1,720,983 research outputs found
Finiteness of K3 surfaces and the Tate conjecture
Get PDFGiven a finite field k of characteristic p ≥ 5, we show that the Tate conjecture holds for K3 surfaces defined over each finite extension of k
Birational Functors in the Derived Category
No full textThesis (Ph.D.)--University of Washington, 2020In this thesis, we study a class of derived equivalences that naturally induce birational maps. We give several equivalent criteria for a birational correspondence to exist, and prove the correspondence induces a -equivalece, extending a result of Kawamata. With the introduction of a canonical natural transformation from the right to left adjoint, we are also able to extend this study to the fully faithful situation. This natural transformation gives us context to provide a new proof of Bridgland's equivalence criterion and of the indecomposability of the derived category in the trivial canonical bundle. Additionally, we construct an algebraic moduli space of birational integral transforms inside of Lieblich's moduli of complexes
Projective Geometry for Perfectoid Spaces
No full textThesis (Ph.D.)--University of Washington, 2019To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of projective geometry, and explore how equipping a perfectoid space with a map to a certain analog of projective space can be a powerful tool to understand its geometric and arithmetic structure. In particular, we show that maps from a perfectoid space X to the perfectoid analog of projective space correspond to line bundles on X together with some extra data, reflecting the classical theory. Along the way we give a complete classification of vector bundles on the perfectoid unit disk, and compute the Picard group of the perfectoid analog of projective space
Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms
No full textThesis (Ph.D.)--University of Washington, 2013In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent sheaves can be described as twisted sheaves on a noncommutative deformation of the variety. This thesis generalizes Toda's work by creating a chain of inclusions from deformations of schemes to commutative deformations to deformations of the category of coherent sheaves. We define projections from coherent deformations to commutative deformations to scheme deformations and show that the fiber of the projection from commutative deformations to schemes is a gerbe. We also prove that for two derived equivalent K3 surfaces in characteristic p and any scheme deformation of one of these, there is a scheme deformation of the other so that the two deformations are also derived equivalent
Arithmetic Properties of the Derived Category for Calabi-Yau Varieties
No full textThesis (Ph.D.)--University of Washington, 2014This thesis develops a theory of arithmetic Fourier-Mukai transforms in order to obtain results about equivalences between the derived category of Calabi-Yau varieties over non-algebraically closed fields. We obtain answers to classical questions from number theory and arithmetic geometry using these results. The main results of this thesis come in three types. The first concerns classifying moduli of vector bundles on genus one curves. Fourier-Mukai equivalences of genus one curves allow us to produce examples of non-isomorphic moduli spaces when a genus one curve has large period. The next result extends the result of Lieblich-Olsson which says that derived equivalent K3 surfaces are moduli spaces of sheaves over algebraically closed fields. We get the same result over arbitrary fields of characteristic different from 2. The last class of results are about finding properties that are preserved under derived equivalence for Calabi-Yau threefolds. Examples of such arithmetic invariants include local Zeta functions, modularity, L-series, and the a-number. We then prove a Serre-Tate theory for liftable, ordinary Calabi-Yau threefolds in positive characteristic in order to show that the derived equivalence induces an isomorphism of their deformation functors that sends the canonical lift of one to the canonical lift of the other
Some Theorems on the Resolution Property and the Brauer map
Get PDFThesis (Ph.D.)--University of Washington, 2018Using formal-local methods, we prove that a separated and normal Deligne-Mumford surface must satisfy the resolution property, this includes the first class of separated algebraic spaces which are not schemes. Our analysis passes through the case of gerbes and an arbitrarily singular Deligne-Mumford curve, each of which we establish independently. Our methods can be extended to give new results on the surjectivity of the Brauer map. For example, we show that on a generically reduced variety, any cohomological Brauer class is represented by an Azumaya algebra away from a closed subset of codimension . We also investigate generically trivial Brauer classes in high codimension which arise from singularities
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
A Functorial Approach to Algebraic Vision
No full textThesis (Ph.D.)--University of Washington, 2019We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli spaces always embed in suitable Hilbert schemes, and that these embeddings are open immersions for more than four views, extending and refining work of Aholt--Sturmfels--Thomas. We also give a new construction of the space of essential matrices from first principles. This construction enables us to re-prove the fundamental results of Demazure and to re-prove the recent description of the essential variety due to Kileel--Fløystad--Ottaviani as well as extend the classical twisted pair covering of the essential variety
- …
