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1
Abou-Rached, John. « Sheaves and schemes : an introduction to algebraic geometry ». Kansas State University, 2016. http://hdl.handle.net/2097/32608.
Texte intégralRésumé :
Master of ScienceDepartment of Mathematics
Roman Fedorov
The purpose of this report is to serve as an introduction to the language of sheaves and schemes via algebraic geometry. The main objective is to use examples from algebraic geometry to motivate the utility of the perspective from sheaf and scheme theory. Basic facts and definitions will be provided, and a categorical approach will be frequently incorporated when appropriate.
2
Bechtold, Benjamin [Verfasser], et Jürgen [Akademischer Betreuer] Hausen. « Cox sheaves on graded schemes, algebraic actions and F1-schemes / Benjamin Bechtold ; Betreuer : Jürgen Hausen ». Tübingen : Universitätsbibliothek Tübingen, 2018. http://d-nb.info/1168803829/34.
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Goward, Russell A. « A simple algorithm for principalization of monomial ideals / ». free to MU campus, to others for purchase, 2001. http://wwwlib.umi.com/cr/mo/fullcit?p3012972.
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4
Cliff, Emily Rose. « Universal D-modules, and factorisation structures on Hilbert schemes of points ». Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:9edee0a0-f30a-4a54-baf5-c833222303ca.
Texte intégralRésumé :
This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.
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Lee, Hwa Young. « The flag Hilbert scheme of points on nodal curves and the punctual Hilbert scheme of points of the cusp curve ». Diss., UC access only, 2009. http://proquest.umi.com/pqdweb?did=1907270841&sid=1&Fmt=7&clientId=48051&RQT=309&VName=PQD.
Texte intégralRésumé :
Thesis (Ph. D.)--University of California, Riverside, 2009.Includes abstract. Includes bibliographical references (leaf 71). Issued in print and online. Available via ProQuest Digital Dissertations.
6
BORGHESI, SIMONE. « Algebraic Morava K-theories and the higher degree formula ». Doctoral thesis, Northwestern University, 2000. http://hdl.handle.net/10281/39205.
Texte intégralRésumé :
This manuscript consists of two parts. In the first, a cohomology theory on the category of algebraic schemes over a field of characteristic zero is provided. This theory shares several properties with the topological Morava K-theories, hence the name. The second part contains a proof of Voevodsky and Rost conjectured degree formulae. The proof uses algebraic Morava K-theories.
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Heinze, Aiso. « Applications of Schur rings in algebraic combinatorics graphs, partial difference sets and cyclotomic schemes / ». [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=962888532.
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Kuang, Yu Rang. « Algebraic coupled-state calculation of positron-hydrogen collisions at low energy, using large coupling schemes ». Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/nq23106.pdf.
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Carissimi, Nicola. « Reconstruction of schemes via the tensor triangulated category of perfect complexes ». Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23343/.
Texte intégralRésumé :
This elaborate consists of a detailed presentation of the construction introduced for the first time by Paul Balmer and aimed to define a locally ringed space associated to a given tensor triangulated category, the so called spectrum of the category. The focus of this thesis is the case of the tensor triangulated category of perfect complexes on a noetherian scheme X, the full triangulated subcategory of the derived category of sheaves of modules consisting of complexes of sheaves locally quasi-isomorphic to complexes of locally free sheaves. This category inherits the structure of derived tensor product of complexes of sheaves of modules, becoming a tensor triangulated category. Its spectrum is a scheme isomorphic to X, providing a powerful reconstruction result.
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ALMashrafi, Mufeed Juma. « Analysis of stream cipher based authenticated encryption schemes ». Thesis, Queensland University of Technology, 2012. https://eprints.qut.edu.au/60916/1/Mufeed_ALMashrafi_Thesis.pdf.
Texte intégralRésumé :
Authenticated Encryption (AE) is the cryptographic process of providing simultaneous confidentiality and integrity protection to messages. This approach is more efficient than applying a two-step process of providing confidentiality for a message by encrypting the message, and in a separate pass providing integrity protection by generating a Message Authentication Code (MAC). AE using symmetric ciphers can be provided by either stream ciphers with built in authentication mechanisms or block ciphers using appropriate modes of operation.
However, stream ciphers have the potential for higher performance and smaller footprint in hardware and/or software than block ciphers. This property makes stream ciphers suitable for resource constrained environments, where storage and computational power are limited. There have been several recent stream cipher proposals that claim to provide AE. These ciphers can be analysed using existing techniques that consider confidentiality or integrity separately; however currently there is no existing framework for the analysis of AE stream ciphers that analyses these two properties simultaneously. This thesis introduces a novel framework for the analysis of AE using stream cipher algorithms.
This thesis analyzes the mechanisms for providing confidentiality and for providing integrity in AE algorithms using stream ciphers. There is a greater emphasis on the analysis of the integrity mechanisms, as there is little in the public literature on this, in the context of authenticated encryption. The thesis has four main contributions as follows.
The first contribution is the design of a framework that can be used to classify AE stream ciphers based on three characteristics. The first classification applies Bellare and Namprempre's work on the the order in which encryption and authentication processes take place. The second classification is based on the method used for accumulating the input message (either directly or indirectly) into the into the internal states of the cipher to generate a MAC. The third classification is based on whether the sequence that is used to provide encryption and authentication is generated using a single key and initial vector, or two keys and two initial vectors.
The second contribution is the application of an existing algebraic method to analyse the confidentiality algorithms of two AE stream ciphers; namely SSS and ZUC. The algebraic method is based on considering the nonlinear filter (NLF) of these ciphers as a combiner with memory. This method enables us to construct equations for the NLF that relate the (inputs, outputs and memory of the combiner) to the output keystream. We show that both of these ciphers are secure from this type of algebraic attack. We conclude that using a keydependent SBox in the NLF twice, and using two different SBoxes in the NLF of ZUC, prevents this type of algebraic attack.
The third contribution is a new general matrix based model for MAC generation where the input message is injected directly into the internal state. This model describes the accumulation process when the input message is injected directly into the internal state of a nonlinear filter generator. We show that three recently proposed AE stream ciphers can be considered as instances of this model; namely SSS, NLSv2 and SOBER-128. Our model is more general than a previous investigations into direct injection. Possible forgery attacks against this model are investigated. It is shown that using a nonlinear filter in the accumulation process of the input message when either the input message or the initial states of the register is unknown prevents forgery attacks based on collisions.
