# Browsing by Subject "Algebraic geometry"

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Item Autoequivalences, stability conditions, and n-gons : an example of how stability conditions illuminate the action of autoequivalences associated to derived categories(2010-05) Lowrey, Parker Eastin; Ben-Zvi, David, 1974-; Freed, Daniel; Uhlenbeck, Karen; Allcock, Daniel; Distler, JacquesShow more 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.Show more Item Coordinate systems and associative algebras(2015-05) Orem, Hendrik Nikolas; Ben-Zvi, David, 1974-; Freed, Daniel; Hadani, Ronny; Nadler, David; Nevins, Thomas; Schedler, TravisShow more This dissertation applies and extends the techniques of formal algebraic geometry in the setting of certain "smooth" associative algebras and their globalizations, noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a principal bundle for an appropriate group of local coordinate changes. These bundles are shown to carry a natural flat connection with properties analogous to the classical Gelfand-Kazhdan structure. Every noncommutative manifold has an underlying smooth variety given by abelianization. A basic question is existence and uniqueness of noncommu- tative thickenings of a smooth variety, i.e., finding noncommutative manifolds abelianizing to a given smooth variety. We obtain new results in this direction by showing that noncommutative coordinate systems always arise as reductions of structure group of the commutative bundle of coordinate systems on the underlying smooth variety; this also explains a relationship between D-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf. The lower central series invariants M[subscript k] of an associative algebra A are the two-sided ideals generated by k-fold nested commutators; the M[subscript k] give a decreasing filtration of A. We study the relationship between the geometry of X = Spec A[subscript ab] and the associated graded components N[subscript k] of this filtration. We show that the N[subscript k] form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. We then construct the N[subscript k] in terms of the bundle of coordinate systems on X and the N[subscript k] invariants for the free associative algebra; in particular, since this is independent of A, we exhibit the N[subscript k] as natural vector bundles on the category of smooth schemes.Show more Item Geometry of integrable hierarchies and their dispersionless limits(2014-05) Safronov, Pavel; Ben-Zvi, David, 1974-Show more This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally.Show more Item Realizability of tropical lines in the fan tropical plane(2013-08) Haque, Mohammad Moinul; Helm, David, doctor of mathematicsShow more In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable.Show more Item Tropical Hurwitz spaces(2011-12) Katz, Brian Paul; Helm, David, doctor of mathematics; Keel, Sean; Katz, Eric; Allcock, Daniel; Ben-Zvi, David; Hassett, BrendanShow more Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli space, called a Hurwitz space. The Hurwitz space has a forgetful morphism to the moduli space of marked, stable curves, and this morphism encodes the Hurwitz numbers. Mikhalkin has constructed a moduli space of tropical marked, stable curves, and this space is a tropical variety. In this paper, I construct a tropical analogue of the Hurwitz space in the sense that it is a connected, polyhedral complex with a morphism to the tropical moduli space of curves such that the degree of the morphism encodes the Hurwitz numbers.Show more Item Wachspress Varieties(2012-11-28) Irving, Corey 1977-Show more Barycentric coordinates are functions on a polygon, one for each vertex, whose values are coefficients that provide an expression of a point of the polygon as a convex combination of the vertices. Wachspress barycentric coordinates are barycentric coordinates that are defined by rational functions of minimal degree. We study the rational map on P2 defined by Wachspress barycentric coordinates, the Wachspress map, and we describe polynomials that set-theoretically cut out the closure of the image, the Wachspress variety. The map has base points at the intersection points of non-adjacent edges. The Wachspress map embeds the polygon into projective space of dimension one less than the number of vertices. Adjacent edges are mapped to lines meeting at the image of the vertex common to both edges, and base points are blown-up into lines. The deformed image of the polygon is such that its non-adjacent edges no longer intersect but both meet the exceptional line over the blown-up corresponding base point. We find an ideal that cuts out the Wachspress variety set-theoretically. The ideal is generated by quadratics and cubics with simple expressions along with other polynomials of higher degree. The quadratic generators are scalar products of vectors of linear forms and the cubics are determinants of 3 x 3 matrices of linear forms. Finally, we conjecture that the higher degree generators are not needed, thus the ideal is generated in degrees two and three.Show more