The number of points on a curve, and applications Arcs and curves: the legacy of Beniamino Segre

Abstract

Curves defined over a finite field have various applications, such as (a) the construction of good error-correcting codes, (b) the correspondence with arcs in a finite Desarguesian plane, (c) the Main Conjecture for maximum-distance-separable (MDS) codes. Bounds for the number of points of such a curve imply results in these cases. For plane curves, there is a variety of bounds that can be considered, such as the Hasse–Weil bound (1934/1948), the St¨ohr–Voloch bound (1986), as well as bounds that depend on the plane embedding. Curves that achieve these bounds can sometimes be characterised. Segre applied bounds for the number of points on a curve to obtain bounds on the sizes of complete arcs. He also considered plane Fermat curves that achieve the Hasse–Weil bound. Various of these results and their applications are surveyed.

Keywords

• algebraic curve
• arc
• code