The last contribution is a new general matrix based model for MAC generation where the input message is injected indirectly into the internal state. This model uses the input message as a controller to accumulate a keystream sequence into an accumulation register. We show that three current AE stream ciphers can be considered as instances of this model; namely ZUC, Grain-128a and Sfinks.
We establish the conditions under which the model is susceptible to forgery and side-channel attacks.
11
Staal, Andrew Philippe. « On the existence of jet schemes logarithmic along families of divisors ». Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/3327.
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A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.Science, Faculty of
Mathematics, Department of
Graduate
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Staal, Andrew Phillipe. « On the existence of jet schemes logarithmic along families of divisors ». Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/3327.
Texte intégralRésumé :
A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.
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Groechenig, Michael. « Autoduality of the Hitchin system and the geometric Langlands programme ». Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:f0a08e96-2f25-4df1-9e56-99931e411f73.
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This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.
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Karadogan, Gulay. « The Moduli Of Surfaces Admitting Genus Two Fibrations Over Elliptic Curves ». Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12606084/index.pdf.
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In this thesis, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1,X(d),n) of morphisms of degree n from elliptic curves to the modular curve X(d), d≤
3. Ultimately, we show that the moduli spaces, considered, are fiber spaces over the affine line A¹
with fibers determined by the components of H (1,X(d),n).
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Wagner, David R. « Schur Rings Over Projective Special Linear Groups ». BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/6089.
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This thesis presents an introduction to Schur rings (S-rings) and their various properties. Special attention is given to S-rings that are commutative. A number of original results are proved, including a complete classification of the central S-rings over the simple groups PSL(2,q), where q is any prime power. A discussion is made of the counting of symmetric S-rings over cyclic groups of prime power order. An appendix is included that gives all S-rings over the symmetric group over 4 elements with basic structural properties, along with code that can be used, for groups of comparatively small order, to enumerate all S-rings and compute character tables for those S-rings that are commutative. The appendix also includes code optimized for the enumeration of S-rings over cyclic groups.
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Leimkuhler, Benedict, et Sebastian Reich. « Symplectic integration of constrained Hamiltonian systems ». Universität Potsdam, 1994. http://opus.kobv.de/ubp/volltexte/2007/1565/.
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A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.
17
Schmidt, Benjamin. « Stability Conditions on Threefolds and Space Curves ». The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1460542777.
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Sebestean, Magda. « Correspondance de McKay et equivalences derivees ». Phd thesis, Université Paris-Diderot - Paris VII, 2005. http://tel.archives-ouvertes.fr/tel-00012064.
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Le premier chapitre montre par des méthodes toriques ($G-$graphes) que pour tout entier positif $n$, le quotient de l'espace affine à $n$ dimensions par le groupe cyclique $G_n$ d'ordre $2^n-1$ admet le $G_n$-schema de Hilbert comme résolution lisse crepante. Le deuxième chapitre contient des résultats sur les champs algébriques (construction du champ algébrique lisse associé à une log-paire). Le troisième chapitre montre l'équivalence entre la catégorie dérivée bornée des faisceaux cohérents $G_n-$équivariants sur l'espace affine et celle des faisceaux cohérents sur la résolution $G_n-$Hilb. Chapitre 4 donne une réalisation géométrique de la conjecture de Broué via la correspondance de McKay. L'annexe contient des résultats sur les groupes trihédraux, y compris un programme magma.
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Prager, Stefan [Verfasser], et Andreas [Akademischer Betreuer] Dreuw. « Development of Frozen-Density Embedded Algebraic Diagrammatic Construction Schemes for Excited States and Quantum-Chemical Investigation of Photophysical Properties of Tetrathiaheterohelicenes / Stefan Prager ; Betreuer : Andreas Dreuw ». Heidelberg : Universitätsbibliothek Heidelberg, 2017. http://d-nb.info/1178008665/34.
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Le, Rudulier Cécile. « Points algébriques de hauteur bornée ». Thesis, Rennes 1, 2014. http://www.theses.fr/2014REN1S073/document.
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L'étude de la répartition des points rationnels ou algébriques d'une variété algébrique selon leur hauteur est un problème classique de géométrie diophantienne. Dans cette thèse, nous nous intéresserons au cardinal asymptotique de l'ensemble des points algébriques de degré fixé et de hauteur bornée d'une variété lisse de Fano définie sur un corps de nombres, lorsque la borne sur la hauteur tend vers l'infini. En particulier nous montrerons que cette question peut-être reliée à la conjecture de Batyrev-Manin-Peyre, c'est-à-dire le cas des points rationnels, sur un schéma de Hilbert ponctuel. Nous en déduisons ainsi la distribution des points algébriques de degré fixé d'une courbe rationnelle. Lorsque la variété de départ est une surface lisse de Fano, notre étude montre que les schémas de Hilbert associés fournissent, sous certaines conditions, de nouveaux contre-exemples à la conjecture de Batyrev-Manin-Peyre. Néanmoins, pour deux surfaces que nous étudions en détail, les schémas de Hilbert associés vérifient une version légèrement affaiblie de la conjecture de Batyrev-Manin-PeyreThe study of the distribution of rational or algebraic points of an algebraic variety according to their height is a classic problem in Diophantine geometry. In this thesis, we will be interested in the asymptotic cardinality of the set of algebraic points of fixed degree and bounded height of a smooth Fano variety defined over a number field, when the bound on the height tends to infinity. In particular, we show that this can be connected to the Batyrev-Manin-Peyre conjecture, i.e. the case of rational points, on some ponctual Hilbert scheme. We thus deduce the distribution of algebraic points of fixed degree on a rational curve. When the variety is a smooth Fano surface, our study shows that the associated Hilbert schemes provide, under certain conditions, new counterexamples to the Batyrev-Manin-Peyre conjecture. However, in two cases detailed in this thesis, the associated Hilbert schemes satisfie a slightly weaker version of the Batyrev-Manin-Peyre conjecture
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CATTANEO, ALBERTO. « NON-SYMPLECTIC AUTOMORPHISMS OF IRREDUCIBLE HOLOMORPHIC SYMPLECTIC MANIFOLDS ». Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/606455.
