On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

Authors: Cossidente A.¹; Siciliano A.²

Source: European Journal of Combinatorics, Volume 22, Number 8, November 2001 , pp. 1047-1058(12)

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Abstract:

Some geometry of Hermitian matrices of order three over GF(q²) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M₇³of PG(8,q) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM₇³turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q²) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q²) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q²) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q²) new mixed partitions of PG(8,q) into caps and linear subspaces are given. Copyright 2001 Academic Press

Language: English

Document Type: Research article

Affiliations: 1: Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy 2: Dipartimento di Matematica, Università di Roma “La Sapienza", Piazzale Aldo Moro, 5, 00185 Roma, Italy

Publication date: 2001-11-01