A Combinatorial Characterization of Hermitian Curves
J.A. Thas
DOI: 10.1023/A:1022437415099
Abstract
A unital U with parameter q is a 2 - ( q 3 + 1, q + 1, 1) design. If a point set U in PG(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 PG(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 PG(2, q 2).
Let H be a Hermitian arc in the projective plane PG(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.
Pages: 97–102
Keywords: Hermitian curve; unital; projective plane
Full Text: PDF
References
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2. F. Buekenhout, "Existence of unitals in finite translation planes of order q2 with kernel of order q," Geometriae Dedicata, vol.5, pp. 189-194, 1976.
3. O. Faina and G. Korchmaros, "A graphic characterization of Hermitian curves,"Annals of Discrete Mathematics, vol.18, pp. 335-342, 1983.
4. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979.
5. J.W.P. Hirschfeld, L. Storme, J.A. Thas, and J.F. Voloch, "A characterization of Hermitian curves," Journal of Geometry, vol.41, pp. 72-78, 1991.
6. C. Lefevre-Percsy, "Characterization of Hermitian curves," Achiv de Mathematik, vol.39, pp.476- 480, 1982.
7. R. Metz, "On a class of unitals," Geometrica Dedicata, vol.8, pp. 125-126, 1979.
8. M. Tallini Scafati, "Caratterizzazione grafica delle forme hermitiane di un Sr,q," Rendiconti di Matematica, vol.26, pp. 273-303, 1967.
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