Geometric and combinatorial aspects of normal rational curves in PG(2,Q) and PG(3,Q)

Official URL: https://risc01.sabanciuniv.edu/record=b2583855_(Table of contents)

In this thesis, firstly, we study the small complete arcs in PG(2,q), for q odd, with at least (q + 1)~2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work (Segre, 1962) using algebraic curves. This also gives an alternative to Pellegrino’s long proof published in a series of works in 1980s. As a corollary of our proof, we obtain example of arcs which give counterexamples to the statement in (Hirschfeld, 1993). This concerns the existence of a line satisfying the hypothesis for the main theorem from (Pellegrino, 1993a) in which Pellegrino studied the complete arcs sharing (any) (q +1)~2 points with a conic but with an extra assumption. Secondly, we study combinatorial invariants of the equivalence classes of pencils of cubics on PG(1,q), for q odd and q not divisible by 3. These equivalence classes are considered as orbits of lines in PG(3,q), under the action of the stabiliser group of the twisted cubic C3. In particular we determine the point orbit distribution and plane orbit distributions of all lines which, are contained in an osculating plane of C3, have non-empty intersection with C3, or are imaginary chords, or axes.