Browsing by Subject "Geometry, Algebraic."

Now showing 1 - 4 of 4

Open Access
Curves in projective space
(2003) Yıldız, Ali
This thesis is mainly concerned with classification of nonsingular projective space curves with an emphasis on the degree-genus pairs. In the first chapter, we present basic notions together with a very general notion of an abstract nonsingular curve associated with a function field, which is necessary to understand the problem clearly. Based on Nagata’s work [25], [26], [27], we show that every nonsingular abstract curve can be embedded in some P N and projected to P 3 so that the resulting image is birational to the curve in P N and still nonsingular. As genus is a birational invariant, despite the fact that degree depends on the projective embedding of a curve, curves in P 3 give the most general setting for classification of possible degree-genus pairs. The first notable attempt to classify nonsingular space curves is given in the works of Halphen [11], and Noether [28]. Trying to find valid bounds for the genus of such a curve depending upon its degree, Halphen stated a correct result for these bounds with a wrong claim of construction of such curves with prescribed degree-genus pairs on a cubic surface. The fault in the existence statement of Halphen’s work was corrected later by the works of Gruson, Peskine [9], [10], and Mori [21], which proved the existence of such curves on quartic surfaces. In Chapter 2, we present how the fault appearing in Halphen’s work has been corrected along the lines of Gruson, Peskine, and Mori’s work in addition to some trivial cases such as genus 0, 1, and 2 together with hyperelliptic, and canonical curves.
Open Access
KO-rings and J-groups of lens spaces
(1998) Kırdar, Mehmet
In this thesis, we make the explicit computation of the real A'-theory of lens spaces and making use of these results and Adams conjecture, we describe their .7-groups in terms of generators and relations. These computations give nice by-products on some geometrical problems related to lens spaces. We show that J-groups of lens spaces approximate localized J-groups of complex projective spaces. We also make connections of the J-cornputations with the classical cross-section problem and the .James numbers conjecture. Many difficult geometric problems remain open. The results are related to some arithmetic on representations of cyclic groups o\er fields and the Atiyah-Segal isomormhisrn. Eventually, we are interested in representations over rings, in connection with Algebraic K-theory. This turns out to lie a very non-trivial arithmetic problem related to number theory.
Open Access
Symplectic geometry and topology of spatial polygons in Euclidean and Minkowski spaces
(2000) Paksoy, Emrah
In this work, we studied the relations between spatial polygons in Euclidean spaces and point configurations in projective line P'. We classi- (i('d all non-singular hexagon spa.ces and modified some methods to evaluate ( 'how(cohomology) rings of'pol3^gon spaces. In addition, we detoi-mine the Fano Hexagon spa.ces. Besides these algeliraic a.nd algebraic topological properties, we aiso gave explicit geometric structures to non-singular polygon spaces and examined their .syrnplectic geornetricai properties. We adapt(>d some previously known results for poly^gons in Euclidean space to polygons in .Vlinkowski space and esta.blished explicit catlculational tobis which are used in showing the integrability of the almost complex structure of moduli spaces of spatial polyygons.
Open Access
Weingarten surfaces arising from soliton theory
(1999) Ceyhan, Özgür
In this work we presented a method for constructing surfaces in associated with the symmetries of Gauss-Mainardi-Codazzi equations. We show that among these surfaces the sphere has a unique role. Under constant gauge transformations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by symmetries of the sine-Gordon equation are homeomorphic to sphere. We also construct some Weingarten surfaces arising from the deformations of sine-Gordon, sinh-Gordon, nonlinear Schrödinger and modified Korteweg-de Vries equations.