The partition of PG(2, q3) arising from an order 3 planar collineation
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S. G. Barwick
and
Wen-Ai Jackson
Abstract
Let ϕ be a collineation of order 3 acting on PG(2, q3) whose fixed points are exactly an 𝔽q-plane P2,q. Let T be a point whose orbit under ϕ is a triangle and let ST be the subgroup of PGL(3, q3) that fixes setwise the 𝔽q-plane P2,q and fixes setwise the line
Funding statement: A.M. W. Hui acknowledges the support of National Natural Science Foundation of China (Grant No. 12071041).
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Communicated by: J. Bamberg
References
[1] R. D. Baker, J. M. N. Brown, G. L. Ebert, J. C. Fisher, Projective bundles. Bull. Belg. Math. Soc. Simon Stevin 1 (1994), 329–336. MR1317131 Zbl 0803.5100910.36045/bbms/1103408578Search in Google Scholar
[2] S. G. Barwick, A. M. W. Hui, W.-A. Jackson, A geometric description of the Figueroa plane. Des. Codes Cryptogr. 91 (2023), 1581–1593. MR4578154 Zbl 1520.5100210.1007/s10623-022-01158-5Search in Google Scholar
[3] S. G. Barwick, W.-A. Jackson, Exterior splashes and linear sets of rank 3. Discrete Math. 339 (2016), 1613–1623. MR3475577 Zbl 1338.5100710.1016/j.disc.2015.12.018Search in Google Scholar
[4] L. M. Batten, P. M. Johnson, The collineation groups of Figueroa planes. Canad. Math. Bull. 36 (1993), 390–397. MR1245311 Zbl 0803.5100210.4153/CMB-1993-053-0Search in Google Scholar
[5] U. Dempwolff, PSL(3, q) on projective planes of order q3 Geom. Dedicata 18 (1985), 101–112. MR787308 Zbl 0563.5100810.1007/BF00221208Search in Google Scholar
[6] T. Grundhöfer, A synthetic construction of the Figueroa planes. J. Geom. 26 (1986), 191–201. MR850165 Zbl 0595.5101010.1007/BF01227843Search in Google Scholar
[7] N. L. Johnson, Planes and processes. Discrete Math. 309 (2009), 430–461. MR2473092 Zbl 1169.5101210.1016/j.disc.2007.12.031Search in Google Scholar
[8] M. Lavrauw, G. Van de Voorde, On linear sets on a projective line. Des. Codes Cryptogr. 56 (2010), 89–104. MR2658923 Zbl 1204.5101310.1007/s10623-010-9393-9Search in Google Scholar
[9] G. Lunardon, G. Marino, O. Polverino, R. Trombetti, Maximum scattered linear sets of pseudoregulus type and the Segre variety Snn J. Algebraic Combin. 39 (2014), 807–831. MR3199027 Zbl 1295.5101110.1007/s10801-013-0468-3Search in Google Scholar
[10] G. Lunardon, O. Polverino, Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004), 663–669. MR2096680 Zbl 1072.5101010.1515/form.2004.029Search in Google Scholar
[11] J. M. Nowlin Brown, Some partitions in Figueroa planes. Note Mat. 29 (2009), 33–43. MR2942757 Zbl 1252.51005Search in Google Scholar
[12] F. A. Sherk, The geometry of GF(q3 Canad. J. Math. 38 (1986), 672–696. MR845672 Zbl 0589.5102210.4153/CJM-1986-035-2Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
