## Unitals in Projective PlanesThis 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

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 |

177 | |

187 | |

189 | |

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### Common terms and phrases

absolute points afﬁne plane afﬁne points automorphism Baer subline Bruck-Bose representation Buekenhout unital called characterization classical unital collinear constructed contains Corollary corresponds cubic surface deﬁned Deﬁnition denote derivation set Desarguesian plane disjoint dual element elliptic cone elliptic quadric equation equivalent ﬁrst ﬁxed GF(q Hall plane Hall(q2 Hence homogeneous coordinates hyperbolic quadric hyperplane incidence structure inequivalent intersection inversive plane irreducible conic isomorphic Lemma line of PG(2 linear code meets nonclassical nondegenerate nondegenerate Hermitian curve nonsingular quadric nonsingular-Buekenhout unital nonzero O’Nan conﬁgurations order q2 orthogonal orthogonal-Buekenhout-Metz unital oval ovoidal cone ovoidal-Buekenhout-Metz unital PG(d plane of order plane of PG(4 projective geometry projective plane quadratic extension regular spread regulus result ruled cubic surface secant line Section semiﬁeld spread line subﬁeld subgroup subplane subspace tangent line translation plane unital in PG(2 unital in PG(2,q2 unital with respect unitals embedded unitary polarity vector space vertex