Surfaces." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/33692235893984153443.
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博士國立臺灣大學
數學研究所
102
The thesis in on the geometry of del Pezzo surfaces. Early researches focused on smooth surfaces, while recently surfaces with singularities have been mostly considered. Consequently, in Chapter 2, different types of singularities are first discussed, and then del Pezzo surfaces can be defined formally in Chapter 3. Research on smooth surfaces are also given there. In Chapter 4, we introduce the complement theory developed by Shokurov, and we give some examples of weighted complete intersection in Chapter 5. Chapter 6 is about the relation between Kahler-Einstein metrics and del Pezzo surfaces. In Chapter 7 and Chapter 8, we introduce our research result. We use Riemann-Roch theorem to calculated Euler characteristics, and then give a special type of nonvanishing theorem.
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Pirozhkov, Dmitrii. "Admissible subcategories of del Pezzo surfaces." Thesis, 2020. https://doi.org/10.7916/d8-4wvy-fe69.
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Admissible subcategories are building blocks of semiorthogonal decompositions. Many examples of them are known, but few general properties have been proved, even for admissible subcategories in the derived categories of coherent sheaves on basic varieties such as projective spaces. We use a relation between admissible subcategories and anticanonical divisors to study admissible subcategories of del Pezzo surfaces. We show that any admissible subcategory of the projective plane has a full exceptional collection, and since all exceptional objects and collections for the projective plane are known, this provides a classification result for admissible subcategories. We also show that del Pezzo surfaces of degree at least three do not contain so-called phantom subcategories. These are the first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms.
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"Rational points on del Pezzo surfaces of degree 1 and 2." Thesis, 2011. http://hdl.handle.net/1911/70318.
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One of the fundamental problems in Algebraic Geometry is to study solutions to certain systems of polynomial equations in several variables, or in other words, find rational points on a given variety which is defined by equations. In this paper, we discuss the existence of del Pezzo surface of degree 1 and 2 with a unique rational point over any finite field [Special characters omitted.] , and we will give a lower bound on the number of rational points to each q. Furthermore, we will give explicit equations of del Pezzo surfaces with a unique rational point. Also we will discuss the rationality property of the del Pezzo surfaces especially in lower degrees.
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Knecht, Amanda Leigh. "Weak approximation for degree 2del Pezzo surfaces at places of bad reduction." Thesis, 2007. http://hdl.handle.net/1911/20618.
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This thesis addresses weak approximation for certain degree 2 del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of singular reductions of the surfaces to find prescribed sections through these fibers.
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Maddock, Zachary Alexander. "Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory." Thesis, 2012. https://doi.org/10.7916/D82B9568.
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This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.