3,610 research outputs found
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
The derived category Db(X) of a variety contains a lot of information about X. If X and X′ are Fano, then an equivalence Db(X) ≃ Db(X′) implies that X and X′ are isomorphic. For prime Fano threefolds X (of Picard rank 1, index 1, and genus g ≥ 6) the derived category decomposes semiorthogonally as ⟨Ku(X), E, OX ⟩, where E is a certain vector bundle on X. Therefore one can ask whether less data (in particular the Kuznetsov component Ku(X)) than the whole of Db(X) determines X isomorphically (or at least birationally). In this thesis, we focus on this question in the case of ordinary Gushel–Mukai threefolds (genus 6 prime Fano threefolds). We show that Ku(X) determines the birational class of X which proves a conjecture of Kuznetsov–Perry in dimension 3. We also prove a refined categorical Torelli theorem for oridnary Gushel–Mukai threefolds. In other words, we show that Ku(X) along with the data of the vector bundle E is enough to determine X up to isomorphism
Bridgeland stability conditions are powerful tools for studying derived categories, with several applications to algebraic geometry. They were introduced by Bridgeland in 2002 [Bri07], who was motivated by Douglas’ work on Π-stability of D-branes [Dou02] in the context of string theory. Bridgeland showed that the set Stab(D) of stability conditions on a triangulated category D is a complex manifold, a result of extreme importance and central to all mathematical applications of this field of study. But in order to use this concept of stability conditions in string theory (as intended by Bridgeland), one needs to prove the existence of stability conditions on the bounded derived category Db(X) of a compact Calabi-Yau threefold X. This task is far from easy, as it took more than a decade before the first example was produced for the smooth quintic threefold by Li in [Li18]. This achievement came into fruition thanks to the extensive amount of work in the domain over this period of time, where the existence of stability conditions was progressively established for arbitrary smooth projective varieties of dimension one [Bri07, Oka06, Mac07], dimension two [Bri08, AB13], and then some dimension three cases (see Section 1.3). One of the main applications of stability conditions on Db(X) (for an arbitrary variety X) is to study the geometry of moduli spaces of coherent sheaves over X with some Chern character v via the strategy known as “wall crossing”. In loose terms, a “wall” is a codimension one submanifold of Stab(Db(X)) such that by changing stability conditions along a continuous path in Stab(Db(X)) that goes through the wall causes the moduli space of sheaves over X with Chern character v to transform. When X is of dimension two, we have a solid control over wall crossing thanks to Bayer–Macr`ı [BM11], who provided a full understanding of how moduli spaces of sheaves change as we cross walls, as well as knowing the exact geometrical relationship these walls have with the underlying surface. In addition the precise structure of the walls is known and there are effective techniques to detect them. This thorough picture of wall crossing in dimension two is demonstrated through various complete studies of moduli spaces of sheaves over surfaces [AB13, ABCH13, Mea12]. [...
We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and Lafforgue. We refer to Taylor’s and Rouquier’s pseudorepresentations as pseudocharacters. They are very closely related, the main difference being that Taylor’s pseudocharacters are defined for a group, where as Rouquier’s pseudocharacters are defined for algebras. Chenevier’s pseudorepresentations are so-called polynomial laws and will be called determinant laws. Lafforgue’s pseudorepresentations are a generalization of Taylor’s pseudocharacters to other reductive groups G, in that the corresponding notion of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters. We survey the known comparison theorems, notably Emerson’s bijection between Chenevier’s determinant laws and Lafforgue’s GL(n)-pseudocharacters and the bijection with Taylor’s pseudocharacters away from small characteristics. We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously, we obtain that tensor products of determinant laws exist and are compatible with tensor products of representations. Further the tensor product of Lafforgue’s pseudocharacters agrees with the tensor product of Taylor’s pseudocharacters. We generalize some of the results of [Che14] to general reductive groups, in particular we show that the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely generated profinite group Γ with values in a finite field (of characteristic p) is noetherian. We also show, that for specific groups G it is sufficient, that Γ satisfies Mazur’s condition Φ_p. One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater of Gal(F) into obstructed subloci Xdec(Θ), Xpair(Θ), Xspcl(Θ) of dimension smaller than the expected dimension n(2n + 1)[F : Qp]. We also prove that Lafforgue’s G-pseudocharacters over algebraically closed fields for possibly nonconnected reductive groups G come from a semisimple representation. We introduce a formal scheme and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to conjugacy and Frobenius automorphisms. At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic groups, which is based on adding a ’Pfaffian polynomial law’ to a determinant law which is invariant under an involution. We prove the expected basic properties in that we show that symplectic determinant laws over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgue’s pseudocharacters and show that it is an isomorphism over reduced rings
openThe body of this thesis unfolds in detail the theory of derived categories in order to exploit and explore the geometry of the derived category of coherent sheaves on a projective variety, and inherently of the variety itself. By tracing the development of Fourier-Mukai transforms, we elucidate some of the different modalities and criteria for establishing derived equivalences between two varieties. In particular we delve into some of the results that bridge the geometry of a variety with that of its derived category---among them---we discuss the Bondal-Orlov reconstruction theorem and the derived equivalence between an abelian variety and its dual