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La tesi si concentra sullo studio degli automorfismi di varietà olomorfe simplettiche irriducibili di tipo K3^[n], ovvero varietà equivalenti per deformazione allo schema di Hilbert di n punti su una superficie K3, per n > 1. Negli ultimi anni, molti teoremi classici riguardanti la classificazione degli automorfismi non-simplettici di superfici K3 sono stati estesi alle varietà di tipo K3^[2]. Siamo quindi interessati a comprendere se tali risultati possono essere ulteriormente generalizzati anche al caso di varietà di tipo K3^[n], per n > 2.
Nella prima parte della tesi descriviamo il gruppo degli automorfismi dello schema di Hilbert di n punti su una superficie K3 proiettiva generica, il cui reticolo di Picard è generato da un singolo fibrato ampio. Mostriamo che, se il gruppo non è triviale, esso è generato da una involuzione non-simplettica, la cui esistenza è determinata da condizioni aritmetiche che coinvolgono il numero n di punti e la polarizzazione della superficie. In aggiunta a tale caratterizzazione numerica, individuiamo anche delle condizioni necessarie e sufficienti per l'esistenza dell'involuzione riguardanti la struttura del reticolo di Picard dello schema di Hilbert.
La seconda parte della tesi è dedicata allo studio degli automorfismi non-simplettici di ordine primo su varietà di tipo K3^[n]. Dopo aver investigato le proprietà del reticolo invariante dell'automorfismo e del suo complemento ortogonale all'interno del secondo reticolo di coomologia della varietà, forniamo una classificazione per le loro classi di isometria. Affrontiamo quindi il problema di individuare varietà di tipo K3^[n] dotate di automorfismi non-simplettici che inducano ognuna delle possibili azioni in coomologia presenti nella nostra classificazione. Nel caso delle involuzioni, e degli automorfismi di ordine primo dispari per n=3, 4, siamo in grado di realizzare tutti i casi ammissibili, presentando una costruzione esplicita della varietà o almeno dimostrandone l'esistenza. Tra i numerosi esempi esibiti, è di particolare rilievo un nuovo automorfismo di ordine tre su una famiglia di dimensione dieci di varietà di Lehn-Lehn-Sorger-van Straten di tipo K3^[4]. Infine, per n < 6 descriviamo le famiglie di deformazione massimali di varietà di tipo K3^[n] dotate di una involuzione non-simplettica.We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution. In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution.
Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1. Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution. Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique.
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Sædén, Ståhl Gustav. « Hilbert schemes and Rees algebras ». Doctoral thesis, KTH, Matematik (Avd.), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-195717.
Texte intégralRésumé :
The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field. The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property. The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack.QC 20161110
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Epple, Alexander. « Methods for increased computational efficiency of multibody simulations ». Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/26532.
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Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2009.Committee Chair: Olivier A. Bauchau; Committee Member: Andrew Makeev; Committee Member: Carlo L. Bottasso; Committee Member: Dewey H. Hodges; Committee Member: Massimo Ruzzene. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Engel, Johannes [Verfasser]. « Hilbert Schemes of Quiver Algebras / Johannes Engel ». Wuppertal : Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1004850492/34.
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Bhattacharyya, Gargi. « Terwilliger algebras of wreath products of association schemes ». [Ames, Iowa : Iowa State University], 2008.
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Ozkan, Engin. « Fixed Point Scheme Of The Hilbert Scheme Under A 1-dimensional Additive Algebraic Group Action ». Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613165/index.pdf.
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In general we know that the fixed point locus of a 1-dimensional additive linear algebraic
group,G_{a}, action over a complete nonsingular variety is connected. In thesis, we explicitly
identify a subset of the G_{a}-fixed locus of the punctual Hilbert scheme of the d points,Hilb^{d}(P^{2}0),in P^{2}. In particular we give an other proof of the fact that Hilb^{d}(P^{2}
0) is connected.
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Kodalen, Brian G. « Cometric Association Schemes ». Digital WPI, 2019. https://digitalcommons.wpi.edu/etd-dissertations/512.
Texte intégralRésumé :
The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes.
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Sædén, Ståhl Gustav. « Rees algebras of modules and Quot schemes of points ». Licentiate thesis, KTH, Matematik (Avd.), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-156636.
Texte intégralRésumé :
This thesis consists of three articles. The first two concern a generalization of Rees algebras of ideals to modules. Paper A shows that the definition of the Rees algebra due to Eisenbud, Huneke and Ulrich has an equivalent, intrinsic, definition in terms of divided powers. In Paper B, we use coherent functors to describe properties of the Rees algebra. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors. In Paper C, we prove a generalization of Gotzmann's persistence theorem to finite modules. As a consequence, we show that the embedding of the Quot scheme of points into a Grassmannian is given by a single Fitting ideal.QC 20141218
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Song, Sung Yell. « The character tables of certain association schemes / ». The Ohio State University, 1987. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487329662147808.
Texte intégral
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Shao, Yijun. « A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian ». Diss., The University of Arizona, 2010. http://hdl.handle.net/10150/194715.
Texte intégralRésumé :
Let Md be the moduli space of algebraic maps (morphisms) of degree d from P^1 to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of Md satisfying the following property: the compactification is a smooth projective variety and the boundary is a simple normal crossing divisor. The main tool of the construction is blowing-up. We start with a smooth compactification given by Quot scheme, which we denote by Qd. The boundary Qd\Md is singular and of high codimension. Next, we give a filtration of the boundary Qd\Md by closed subschemes: Zd,0 subset Zd,1 subset ... Zd,d-1=Qd\Md. Then we blow up the Quot scheme Qd along these subschemes succesively, and prove that the final outcome is a compactification satisfying the desired properties. The proof is based on the key observation that each Zd,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Qd,r, is a relative Quot scheme over the Quot-scheme compactification Qr for Mr. The map from Qd,r to Zd,r is an isomorphism when restricted to the preimage of Zd,r\ Zd,r-1. With the help of the Qd,r's, one can show that the final outcome of the successive blowing-up is a smooth compactification whose boundary is a simple normal crossing divisor.
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Blanc, Anthony. « Invariants topologiques des espaces non-commutatifs ». Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2013. http://tel.archives-ouvertes.fr/tel-01012109.
