Abstract
We classify nets of conics of rank one in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in \(\mathrm {PG}(5,q)\) which meet the quadric Veronesean in at least one point, under the action of \(\mathrm {PGL}(3,q) \leqslant \mathrm {PGL}(6,q)\) (for q odd). Our results complete a partial classification of nets of conics of rank one obtained by Wilson (Am J Math 36:187–210, 1914).
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Acknowledgements
The first author acknowledges the support of The Scientific and Technological Research Council of Turkey TÜBİTAK (Project No. 118F159). We thank the referee for a careful reading and several helpful remarks.
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Lavrauw, M., Popiel, T. & Sheekey, J. Nets of conics of rank one in \(\mathrm {PG}(2,q)\), q odd. J. Geom. 111, 36 (2020). https://doi.org/10.1007/s00022-020-00548-1
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DOI: https://doi.org/10.1007/s00022-020-00548-1