The exterior splash in PG(6, q): carrier conics
-
Susan G. Barwick
and
Wen-Ai Jackson
Abstract
Let π be a subplane of PG(2,q3) of order q that is exterior to ℓ∞. The exterior splash of π is the set of q2+q+1 points on ℓ∞ that lie on the extended lines of π. Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry CG(3,q), and hyper-reguli of PG(5,q). In this article we use the Bruck–Bose representation in PG(6,q) to give a geometric characterisation of carrier conics of π in terms of the covers of the exterior splash of π. We also investigate properties of subplanes of order q with a common exterior splash, and study the intersection of two exterior splashes.
References
[1] J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60 (1954), 156–186. MR0063056 Zbl 0056.3850310.1007/BF01187370Search in Google Scholar
[2] R. D. Baker, J. M. N. Brown, G. L. Ebert, J. C. Fisher, Projective bundles. Bull. Belg. Math. Soc. Simon Stevin1 (1994), 329–336. MR1317131 Zbl 0803.5100910.36045/bbms/1103408578Search in Google Scholar
[3] S. G. Barwick, W.-A. Jackson, Sublines and subplanes of PG(2, q3) in the Bruck-Bose representation in PG(6, q). Finite Fields Appl. 18 (2012), 93–107. MR2874908 Zbl 1261.5100610.1016/j.ffa.2011.07.001Search in Google Scholar
[4] S. G. Barwick, W.-A. Jackson, A characterisation of tangent subplanes of PG(2, q3). Des. Codes Cryptogr. 71 (2014), 541–545. MR3189132 Zbl 1296.5100810.1007/s10623-012-9754-7Search in Google Scholar
[5] S. G. Barwick, W.-A. Jackson, An investigation of the tangent splash of a subplane of PG(2, q3). Des. Codes Cryptogr. 76 (2015), 451–468. MR3375571 Zbl 1346.5100110.1007/s10623-014-9971-3Search in Google Scholar
[6] 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
[7] S. G. Barwick, W. A. Jackson, The exterior splash in PG(6, q): Transversals. Preprint arXiv:1409.6794 [math.CO]Search in Google Scholar
[8] S. G. Barwick, W. A. Jackson, Hyper-reguli in PG(5, q). Preprint arXiv:1409.6795 [math.CO]Search in Google Scholar
[9] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. MR1484478 Zbl 0898.6803910.1006/jsco.1996.0125Search in Google Scholar
[10] R. H. Bruck, Construction problems of finite projective planes. In: Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967), 426–514, Univ. North Carolina Press, Chapel Hill, N.C. 1969. MR0250182 Zbl 0206.23402Search in Google Scholar
[11] R. H. Bruck, Circle geometry in higher dimensions. In: A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), 69–77, North-Holland 1973. MR0365330 Zbl 0274.5001910.1016/B978-0-7204-2262-7.50011-9Search in Google Scholar
[12] R. H. Bruck, Circle geometry in higher dimensions. II. Geometriae Dedicata2 (1973), 133–188. MR0420430 Zbl 0279.5002110.1007/BF00147854Search in Google Scholar
[13] R. H. Bruck, R. C. Bose, The construction of translation planes from projective spaces. J. Algebra1 (1964), 85–102. MR0161206 Zbl 0117.3740210.1016/0021-8693(64)90010-9Search in Google Scholar
[14] R. H. Bruck, R. C. Bose, Linear representations of projective planes in projective spaces. J. Algebra4 (1966), 117–172. MR0196590 Zbl 0141.3680110.1016/0021-8693(66)90054-8Search in Google Scholar
[15] C. Culbert, G. L. Ebert, Circle geometry and three-dimensional subregular translation planes. Innov. Incidence Geom. 1 (2005), 3–18. MR2213951 Zbl 1110.5100210.2140/iig.2005.1.3Search in Google Scholar