Texte intégralRésumé :
Dans cette thèse, on donne une définition de la K-théorie topologique des espaces non-commutatifs de Kontsevich (c'est-à-dire des dg-catégories) définis sur les nombres complexes. L'introduction de ce nouvel invariant initie la recherche des invariants de nature topologique des espaces non-commutatifs, comme "simplifications" des invariants algébriques (K-théorie algébrique, homologie cyclique, périodique comme étudiés dans les travaux de Tsygan, Keller). La motivation principale vient de la théorie de Hodge non-commutative au sens de Katzarkov--Kontsevich--Pantev. En géométrie algébrique, la partie rationnelle de la structure de Hodge est donnée par la cohomologie de Betti rationnelle, qui est la cohomologie rationnelle de l'espace des points complexes du schéma. La recherche d'un espace associé à une dg-catégorie trouve une première réponse avec le champ (défini par Toën--Vaquié) classifiant les dg-modules parfaits sur cette dg-catégorie. La définition de la K-théorie topologique a pour ingrédient essentiel le foncteur de réalisation topologique des préfaisceaux en spectres sur le site des schémas de type fini sur les complexes. La partie connective de la K-théorie semi-topologique peut être définie comme la réalisation topologique du champ en monoïdes commutatifs des dg-modules parfaits. Cependant pour atteindre la K-théorie négative, on réalise le préfaisceau donné par la K-théorie algébrique non-connective. Un de nos résultats principaux énonce l'existence d'une équivalence naturelle entre ces deux définitions dans le cas connectif. On montre que la réalisation topologique du préfaisceau de K-théorie algébrique connective pour la dg-catégorie unité donne le spectre de K-théorie topologique usuel. Puis que c'est aussi vrai pour la K-théorie algébrique non-connective, en utilisant la propriété de restriction aux lisses de la réalisation topologique. En outre, cette propriété de restriction aux schémas lisses nécessite de montrer une généralisation de la descente propre cohomologique de Deligne, dans le cadre homotopique non-abélien.La K-théorie topologique est alors définie en localisant par rapport à l'élément de Bott. Cette définition repose donc sur des résultats non-triviaux. On montre alors que le caractère de Chern de la K-théorie algébrique vers l'homologie périodique se factorise par la K-théorie topologique, donnant un candidat naturel pour la partie rationnelle d'une structure de Hodge non-commutative sur l'homologie périodique, ceci étant énoncé sous la forme de la conjecture du réseau. Notre premier résultat de comparaison concerne le cas d'un schéma lisse de type fini sur les complexes -- la conjecture du réseau est alors vraie pour de tels schémas. On montre ensuite que cette conjecture est vraie dans le cas des algèbres associatives de dimension finie.
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Kioulos, Charalambos. « From Flag Manifolds to Severi-Brauer Varieties : Intersection Theory, Algebraic Cycles and Motives ». Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40716.
Texte intégralRésumé :
The study of algebraic varieties originates from the study of smooth manifolds. One
of the focal points is the theory of differential forms and de Rham cohomology. It’s
algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing
and taking the pseudo-abelian envelope of the category of smooth projective varieties,
one obtains the category of pure motives.
In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer
varieties. This has been a subject of intensive investigation for the past twenty years,
with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin,
[Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2];
Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen];
Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the
thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in
the paper [Cal] by providing new examples of motivic decompositions of generalized
Severi-Brauer varieties.
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Daqqa, Ibtisam. « Subconstituent Algebras of Latin Squares ». Scholar Commons, 2007. https://scholarcommons.usf.edu/etd/199.
Texte intégralRésumé :
Let n be a positive integer. A Latin square of order n is an n×n array L such that each element of some n-set occurs in each row and in each column of L exactly once. It is well-known that one may construct a 4-class association scheme on the positions of a Latin square, where the relations are the identity, being in the same row, being in the same column, having the same entry, and everything else. We describe the subconstituent (Terwilliger) algebras of such an association scheme. One also may construct several strongly regular graphs on the positions of a Latin square, where adjacency corresponds to any subset of the nonidentity relations described above. We describe the local spectrum and subconstituent algebras of such strongly regular graphs. Finally, we study various notions of isomorphism for subconstituent algebras using Latin squares as examples.
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Lourdeaux, Alexandre. « Sur les invariants cohomologiques des groupes algébriques linéaires ». Thesis, Lyon, 2020. http://www.theses.fr/2020LYSE1044.
Texte intégralRésumé :
Notre thèse s'intéresse aux invariants cohomologiques des groupes algébriques linéaires, lisses et connexes sur un corps quelconque. Plus spécifiquement on étudie les invariants de degré 2 à coefficients dans le complexe de faisceaux galoisiens Q/Z(1), c'est-à-dire des invariants à valeurs dans le groupe de Brauer. Pour se faire on utilise la cohomologie étale des faisceaux sur les schéma simpliciaux. On obtient une description de ces invariants pour tous les groupes linéaires, lisses et connexes, notamment les groupes non réductifs sur un corps imparfait (par exemple les groupes pseudo-réductifs ou unipotents).On se sert de la description établie pour étudier le comportement du groupe des invariants à valeurs dans le groupe de Brauer par des opérations sur les groupes algébriques. On explicite aussi ce groupe d'invariants pour certains groupes algébriques non réductifs sur un corps imparfaitOur thesis deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree 2 invariants with coefficients Q/Z(1), that is invariants taking values in the Brauer group. Our main tool is the étale cohomology of sheaves on simplicial schemes. We get a description of these invariants for every smooth and connected linear groups, in particular for non reductive groups over an imperfect field (as pseudo-reductive or unipotent groups for instance).We use our description to investigate how the groups of invariants with values in the Brauer group behave with respect to operations on algebraic groups. We detail this group of invariants for particular non reductive algebraic groups over an imperfect field
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Xiao, Xinli. « The double of representations of Cohomological Hall algebras ». Diss., Kansas State University, 2016. http://hdl.handle.net/2097/32900.
Texte intégralRésumé :
Doctor of PhilosophyDepartment of Mathematics
Yan Soibelman
Given a quiver Q with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of Q. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver Q, and discuss the moduli space of the stable framed representations of Q. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In this dissertation, we focus on the quiver without potential case. We first define Cohomological Hall algebras, and then the above construction is stated under some assumptions. We computed two examples in detail: A₁-quiver and Jordan quiver. It turns out that A₁-COHA and its double representations are related to the half infinite Clifford algebra, and Jordan-COHA and its double representations are related to the infinite Heisenberg algebra. Then by the fact that the underlying vector spaces of these two COHAs are isomorphic to each other, we get a COHA version of Boson-Fermion correspondence.
