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

✍ Scribed by Antonio Cossidente; Alessandro Siciliano

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
139 KB
Volume
22
Category
Article
ISSN
0195-6698

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✦ Synopsis

Some geometry of Hermitian matrices of order three over GF(q 2 ) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M 3 7 of PG(8, q) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. Beside M 3 7 turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q 2 ) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q 2 ) is a subgroup of all linear automorphisms of H.

Further, the Hermitian lifting of a collineation of PG(2, q 2 ) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q 2 ) new mixed partitions of PG(8, q) into caps and linear subspaces are given.

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