On the Geometry of Hermitian Matrices of Order Three Over Finite Fields |
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Authors: | Antonio Cossidente |
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Institution: | Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italyf1;Dipartimento di Matematica, Università di Roma “La Sapienza" Piazzale Aldo Moro, 5, 00185 Roma, Italy, f2 |
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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. BesideM73 turns 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. |
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