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Bastian, Nicholas Lee. « Terwilliger Algebras for Several Finite Groups ». BYU ScholarsArchive, 2021. https://scholarsarchive.byu.edu/etd/8897.
Texte intégralRésumé :
In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.
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Dyer, Ben. « NC-algebroid thickenings of moduli spaces and bimodule extensions of vector bundles over NC-smooth schemes ». Thesis, University of Oregon, 2018. http://hdl.handle.net/1794/23168.
Texte intégralRésumé :
We begin by reviewing the theory of NC-schemes and NC-smoothness, as introduced by Kapranov in \cite{Kapranov} and developed further by Polishchuk and Tu in \cite{PT}.
For a smooth algebraic variety $X$ with a torsion-free connection $\nabla$, we study modules over the NC-smooth thickening $\tw \O_X$ of $X$ constructed in \cite{PT} via NC-connections. In particular we show that the NC-vector bundle $\tw E_{\bar\nabla}$ constructed via mNC-connections in \cite{PT} from a vector bundle $(E,\bar\nabla)$ with connection additionally admits a bimodule extension at least to nilpotency degree 3.
Next, in joint work with A. Polishchuk \cite{DP}, we show that the gap, as first noticed in \cite{PT}, in the proof from \cite{Kapranov} that certain functors are representable by NC-smooth thickenings of moduli spaces of vector bundles is unfixable. Although the functors do not represent NC-smooth thickenings, they lead to a weaker structure of \textit{NC-algebroid thickening}, which we define. We also consider a similar construction for families of quiver representations, in particular upgrading some of the quasi-NC-structures of \cite{Toda1} to NC-smooth algebroid thickenings.
This thesis includes unpublished co-authored material.
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Harris, Twyla. « The Effects of Two Homework Assessment Schemes on At-Risk Student Performance in College Algebra ». TopSCHOLAR®, 2008. http://digitalcommons.wku.edu/theses/400.
Texte intégralRésumé :
This project is the result of a yearlong study documenting the comparative effectiveness of two homework assessment schemes. While both schemes assess completeness and accuracy, one scheme was more traditional and one was more nontraditional in nature. The more traditional method required students to complete homework assignments that were constructed from problems found in the textbook that accompanied the course. These assignments were monitored and were checked weekly for accuracy. The non-traditional method utilized the on-line assessment tool called Math XLR. The effectiveness of these two methods was compared through analysis of differences in student persistence on homework, student performance on tests, and final course grades. After analyzing the material, this study suggested that using MathXLR as a tool versus the traditional book and paper/pencil method does not lead to significant increases in persistence or success. Thus, it seems that using Math XLR should be a personal choice of the instructor for the purpose of convenience. The study concludes with a discussion of findings and study limitations, as well as suggestions for future research.
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Tran, Nguyen Khanh Linh [Verfasser], et Martin [Akademischer Betreuer] Kreuzer. « Kähler Differential Algebras for 0-Dimensional Schemes and Applications / Nguyen Khanh Linh Tran. Betreuer : Martin Kreuzer ». Passau : Universität Passau, 2015. http://d-nb.info/1079066950/34.
Texte intégral
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Rydh, David. « Families of cycles and the Chow scheme ». Doctoral thesis, Stockholm : Matematik, Mathematics, Kungliga Tekniska högskolan, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4813.
Texte intégral
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TOSSICI, DAJANO. « Group schemes of order p^2 and extension of Z/p^2Z-torsors ». Doctoral thesis, Università di Roma Tre, 2008. http://hdl.handle.net/10281/20961.
Texte intégralRésumé :
In this work we study finite group schemes over a discrete valuation ring of unequal characteristic which are isomorphic to the group scheme of p^2-roots of unity, where p is the characteristic of the residue field of R, on the generic fiber. And we apply this to the study of the degeneration, from caracteristic p to caracteristic 0, of torsors under the cyclic group of order p^2.
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Ngo, Long. « Computationally sound automated proofs of cryptographic schemes ». Thesis, Queensland University of Technology, 2012. https://eprints.qut.edu.au/54668/1/Long_Ngo__Thesis.pdf.
Texte intégralRésumé :
Proving security of cryptographic schemes, which normally are short algorithms, has been known to be time-consuming and easy to get wrong. Using computers to analyse their security can help to solve the problem. This thesis focuses on methods of using computers to verify security of such schemes in cryptographic models.
The contributions of this thesis to automated security proofs of cryptographic schemes can be divided into two groups: indirect and direct techniques. Regarding indirect ones, we propose a technique to verify the security of public-key-based key exchange protocols. Security of such protocols has been able to be proved automatically using an existing tool, but in a noncryptographic model. We show that under some conditions, security in that non-cryptographic model implies security in a common cryptographic one, the Bellare-Rogaway model [11]. The implication enables one to use that existing tool, which was designed to work with a different type of model, in order to achieve security proofs of public-key-based key exchange protocols in a cryptographic model.
For direct techniques, we have two contributions. The first is a tool to verify Diffie-Hellmanbased key exchange protocols. In that work, we design a simple programming language for specifying Diffie-Hellman-based key exchange algorithms. The language has a semantics based on a cryptographic model, the Bellare-Rogaway model [11]. From the semantics, we build a Hoare-style logic which allows us to reason about the security of a key exchange algorithm, specified as a pair of initiator and responder programs.
The other contribution to the direct technique line is on automated proofs for computational indistinguishability. Unlike the two other contributions, this one does not treat a fixed class of protocols. We construct a generic formalism which allows one to model the security problem of a variety of classes of cryptographic schemes as the indistinguishability between two pieces of information. We also design and implement an algorithm for solving indistinguishability problems. Compared to the two other works, this one covers significantly more types of schemes, but consequently, it can verify only weaker forms of security.
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Nogueira, Davi Maximo Alexandrino. « Colagem de espaÃos anelados e um esquema sem pontos fechados ». Universidade Federal do CearÃ, 2007. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=2847.
