published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

published_or_final_version / Mathematics / Master / Master of Philosophy

A (finite) projective plane of order m, m an integer greater than 1, is a 2- (m^2 + m + 1, m + 1, 1) design. A unital (or a unitary block design) U of order n, n an integer greater than 2, is a 2-(n^3 + 1, n + 1, 1) design. An embedded unital is one that is a subdesign of a projective plane. If a unital of order n is embedded in a projective planeπ, then the order of π is at least n^2. A unital of order n embedded in a projective planeπ of order n^2 is called a polar unital if it consists of the absolute points and non-absolute lines of a unitary polarity of π. In particular, if π is the classical (Desarguesian) plane PG(2, q^2) coordinatized by the finite field Fq^2, then the polar unital is called a classical unital. The main problem in the study of unitals is their characterization and classification. The classical unital does not contain a configuration of four lines meeting in six points (an O'Nan configuration) [O'Nan, 1972]. It is conjectured that this property characterizes the classical unital [Piper, 1979]. The classical unital is characterized by three conditions (I), (II) and (III): (I) is the absence of O'Nan configurations; (II) and (III) are further configurational requirements [Wilbrink, 1983]. The result depends on the classification of finite doubly transitive groups. Furthermore, when the order of a unital is even, (III) is a necessary condition of (I) and (II) [Wilbrink, 1983]. As for group theoretic characterizations, the only unitals that admit doubly transitive automorphism groups are the classical unitals and the Ree unitals [Kantor, 1985]. The classical unital is also characterized by the existence of sufficiently many translations [Grundhöfer, Stroppel, Van Maldeghem, 2013]. In this thesis, a necessary and sufficient condition is given for embedding a unital into a projective plane as a polar unital. A strengthened version of the condition is introduced and is shown to be necessary for a classical unital. Using the strengthened condition and results of [Wilbrink,1983] and [Grundhöfer, Stroppel and Van Maldeghem, 2013], a new intrinsic characterization of the classical unital is given without assuming the absence of O'Nan configurations. Finally, a unital of even order satisfying the first two intrinsic characterization conditions of Wilbrink is shown to be classical without invoking deep results from group theory. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Wisconsin--Madison, 1975. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 85-86).

Thesis (Ph. D.)--University of Wyoming, 2006. / Title from PDF title page (viewed on Dec. 20, 2007). Includes bibliographical references (p. 201-202).

Using tropical methods, we study the problem of compactifying the moduli space of six general lines in the projective plane. In particular, we show that the log canonical compactification is Kapranov’s Chow quotient compactification. / text

abstract / Mathematics / Doctoral / Doctor of Philosophy

Mode of access: World Wide Web. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 160-161). / Electronic reproduction. / Also available by subscription via World Wide Web / viii, 161 leaves, bound ill. 29 cm

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Qing Xiang, Dept. of Mathematical Sciences. Includes bibliographical references.