Unitals in Projective Planes

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Springer Science & Business Media, Apr 3, 2009 - Mathematics - 196 pages
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This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2,q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds,H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil bound.
 

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Contents

Preliminaries
1
Hermitian Curves and Unitals
21
Translation Planes
32
Unitals Embedded in Desarguesian Planes
59
Unitals Embedded in NonDesarguesian Planes
88
Combinatorial Questions and Associated Configurations
109
Characterization Results
133
Open Problems
167
Nomenclature of Unitals
170
Group Theoretic Characterizations of Unitals
173
References
177
Notation Index
187
Index
189
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