Texte intégralRésumé :
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgicoNeste trabalho mostraremos resultados sobre a colagem de espaÃos anelados e suas aplicacÃes a teoria de Esquemas, seguindo a linha de [5]. O principal resultado sobre espaÃos à o teorema 2.1: Teorema Suponha que W à um espaÃo anelado e que para cada i ∈ I existem mapas. No final, usamos um resultado para construir um esquema sem pontosfechados. Uma outra construÃÃo usando anÃis de valorizaÃÃo tambÃm à apresentada.
In this paper, we present results concerning gluing of ringed spaces and its applications to Schemes, following [5]. Our principal result about ringed spaces is theorem 2.1: Theorem Assume W is a ringed space and also that for each i ∈ I there exists maps. In the end, we use this last result to construct an scheme without closed points. Another construction is given using valuation rings.
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NOVARIO, SIMONE. « LINEAR SYSTEMS ON IRREDUCIBLE HOLOMORPHIC SYMPLECTIC MANIFOLDS ». Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/886303.
Texte intégralRésumé :
In questa tesi studiamo alcuni sistemi lineari completi associati a divisori di schemi di Hilbert di 2 punti su una superficie K3 proiettiva complessa con gruppo di Picard di rango 1, e le mappe razionali indotte. Queste varietà sono chiamate quadrati di Hilbert su superfici K3 generiche, e sono esempi di varietà irriducibili olomorfe simplettiche (varietà IHS).
Nella prima parte della tesi, usando la teoria dei reticoli, gli operatori di Nakajima e il modello di Lehn–Sorger, diamo una base per il sottospazio vettoriale dell’anello di coomologia singolare a coefficienti razionali generato dalle classi di Hodge razionali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. In seguito sfruttiamo un teorema di Qin e Wang insieme a un risultato di Ellingsrud, Göttsche e Lehn per ottenere una base del reticolo delle classi di Hodge integrali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva.
Nella seconda parte della tesi studiamo il problema seguente: se X è il quadrato di Hilbert di una superficie K3 generica che ammette un divisore ampio D con q(D) = 2, dove q è la forma quadratica di Beauville-Bogomolov-Fujiki, descrivere geometricamente la mappa razionale indotta dal sistema lineare completo |D|. Il risultato principale della tesi mostra che tale X, tranne nel caso del quadrato di Hilbert di una superficie quartica generica di P^3, è una doppia EPW sestica, cioè il ricoprimento doppio di una EPW sestica, una ipersuperficie normale di P^5, ramificato nel suo luogo singolare. Inoltre la mappa razionale indotta da |D| coincide proprio con tale ricoprimento doppio. Gli strumenti principali per ottenere questo risultato sono la descrizione del reticolo delle classi integrali di Hodge di tipo (2, 2) della prima parte della tesi e l’esistenza di un’involuzione anti-simplettica su tali varietà per un teorema di Boissière, Cattaneo, Nieper-Wißkirchen e Sarti.In this thesis we study some complete linear systems associated to divisors of Hilbert schemes of 2 points on complex projective K3 surfaces with Picard group of rank 1, together with the rational maps induced. We call these varieties Hilbert squares of generic K3 surfaces, and they are examples of irreducible holomorphic symplectic (IHS) manifold. In the first part of the thesis, using lattice theory, Nakajima operators and the model of Lehn–Sorger, we give a basis for the subvector space of the singular cohomology ring with rational coefficients generated by rational Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. We then exploit a theorem by Qin and Wang together with a result by Ellingsrud, Göttsche and Lehn to obtain a basis of the lattice of integral Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. In the second part of the thesis we study the following problem: if X is the Hilbert square of a generic K3 surface admitting an ample divisor D with q(D)=2, where q is the Beauville–Bogomolov–Fujiki form, describe geometrically the rational map induced by the complete linear system |D|. The main result of the thesis shows that such an X, except on the case of the Hilbert square of a generic quartic surface of P^3, is a double EPW sextic, i.e., the double cover of an EPW sextic, a normal hypersurface of P^5, ramified over its singular locus. Moreover, the rational map induced by |D| is a morphism and coincides exactly with this double covering. The main tools to obtain this result are the description of integral Hodge classes of type (2, 2) of the first part of the thesis and the existence of an anti-symplectic involution on such varieties due to a theorem by Boissière, Cattaneo, Nieper-Wißkirchen and Sarti.
Dans cette thèse, nous étudions certains systèmes linéaires complets associés aux diviseurs des schémas de Hilbert de 2 points sur des surfaces K3 projectives complexes avec groupe de Picard de rang 1, et les fonctions rationnelles induites. Ces variétés sont appelées carrés de Hilbert sur des surfaces K3 génériques, et sont un exemple de variété symplectique holomorphe irréductible (variété IHS). Dans la première partie de la thèse, en utilisant la théorie des réseaux, les opérateurs de Nakajima et le modèle de Lehn–Sorger, nous donnons une base pour le sous-espace vectoriel de l’anneau de cohomologie singulière à coefficients rationnels engendré par les classes de Hodge rationnels de type (2, 2) sur le carré de Hilbert de toute surface K3 projective. Nous exploitons ensuite un théorème de Qin et Wang ainsi qu’un résultat de Ellingsrud, Göttsche et Lehn pour obtenir une base du réseau des classes de Hodge intégraux de type (2, 2) sur le carré de Hilbert d’une surface K3 projective quelconque. Dans la deuxième partie de la thèse, nous étudions le problème suivant : si X est le carré de Hilbert d’une surface K3 générique tel que X admet un diviseur ample D avec q(D) = 2, où q est la forme quadratique de Beauville–Bogomolov–Fujiki, on veut décrire géométriquement la fonction rationnelle induite par le système linéaire complet |D|. Le résultat principal de la thèse montre qu’une telle X, sauf dans le cas du carré de Hilbert d’une surface quartique générique de P^3, est une double sextique EPW, c’est-à-dire le revêtement double d’une sextique EPW, une hypersurface normale de P^5, ramifié sur son lieu singulier. En plus la fonction rationnelle induite par |D| est exactement ce revêtement double. Les outils principaux pour obtenir ce résultat sont la description des classes de Hodge intégraux de type (2, 2) de la première partie de la thèse et l’existence d’une involution anti-symplectique sur de telles variétés par un théorème de Boissière, Cattaneo, Nieper-Wißkirchen et Sarti.
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Cattaneo, Alberto. « Non-symplectic automorphisms of irreducible holomorphic symplectic manifolds ». Thesis, Poitiers, 2018. http://www.theses.fr/2018POIT2322/document.
