On the genus of a cyclic plane curve over a finite field

TL;DR

This paper computes the genus of cyclic curves derived from cyclic k-arcs of Singer type in Desarguesian projective planes, advancing understanding of their algebraic and geometric properties.

Contribution

It provides a new explicit calculation of the genus for cyclic curves associated with Singer type arcs, linking group actions to algebraic curve invariants.

Findings

Genus of cyclic curves is explicitly determined.

Connects cyclic collineation groups to algebraic curve properties.

Enhances understanding of cyclic arcs in finite projective planes.

Abstract

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.