This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

In [Bri07], Bridgeland introduced the notion of stability conditions on the bounded derived category D(X) of coherent sheaves on an algebraic variety X. This topic is originally inspired by concepts in string theory and mathematical physics and has many interesting applications in algebraic geometry. In the first part of the thesis, we provide a direct proof of an important result in [Bri08, BMS16] which states there is a two dimensional family of weak Bridgeland stability conditions on the bounded derived category D(X) of coherent sheaves on a variety X. As a first application of this result, we prove an effective restriction theorem which provides sufficient conditions on a stable locally free sheaf on a projective variety such that its restriction to a hypersurface remains stable. Secondly, we extend and complete Mukai's program to reconstruct a K3 surface from a curve on that surface. We show that the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of a suitable Brill-Noether locus of vector bundles on the curve.

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.

Soit $X$ une variété algébrique de Gorenstein à singularités rationnelles. Une résolution des singularités crépante de $X$ est souvent considérée comme une résolution des singularités minimales de $X$. Malheureusement, les résolutions crépantes sont très rares. Ainsi, les variétés déterminantielles de matrices anti-symétriques n'admettent jamais de résolution crépante des singularités. Dans cette thèse, on discutera de diverses notions de résolutions catégoriques crépantes développées par Alexander Kuznetsov. Conjecturalement, ces résolutions doivent être minimale du point de vue catégorique. On introduit dans ce manuscrit la notion de résolution magnifiques des singularités et on montre que tout variété munie d'une telle résolution admet une résolution catégorique faiblement crépante. On en déduit que toutes les variétés déterminantielles (carrées, symétriques et anti-symétriques) admettent des résolutions catégoriques faiblement crépantes. Finalement, on s'intéressera à des hypersurfaces quartiques issues du carré magique de Tits-Freudenthal. On ne peut pas construire de résolution magnifique des singularités pour de telles hypersurfaces, mais on montrera qu'elles admettent tout de même des résolutions catégorique faiblement crépantes des singularités. Ce résultat devrait s'avérer intéressant pour la construction de duales projectives homologiques de certaines Grassmaniennes symplectiques sur les algèbres de composition. / Let $X$ be an algebraic variety with Gorenstein rational singularities. A crepant resolution of $X$ is often considered to be a minimal resolution of singularities for $X$. Unfortunately, crepant resolution of singularities are very rare. For instance, determinantal varieties of skew-symmetric matrices never admit crepant resolution of singularities. In this thesis, we discuss various notions of categorical crepant resolution of singularities as defined by Alexander Kuznetsov. Conjecturally, these resolutions are minimal from the categorical point of view. We introduce the notion of wonderful resolution of singularities and we prove that a variety endowed with such a resolution admits a weakly crepant resolution of singularities. As a corollary, we prove that all determinantal varieties (square, as well as symmetric and skew-symmetric) admit weakly crepant resolution of singularities. Finally, we study some quartics hypersurfaces which come from the Tits-Freudenthal magic square. Though they do no admit any wonderful resolution of singularities, we are still able to prove that they have a weakly crepant resolution of singularities. This last result should be of interest in order to construct homological projective duals for some symplectic Grassmannians over the composition algebras.

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Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T16:48:31Z (GMT). No. of bitstreams: 1 Cardoso_NunoFilipedeAndrade_M.pdf: 1858794 bytes, checksum: bbe47182338feb3de60b480df87b52a7 (MD5) Previous issue date: 2014 / Resumo: Os teoremas de degenerescência de Hodge e de anulamento de Kodaira, Akizuki e Nakano são de suma importância na teoria de variedades complexas. Usando o teorema de comparação de Serre, ambos podem ser traduzidos para o contexto de esquemas projetivos e suaves sobre um corpo de característica zero. Para corpos de característica positiva, no entanto, os dois deixam de valer sem hipóteses adicionais, sendo que os primeiros contra-exemplos foram encontrados por Mumford e Raynaud. O objetivo desta dissertação é apresentar um teorema devido a Deligne e Illusie que assegura a degenerescência da seqüência espectral de Hodge-de Rham e uma versão do teorema de Kodaira, Akizuki e Nakano para certos esquemas projetivos e suaves sobre um corpo perfeito de característica positiva. Nos propusemos a dar um tratamento, na medida do possível, auto-suficiente / Abstract: The Hodge degeneration theorem and the Kodaira, Akizuki and Nakano's vanishing theorem are of paramount importance in the theory of complex manifolds. Using Serre's comparison theorem, both can be translated to the context of smooth projective schemes over a field of characteristic zero. For fields of positive characteristic, however, both fail to hold without additional hypothesis, and the first counterexamples were found by Mumford and Raynaud. Our goal in this dissertation is to present a theorem due to Deligne and Illusie that ensures the degeneration of the Hodge-de Rham spectral sequence and a version of the theorem of Kodaira, Akizuki and Nakano for certain smooth projective schemes over a perfect field of positive characteristic. We tried to keep the treatment as self-contained as possible / Mestrado / Matematica / Mestre em Matemática

