Flocks of ovoids and hyperbolic quadrics in PG (3,Q)

• Main Author: Go, Junie T.
• Format: text
• Language: English
• Published: Animo Repository 1993
• Subjects: Quadrics; Ovals; Mathematics > Formulae; Geometry; Hyperbolic; xx1 Surfaces; xx3 Hyperbolic geometry; Algebraic Geometry; Educational Assessment, Evaluation, and Research; Educational Methods
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/1510
• Institution: De La Salle University

This thesis gives a comprehensive account of the geometric structures and combinatorial properties of ovoids and hyperbolic quadrics in PG(3,q), a three-dimensional projective space of order q where q is a prime power. An ovoid of PG(3,q), q 2 is a set of q2 + 1 points such that no three of which are collinear. Moreover, a nondegenerate quadric in PG(3,q) is a set of points which satisfies a second degree homogeneous equation. There are three types of quadrics namely elliptic, hyperbolic and cone. The first two types are thoroughly discussed in this paper. Furthermore, this study deals with flocks of ovoids and hyperbolic quadrics. Finally, there are two types of flocks namely linear flock and nonlinear flock. This thesis gives an exposition of the theorems about flocks found in [5]. These main results are:1. Every flock of an ovoid in PG(3,q) with q even is linear.2. Any flock of an elliptic quadric in PG(3,q) with q odd is linear.3. Every flock of a hyperbolic quadric in PG(3,q) with q even is linear.4. In PG(3,q) where q is odd, every hyperbolic quadric has nonlinear flocks. This thesis gives a comprehensive account of the geometric structures and combinatorial properties of ovoids and hyperbolic quadrics in PG(3,q), a three-dimensional projective space of order q where q is a prime power. An ovoid of PG(3,q), q 2, is a set of q2 + 1 points such that no three of which are collinear. Moreover, a nondegenerate quadric in PG(3,q) is a set of points which satisfies a second degree homogeneous equation. There are three types of quadrics, namely, elliptic, hyperbolic and cone. The first two types are thoroughly discussed in this paper. Furthermore, this study deals with flocks of ovoids and hyperbolic quadrics. Finally, there are two types of flocks, namely, linear flock and non-linear flock. This thesis gives an exposition of the theorems about flocks. The main results are the following:1. Every flock of an ovoid in PG(3,q) with q even is linear.2. Any flock of an elliptic quadric in PG(3,q) wit