Please use this identifier to cite or link to this item: https://hdl.handle.net/2440/61625
Type: Thesis
Title: Conics, unitals and net replacement.
Author: Marshall, Daniel
Issue Date: 2010
School/Discipline: School of Mathematical Sciences : Pure Mathematics
Abstract: The main concerns of this thesis are inherited unitals and conics in finite translation planes. Translation planes may be constructed from particular incidences in other translation planes. One method for doing this is "net-derivation" or the corresponding operation "net replacement". We consider conics and unitals of the finite projective plane PG(2,q²) and observe the effect of net-derivation on their pointsets. Our aim is to determine when the pointsets of conics and unitals of PG(2,q²) are conics and unitals respectively in the translation planes formed after net-derivation. In particular, we focus on t-nest replacement and the corresponding nest-derivation sets. Chapter one introduces all the necessary background on finite affine and projective planes. We consider all relevant substructures and concepts. Of major importance are the definitions of unitals, quadrics, Baer subplanes, Baer sublines and derivation. Chapter two introduces the Bruck-Bose correspondence. We use the Bruck-Bose correspondence extensively in chapters three and four. The Bruck-Bose correspondence is a correspondence between PG(2,q²) and certain incidences in PG(4,q). The key elements are spreads of PG(3,q) as a subspace of PG(4,q). We also detail the known correspondences for Baer sublines, Baer subplanes and unitals as well as the equivalent operation for derivation. Chapter three is where we begin our main work. Here we define net replacement in spreads and show the equivalence to net-derivation sets in PG(2,q²). We look at t-nests in depth, which are an example of net replacement. We prove several known results as well as a host of new geometric and combinatorial properties about t-nests. We show a detailed example of a known t-nest and also define a particular type of replacement set that is common to most t-nests. We finish with examples of different kinds of net-derivation. Chapter four looks at unitals of PG(2,q²) and the effect of general net-derivation. Given a unital of PG(2,q²), suppose we perform net-derivation in PG(2,q²) to form a new translation plane. Can we complete the affine points of the unital to a unital in the new translation plane? We first detail the known results for unitals and derivation. We then prove results for unitals and general net-derivation for all known cases where the net-derivation set lies on l8. The particular case for t-nests was published separately by the author in [9]. We prove a new result for when the net-derivation set is not on l8 which is also a new result when just considering derivation. Next, we generalise several other results about derivation and unitals to include general net-derivation. We show the existence of non-inherited unitals in translation planes formed by t-nest replacement of a type that are not present in translation planes formed by derivation. We finish by considering O'Nan configurations contained in unitals in PG(2,q²) and planes formed by net-derivation. Chapter five considers conics and the effect of multiple derivation. Given a conic of PG(2,q²), suppose we perform net-derivation in PG(2,q²) to form a new translation plane. Can we complete the affine points of the conic to a conic in the new translation plane? In particular, we focus on inherited conics with respect to multiple derivation. We begin by defining notation and present a new corollary on nest-derivation and conics, followed by several basic theorems on conics and derivation. We then present, in three stages, a novel characterisation of the equations of conics that are not arcs after derivation with the real derivation set. Next we provide a brief survey of the known results for inherited conics and derivation. We then restrict our attention to conics contained in a particular family Cc,d. Using this family, we prove several new theorems on the existence of inherited (q²+1)-arcs in a class of planes formed by double derivation in PG(2,q²), where q is odd. We follow this by computing an example of a complete 24-arc in a particular translation plane of order 25. Finally, we show the existence of a family of inherited arcs in a class of Andre planes which includes the regular Nearfield planes of odd order.
Advisor: Barwick, Susan Gay
Quinn, Catherine
Brown, Matthew
Wolff, Alison
Casse, Rey
Dissertation Note: Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2010
Keywords: finite geometry; projective geometry
Appears in Collections:Research Theses

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