Texte intégralRésumé :
Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectiqueWe study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution
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Graziani, Giacomo. « Modular sheaves of de Rham classes on Hilbert formal modular schemes for unramified primes ». Doctoral thesis, Università degli studi di Padova, 2020. http://hdl.handle.net/11577/3425908.
Texte intégralRésumé :
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a p-adic interpolation of the usual Hodge filtration. We define an U_p-operator on this sheaf and relate it with the sheaf of overconvergent Hilbert modular forms.
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Tari, Kévin. « Automorphismes des variétés de Kummer généralisées ». Thesis, Poitiers, 2015. http://www.theses.fr/2015POIT2301/document.
Texte intégralRésumé :
Dans ce travail, nous classifions les automorphismes non-symplectiques des variétés équivalentes par déformations à des variétés de Kummer généralisées de dimension 4, ayant une action d'ordre premier sur le réseau de Beauville-Bogomolov. Dans un premier temps, nous donnons les lieux fixes des automorphismes naturels de cette forme. Par la suite, nous développons des outils sur les réseaux en vue de les appliquer à nos variétés. Une étude réticulaire des tores complexes de dimension 2 permet de mieux comprendre les automorphismes naturels sur les variétés de type Kummer. Nous classifions finalement tous les automorphismes décrits précédemment sur ces variétés. En application de nos résultats sur les réseaux, nous complétons également la classification des automorphismes d'ordre premier sur les variétés équivalentes par déformations à des schémas de Hilbert de 2 points sur des surfaces K3, en traitant le cas de l'ordre 5 qui restait ouvertLn this work, we classify non-symplectic automorphisms of varieties deformation equivalent to 4-dimensional generalized Kummer varieties, having a prime order action on the Beauville-Bogomolov lattice. Firstly, we give the fixed loci of natural automorphisms of this kind. Thereafter, we develop tools on lattices, in order to apply them to our varieties. A lattice-theoritic study of 2-dimensional complex tori allows a better understanding of natural automorphisms of Kummer-type varieties. Finaly, we classify all the automorphisms described above on thos varieties. As an application of our results on lattices, we complete also the classification of prime order automorphisms on varieties deformation-equivalent to Hilbert schemes of 2 points on K3 surfaces, solving the case of order 5 which was still open
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Prado, Laerte Gomes. « Teoria dos esquemas e a invariÃncia birracional do gÃnero geomÃtrico ». Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11190.
Texte intégralRésumé :
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgicoO objetivo deste trabalho à desenvolver a teoria bÃsica de esquemas e mostrar que duas variedades projetivas birracionalmente equivalentes e nÃo-singulares sobre um corpo algebricamente fechado possuem um mesmo gÃnero geomÃtrico. Um resultado relacionado permite determinar se uma hipersuperfÃcie nÃo-singular de grau d em um espaÃo projetivo Pn à uma variedade nÃo-racional.
This work aims to develop basic scheme theory and show that two projective, non-singular and birationally equivalent varieties over an algebraically closed field have same geometric genus. A related result allows to check whether a non-singular hipersurface of degree d in a projective space Pn is a non-rational variety.
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N'guessan, Marc-Arthur. « Space adaptive methods with error control based on adaptive multiresolution for the simulation of low-Mach reactive flows ». Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASC017.
Texte intégralRésumé :
Ce travail vise au développement de nouvelles méthodes numériques adaptatives pour la simulation numérique de phénomènes physiques multi-échelles en temps et en espace. Nous nous concentrons sur les écoulements réactifs à faible nombre de Mach, caractéristiques d'un grand nombre de configurations industrielles telles que la convection naturelle, la dynamique de fronts de flamme ou encore les décharges plasmas. La raideur associée à ce type de problèmes, que ce soit via le terme source chimique qui présente un large spectre d'échelles de temps caractéristiques ou encore via la présence de forts gradients très localisés associés aux fronts de réaction, génère des difficultés numériques considérables. Il est donc nécessaire de concevoir des méthodes sur mesure pour traiter la raideur de telles applications, afin d'obtenir des résultats d'une grande précision avec un coût calcul raisonnable. Dans ce cadre général, nous introduisons de nouvelles méthodes numériques pour la résolution des équations de Navier-Stokes incompressibles, une étape importante dans la réalisation d'un solveur hydrodynamique pour les écoulements à faible nombre de Mach. Nous construisons un solveur volumes finis avec adaptation de maillage par l'analyse de multirésolution, qui permet un contrôle a priori des erreurs générées par l'adaptation de maillage. Pour ce faire, nous développons un nouveau schéma de volumes finis collocalisé, avec un traitement original des modes de pression et de vitesse parasites qui n'affecte pas la précision de la discrétisation spatiale. Cette dernière est couplée à un nouveau schéma de Runge-Kutta additif d'ordre 3 pour les écoulements incompressibles, qui présente des propriétés de stabilité adaptées à la raideur des équations différentielles algébriques semi-explicites d'index 2. L'ensemble de cette stratégie est implémentée dans le code de calcul scientifique mrpy. Ce dernier est écrit en Python, et repose sur la librairie PETSc, écrite en C, pour le traitement des opérations d'algèbre linéaire. Nous évaluons l'efficacité algorithmique de cette stratégie par la simulation numérique d'un transport de scalaire passif dans un écoulement incompressible sur maillage adaptatif. Ce travail présente donc un nouveau solveur hydrodynamique d'ordre élevé pour les écoulements incompressibles, avec adaptation de maillage par multirésolution et contrôle d'erreur, qui peut être étendu aux écoulements à faible nombre de MachWe address the development of new numerical methods for the efficient resolution of stiff Partial Differential Equations modelling multi-scale time/space physical phenomena. We are more specifically interested in low Mach reacting flow processes, that cover various real-world applications such as flame dynamics at low gas velocity, buoyant jet flows or plasma/flow interactions. It is well-known that the numerical simulation of these problems is a highly difficult task, due to the large spectrum of spatial and time scales caused by the presence of nonlinear The adaptive spatial discretization is coupled to a new 3rd-order additive Runge-Kutta method for the incompressible Navier-Stokes equations, combining a 3rd-order, A-stable, stiffly accurate, 4-stage ESDIRK method for the algebraic linear part of these equations, and a 4th-order explicit Runge-Kutta scheme for the nonlinear convective part. This numerical strategy is implemented from scratch in the in-house numerical code mrpy. This software is written in Python, and relies on the PETSc library, written in C, for linear algebra operations. We assess the capabilities of this mechanisms taking place into dynamic fronts. In this general context, this work introduces dedicated numerical tools for the resolution of the incompressible Navier-Stokes equations, an important first step when designing an hydrodynamic solver for low Mach flows. We build a space adaptive numerical scheme to solve incompressible flows in a finite-volume context, that relies on multiresolution analysis with error control. To this end, we introduce a new collocated finite-volume method on adaptive rectangular grids, with an original treatment of the spurious pressure and velocity modes that does not alter the precision of the discretization technique. new hydrodynamic solver in terms of speed and efficiency, in the context of scalar transport on adaptive grids. Hence, this study presents a new high-order hydrodynamics solver for incompressible flows, with grid adaptation by multiresolution, that can be extended to the more general low-Mach flow configuration
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Codorniu, Rodrigo. « Schéma en groupes fondamental de quelques variétés connexes par courbes et associées ». Thesis, Université Côte d'Azur, 2021. http://www.theses.fr/2021COAZ4036.