[16] G. Donati, N. Durante, Scattered linear sets generated by collineations between pencils of lines. J. Algebraic Combin. 40 (2014), 1121–1134. MR3273404 Zbl 1320.5100710.1007/s10801-014-0521-xSearch in Google Scholar
[17] S. Ferret, L. Storme, Results on maximal partial spreads in PG(3, p3) and on related minihypers. In: Proceedings of the Conference on Finite Geometries (Oberwolfach, 2001), volume 29, 105–122, 2003. MR1993160 Zbl 1032.51005Search in Google Scholar
[18] J. W. P. Hirschfeld, Finite projective spaces of three dimensions. Oxford Univ. Press 1985. MR840877 Zbl 0574.51001Search in Google Scholar
[19] J. W. P. Hirschfeld, J. A. Thas, General Galois geometries. Oxford Univ. Press 1991. MR1363259 Zbl 0789.51001Search in Google Scholar
[20] M. Lavrauw, G. Marino, O. Polverino, R. Trombetti, Solution to an isotopism question concerning rank 2 semifields. J. Combin. Des. 23 (2015), 60–77. MR3291847 Zbl 1309.1201110.1002/jcd.21382Search in Google Scholar
[21] 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
[22] M. Lavrauw, G. Van de Voorde, Scattered linear sets and pseudoreguli. Electron. J. Combin. 20 (2013), Paper 15, 14. MR3035025 Zbl 1270.5101110.37236/2871Search in Google Scholar
[23] G. Lunardon, G. Marino, O. Polverino, R. Trombetti, Maximum scattered linear sets of pseudoregulus type and the Segre variety 𝒮n, n. J. Algebraic Combin. 39 (2014), 807–831. MR3199027 Zbl 1295.5101110.1007/s10801-013-0468-3Search in Google Scholar
[24] G. Marino, O. Polverino, On translation spreads of H(q). J. Algebraic Combin. 42 (2015), 725–744. MR3403178 Zbl 1331.5100710.1007/s10801-015-0599-9Search in Google Scholar
[25] G. Marino, O. Polverino, R. Trombetti, On 𝔽q-linear sets of PG(3, q3) and semifields. J. Combin. Theory Ser. A114 (2007), 769–788. MR2333132 Zbl 1118.5100610.1016/j.jcta.2006.08.012Search in Google Scholar
[26] T. G. Ostrom, Hyper-reguli. J. Geom. 48 (1993), 157–166. MR1242708 Zbl 0792.5100110.1007/BF01226806Search in Google Scholar
[27] R. Pomareda, Hyper-reguli in projective space of dimension 5. In: Mostly finite geometries (Iowa City, IA, 1996), 379–381, Dekker 1997. MR1463998 Zbl 0888.51002Search in Google Scholar
© 2017 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Ricci solitons on four-dimensional Lorentzian Walker manifolds
- The exterior splash in PG(6, q): carrier conics
- Central figure-8 cross-cuts make surfaces cylindrical
- Vieta’s formulae for regular polynomials of a quaternionic variable
- On characterizations of sausages via inequalities and roots of Steiner polynomials
- On maximal symplectic partial spreads
- Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
- A maximum problem of S.-T. Yau for variational p-capacity
- Drapeable unit arcs fit in the unit 30° sector
- Uniform Lie algebras and uniformly colored graphs
- A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes
Articles in the same Issue
- Frontmatter
- Ricci solitons on four-dimensional Lorentzian Walker manifolds
- The exterior splash in PG(6, q): carrier conics
- Central figure-8 cross-cuts make surfaces cylindrical
- Vieta’s formulae for regular polynomials of a quaternionic variable
- On characterizations of sausages via inequalities and roots of Steiner polynomials
- On maximal symplectic partial spreads
- Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
- A maximum problem of S.-T. Yau for variational p-capacity
- Drapeable unit arcs fit in the unit 30° sector
- Uniform Lie algebras and uniformly colored graphs
- A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes
