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Applications of line geometry over finite fields I: The twisted cubic

✍ Scribed by A. A. Bruen; J. W. P. Hirschfeld

Publisher
Springer
Year
1977
Tongue
English
Weight
528 KB
Volume
6
Category
Article
ISSN
0046-5755

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Let Y & P n be a cubic hypersurface defined over GFðqÞ. Here, we study the Finite Field Nullstellensatz of order ½q=3 for the set Y ðqÞ of its GFðqÞ-points, the existence of linear subspaces of PGðn; qÞ contained in Y ðqÞ and the possibility to join any two points of Y ðqÞ by the union of two lines

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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.