Texte intégralRésumé :
Dans ce travail de thèse on étudie le schéma en groupes fondamental des variétés connexes par courbes ou qui sont associées à ces variétés. Les variétés connexes par courbes sont la généralisation des variétés rationnellement connexes, dont la définition a été conçu par J. Kollár. Ces notions sont les plus proches en géométrie algébrique à la notion de connexité par arcs en topologie, car sur un corps algébriquement clos (non dénombrable), par deux points très généraux d'une variété connexe par courbes (par chaînes resp.) il existe une courbe (chaîne de courbes resp.) avec un morphisme vers la variété dont l'image contient les deux points-ci considérés. En dépendant du type de courbes qu'on considère, on a les notions de g-connexité (par chaînes resp.) où on considère exclusivement des courbes (chaînes de courbes resp.) où chaque composante irréductible est une courbe lisse et projective de genre g, et la notion de la C-connexité pour une courbe fixe C où par deux points très généraux on peut faire passer l'image d'un morphisme depuis la courbe C.En utilisant des résultats classiques et récents de la théorie des schémas en groupes fondamentaux, qui classifient des torseurs sous l'action d'un schéma en groupes affine, notamment le schéma en groupes fondamental de Nori et le S-schéma en groupes fondamental, on essaie à décrire le schéma en groupes fondamental de Nori des certains types des variétés connexes par courbes, dont le cas rationnellement connexe est déjà connu, et ceux des certaines variétés associées.Pour obtenir ces résultats, on utilise tous les aspects qui interviennent dans la théorie du schéma en groupes fondamental : les schémas en groupes affines, les catégories tannakiennes des fibrés vectoriels sur des variétés propres et la théorie des torseurs affines. En plus, on construit des nouveaux schémas en groupes fondamentaux associés aux catégories tannakiennes des fibrés pour des variétés où tout pair de points peut être connecté par des chaînes de courbes appartenant à des familles arbitraires de courbes, ce qui généralise une construction récente de I. Biswas, P.H. Hai et J.P. Dos Santos et qui pourrait fournir un nouveau cadre pour l'étude des schémas en groupes fondamentaux des variétés connexes par courbes. Plus spécifiquement, on propose deux approches différentes pour décrire ces schémas en groupes fondamentaux, appliquer le nouveau cadre des schémas en groupes fondamentaux décrit dans le paragraphe précédent aux variétés g-connexes, et utiliser la fibration rationnellement connexe maximale et décrire le comportement du schéma en groupes fondamental sur cette fibration. Inspiré par la deuxième approche, on décrit le schéma en groupes fondamental des fibrations sur des variétés abéliennes avec fibres rationnellement connexes, inspiré par la description des variétés elliptiquement connexes en caractéristique zéro par F. Gounelas. Ces variétés ne sont pas nécessairement elliptiquement connexes en caractéristique positive, mais la description de ses schémas en groupes fondamentaux est possible avec la suite exacte d’homotopieIn this thesis work we study the fundamental group-scheme of curve-connected varieties or associated to them. Curve-connected varieties are the generalization of rationally connected varieties, whose definition was conceived by J. Kollár. These notions are the closest ones in algebraic geometry, to the notion of arc connectedness in topology, because over an algebraically closed field (uncountable), over any pair of two very general points in a curve-connected variety (resp. chain-connected), there exists a curve (resp. chain of curves) with a morphism to the variety whose image contains the two points mentioned before. Depending on the type of curves we consider, we have the notions of g-connectedness (resp. chain g-connectedness) where we consider exclusively curves (resp. chains of curves) with irreducible components are smooth and projective curves of genus g, and the notion of C-connectedness for a fixed curve C where over any two very general points, we can contain them in the image of a morphism from C to the variety.Using classical and recent results from the theory of fundamental group-schemes, which classifies torsors under the action of an affine group-scheme, notably Nori fundamental group-scheme and the S-fundamental group-scheme, we try to describe the Nori fundamental group-scheme of certain types of curve-connected varieties, for which the rationally connected case is known, and some associated varieties.To obtain these results, we use all the aspects that play a role in the theory of the fundamental group-scheme: affine group-schemes, tannakian categories of vector bundles over proper varieties, and the theory of affine torsors. Moreover, we build new fundamental group-schemes associated to tannakian categories of vector bundles over varieties where we can join any pair of points by a chain of curves belonging to arbitrary families of curves, generalizing a recent construction of I.Biswas, P.H. Hai and J.P. Dos Santos which could provide a new framework for the study of fundamental group-schemes of curve-connected varieties.More specifically, we propose two different approaches to understand these fundamental group-schemes, apply the new framework for fundamental group-schemes described in the paragraph above for g-connected varieties and to utilize the maximal rationally connected fibration and describe the behaviour of the fundamental group over it. Inspired by the second approach, we describe the fundamental group-scheme of fibrations over elliptic curves with rationally connected fibers, inspired by the description of elliptically connected varieties in characteristic zero made by F. Gounelas. These varieties are not necessarily elliptically connected in positive characteristic, but the description of their fundamental group-schemes is possible with the homotopy exact sequence