Partial flocks of the quadratic cone yielding Mathon maximal arcs

TL;DR

This paper explores the geometric relationship between maximal arcs of Mathon type and partial flocks of the quadratic cone, introducing a composition on flock planes and an analogue of Mathon's Theorem for constructing specific maximal arcs.

Contribution

It establishes a geometric link between Mathon maximal arcs and partial flocks, and develops a composition method to construct new maximal arcs from existing structures.

Findings

Established a geometric connection between Mathon arcs and partial flocks.

Introduced a composition on flock planes for constructing arcs.

Provided a method to build maximal arcs of degree 2d containing Denniston arcs.

Abstract

N. Hamilton and J. A. Thas describe a link between maximal arcs of Mathon type and partial flocks of the quadratic cone. This link is of a rather algebraic nature. In this paper we establish a geometric connection between these two structures. We also define a composition on the flock planes and use this to work out an analogue of the synthetic version of Mathon's Theorem. Finally, we show how it is possible to construct a maximal arc of Mathon type of degree 2d, containing a Denniston arc of degree d provided that there is a solution to a certain given system of trace conditions.