A combinatorial characterization of Hermitian curves

Thas, J.A.

Journal of Algebraic Combinatorics, Tome 1 (1992) no. 1, pp. 97-102.

Résumé

Summary: A unital $U$ with parameter $q$ is a $2 - ( q ^{3} + 1, q + 1, 1)$ design. If a point set $U$ in $P$G$(2, q ^{2})$ together with its $( q + 1)$-secants forms a unital, then $U$ is called a Hermitian arc. Through each point $p$ of a Hermitian arc $H$ there is exactly one line $L$ having with $H$ only the point $p$ in common; this line $L$ is called the tangent of $H$ at $p$. For any prime power $q$, the absolute points and nonabsolute lines of a unitary polarity of $P$G$(2, q ^{2})$ form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in $P$G$(2, q ^{2})$. Let $H$ be a Hermitian arc in the projective plane $P$G$(2, q ^{2})$. If tangents of $H$ at collinear points of $H$ are concurrent, then $H$ is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.

Détail

Keywords: Hermitian curve, unital, projective plane