Spectral sequences were developed during the mid-twentieth century as a way of computing (co)homology, and have wide uses in both algebraic topology and algebraic geometry.  Grothendieck introduced in his Tôhoku paper the Grothendieck spectral sequence, which given left exact functors $F$ and $G$ between abelian categories, uses the right-derived functors of $F$ and $G$ as initial data and converges to the right-derived functors of the composition $G\circ F.$  This thesis focuses on instead constructing a spectral sequence that uses the derived functors of $G$ and $G\circ F$ as initial data and converges to the derived functors of $F.$ Our approach takes inspiration from the construction of the Eilenberg-Moore spectral sequence, which given a fibration of topological spaces can calculate the singular cohomology of the fiber from the singular cohomology of the base space and total space. The Eilenberg-Moore spectral sequence can be constructed through the use of differential graded algebras and their bar construction, since this defines a double complex for which the column-wise filtration of the corresponding total complex induces the spectral sequence. The correct analogue of this with respect to composite functors is the bar construction for monads. Specifically, we let $G$ have an exact left adjoint $H$, which makes $G\circ H$ into a monad. Then, we extend our adjunction so that the derived functor $RG$ has left adjoint $RH$ in the corresponding derived categories, making $RG\circ RH$ into a monad. This allows us to apply the bar construction in the derived category, but we show that there emerge issues in obtaining a double complex and subsequent total complex from this construction.  Additionally, we present the essential theory of spectral sequences in general, and of the Serre, Eilenberg-Moore and Grothendieck spectral sequences in particular. / Spektralsekvenser utvecklades under mitten av 1900-talet som ett verktyg för att beräkna (ko)homologi, och har många användningsområden inom både algebraisk topologi och algebraisk geometri. Grothendieck introducerade i sin Tôhoku-artikel Grothendieck-spektralsekvensen, som givet vänsterexakta funktorer $F$ och $G$ mellan abelska kategorier använder de högerderiverade funktorerna av $F$ och $G$ som initialdata och som konvergerar till de högerderiverade funktorerna av kompositionen $G\circ F$. Denna masteruppsats fokuserar på att istället konstruera en spektralsekvens som använder de deriverade funktorerna av $G$ och $G\circ F$ som initialdata och konvergerar till de deriverade funktorerna av $F$. Vår metod tar inspiration från konstruktionen av Eilenberg-Moore-spektralsekvensen, som givet en fibrering av topologiska rum kan beräkna den singulära kohomologin av fibern från den singulära kohomologin av basrummet och totalrummet. Eilenberg-Moore spektralsekvensen kan konstrueras genom användningen av graderade differentialalgebror och deras bar-konstruktion, eftersom detta definierar ett dubbelkomplex vars kolumnvisa filtrering av det resulterande totalkomplexet inducerar spektralsekvensen. Vad gäller kompositioner av funktorer så är den korrekta analogin till detta bar-konstruktionen för monader. Specifikt så låter vi $G$ ha en exakt vänsteradjungerad funktor $H$, vilket gör $G\circ H$ till en monad. Sedan utvidgar vi denna adjunktion sådant att den deriverade funktorn $RG$ har vänsteradjunkt $RH$ i den deriverade kategorin, vilket gör $RG\circ RH$ till en monad. Detta ger oss möjligheten att använda bar-konstruktionen i den deriverade kategorin, men vi visar att det uppstår problem när vi ska definiera ett dubbelkomplex och resulterande totalkomplex från denna konstruktion. Utöver detta så innehåller denna uppsats en genomgång av den viktigaste teorin om spektralsekvenser i allmänhet, och om Serre-, Eilenberg-Moore- och Grothendieck-spektralsekvensen i synnerhet.