Pour toute courbe projective lisse C et théorie homologique orientée de Borel-Moore libre A, on construit un produit associatif de type Hall sur les A-groupes du champ de modules des faisceaux de Higgs de torsion sur C.On montre que l'algèbre AHa0C qu'on obtient admet une présentation de battage naturelle, qui est fidèle dans le cas où A est l'homologie de Borel-Moore usuelle.On introduit de plus les espaces de modules des triplets stables M(d,n), fortement inspirés par les variétés de carquois de Nakajima.Ces espaces de modules sont des variétés lisses symplectiques, et admettent une autre caractérisation comme les espaces de modules de faisceaux sans torsion stables encadrés sur P(T*C)$.De plus, on munit leurs A-groupes avec une action de AHa0C, qui généralise les opérateurs de modification ponctuelle de Nakajima sur l'homologie des schémas de Hilbert de T*C. / For a smooth projective curve C and a free oriented Borel-Moore homology theory A, we construct a Hall-like associative product on the A-theory of the moduli stack of Higgs torsion sheaves on C.We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel-Moore homology groups.We also introduce moduli spaces of stable triples M(d,n), heavily inspired by Nakajima quiver varieties.These moduli spaces are shown to be smooth symplectic varieties, which admit another characterization as moduli of framed stable torsion-free sheaves on P(T*C).Moreover, we equip their A-theory with an AHa0C-action, which generalizes Nakajima's raising operators on the homology of Hilbert schemes of points on T*C.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer. Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van Hecke eievorme en hul eiewaardes.

Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion. / text

This thesis is on moduli spaces of complexes of sheaves and diagrams of such moduli spaces. The objects in these diagrams are constructed as geometric invariant theory quotients and the points in these quotients correspond to certain equivalence classes of complexes. The morphisms in these diagrams are constructed by taking direct sums with acyclic complexes. We then study the colimit of such a diagram and in particular are interested in studying the images of quasi-isomorphic complexes in the colimit. As part of this thesis we construct categorical quotients of a group action on unstable strata appearing in a stratification associated to a complex projective scheme with a reductive group action linearised by an ample line bundle. We study this stratification for a closed subscheme of a quot scheme parametrising quotient sheaves over a complex projective scheme and relate the Harder-Narasimhan types of unstable sheaves with the unstable strata in the associated stratification. We also study the stratification of a parameter space for complexes with respect to a linearisation determined by certain stability parameters and show that a similar result holds in this case. The objects in these diagrams are indexed by different Harder-Narasimhan types for complexes and are quotients of parameter schemes for complexes of this fixed Harder-Narasimhan type. This quotient is given by a choice of linearisation of the action and so the diagrams depend on these choices. We conjecture that these choices can be made so that for any quasi-isomorphism between complexes representing points in this diagram both complexes are identified in the colimit of this diagram.

The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations). As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.

This thesis mainly concerns the cohomology of the moduli spaces ℳ3[2] and ℳ3,1[2] of genus 3 curves with level 2 structure without respectively with a marked point and some of their natural subspaces. A genus 3 curve which is not hyperelliptic can be realized as a plane quartic and the moduli spaces 𝒬[2] and 𝒬1[2] of plane quartics without respectively with a marked point are given special attention. The spaces considered come with a natural action of the symplectic group Sp(6,𝔽2) and their cohomology groups thus become Sp(6,𝔽2)-representations. All computations are therefore Sp(6,𝔽2)-equivariant. We also study the mixed Hodge structures of these cohomology groups. The computations for ℳ3[2] are mainly via point counts over finite fields while the computations for ℳ3,1[2] primarily uses a description due to Looijenga in terms of arrangements associated to root systems. This leads us to the computation of the cohomology of complements of toric arrangements associated to root systems. These varieties come with an action of the corresponding Weyl group and the computations are equivariant with respect to this action.

Wir studieren Moduln über Maximalordnungen auf glatten projektiven Flächen und ihre Modulräume. Wir untersuchen null- und zweidimensionale Modulräume auf K3 und abelschen Flächen für unverzweigte Ordnungen, den sogenannten Azumaya Algebren. Weiterhin untersuchen wir Modulräume für spezielle verzweigte Ordnungen auf der projektiven Ebene. Wir beweisen das diese Räume immer glatt sind. Mit Hilfe dieses Ergebnisses studieren wir die Deformationstheorie der betrachteten Moduln. Im letzten Kapitel konstruieren wir explizite Ordnungen und berechnen einige Modulräume.

No description available.

L'espace de modules Mgbar des courbes stables de genre g est un object central en géométrie algébrique. Du point de vue de la géométrie birationelle, il apparaît naturel se demander si Mgbar est de type générale. Harris-Mumford et Eisenbud-Harris ont montré que Mgbar est de type générale pour un genre g>=24 et g=22. Le cas g=23 est encore misterieux. Dans les dix dernières années une nouvelle approche a émergé, dans l'essai de clarifier ça : l'idée est celle de considérer de recouvrement fini de Mgbar qui sont des espaces de modules de courbes stables munies d'une structure additionnelle comme un l-recouvrement (racine l-ième du fibré trivial) ou un fibré l-spin (racine l-ième du fibré canonique). Ces espaces ont la propriété que la transition au type générale se produit à un genre inférieur. Dans ce travail nous voulons généraliser cette approche de deux façons : - un étude de l'espace de modules des courbes avec une racine d'une puissance quelconque du fibré canonique ; - un étude de l'espace de modules des courbes avec un G-recouvrement pour un quelconque G groupe fini. Pour définir ces espaces de modules nous utilisons la notion de courbe twisted (voir Abramovich-Corti-Vistoli). Le résultat fondamental obtenu est qu'il est possible de décrire le lieu singulier de ces espaces de modules par la notion de graphe dual d'une courbe. Grace à cette analyse, nous pouvons developper des calculs dans l'anneau tautologique des espaces, et en particulier nous conjecturons que l'espace de modules des courbes avec un S3-recouvrement est de type générale pour genre impaire g>=13. / The moduli space Mgbar of genus g stable curves is a central object in algebraic geometry. From the point of view of birational geometry, it is natural to ask if Mgbar is of general type. Harris-Mumford and Eisenbud-Harris found that Mgbar is of general type for genus g>=24 and g=22. The case g=23 keep being mysterious. In the last decade, in an attempt to clarify this, a new approach emerged: the idea is to consider finite covers of Mgbar that are moduli spaces of stable curves equipped with additional structure as l-covers (l-th roots of the trivial bundle) or l-spin bundles (l-th roots of the canonical bundle). These spaces have the property that the transition to general type happens to a lower genus. In this work we intend to generalize this approach in two ways: - a study of moduli space of curves with any root of any power of the canonical bundle; - a study of the moduli space of curves with G-covers for any finite group G. In order to define these moduli spaces we use the notion of twisted curve (see Abramovich-Corti-Vistoli). The fundamental result obtained is that it is possible to describe the singular locus of these moduli spaces via the notion of dual graph of a curve. Thanks to this analysis, we are able to develop calculations on the tautological rings of the spaces, and in particular we conjecture that the moduli space of curves with S3-covers is of general type for odd genus g>=13.

Thesis advisor: Maksym Fedorchuk / A general smooth curve of genus six lies on a quintic del Pezzo surface. In [AK11], Artebani and Kondō construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this dissertation, we construct a smooth Deligne-Mumford stack P₀ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in P₀ containing special curves. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.