Curves containing all points of a finite projective Galois plane

TL;DR

This paper studies Tallini curves in finite projective planes, focusing on their automorphism groups, Weierstrass semigroups, Hasse-Witt invariants, and quotient curves, revealing new algebraic and geometric properties.

Contribution

It provides new insights into the structure and invariants of Tallini curves, extending previous work by analyzing their automorphisms and related algebraic features.

Findings

Determined automorphism groups of Tallini curves

Analyzed Weierstrass semigroups and Hasse-Witt invariants

Explored quotient curves of Tallini curves

Abstract

In the projective plane PG(2,q) over a finite field of order q, a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree q+2 containing all points of PG(2,q). Such curves were investigated by G. Tallini in 1961, and by Homma and Kim in 2013. Our results concern the automorphism groups, the Weierstrass semigroups, the Hasse-Witt invariants, and quotient curves of the Tallini curves.