Abstract
We prove that a parabolic unitalU in a translation plane π of orderq 2 with kernel containing GF(q) is a Buekenhout-Metz unital if and only if certain Baer subplanes containing the translation line of π meetU in 1 moduloq points. As a corollary we show that a unital 16-03 in PG(2,q 2) is classical if and only if it meets each Baer subplane of PG(2,q 2) in 1 moduloq points.
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References
J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,Math. Z. 60 (1954), 156–186.
R.H. Bruck andR.C. Böse, The construction of translation planes from projective spaces,J. Algebra 1 (1964), 85–102.
R.H. Brück andR.C. Bose, Linear representations of projective planes in projective spaces,J. Algebra 4 (1966), 117–172.
A.A. Bruen andJ.W.P. Hirschfeld, Intersections in projective spaces I: Combinatorics,Math. Z. 193 (1986), 215–225.
F. Buekenhout, Existence of unitals in finite translation planes of orderq 2 with a kernel of orderq, Geom. Dedicata 5 (1976), 189–194.
L.R.A. Casse, C.M. O'Keefe andT. Penttila, Characterizations of Buekenhout-Metz Unitals,Geom. Dedicata 59 (1996), 29–42.
G. Faina andG. Korchmáros, A graphic characterization of Hermitian curves,Ann. Discrete Math. 18 (1983), 335–342.
J.W. Freeman, Reguli and pseudo-reguli in PG(3,s 2),Geom. Dedicata 9 (1980), 267–280.
K. Grüning, A class of unitals of orderq which can be embedded in two different translation planes of orderq 2,J. Geom. 29 (1987), 61–77.
J.W.P. Hirschfeld,Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.
B. Larato, Una caratterizzazione degli unitals parabolici di Buekenhout-Metz,Matematiche (Catania) 38 (1983), 95–98.
C. Lefévre-Percsy, Characterization of Buekenhout-Metz unitals,Arch. Math. 36 (1981), 565–568.
C. Lefévre-Percsy, Characterization of Hermitian curves,Arch. Math. 39 (1982), 476–480.
R. Metz, On a class of unitals,Geom,. Dedicata 8 (1979), 125–126.
C.M. O'Keefe andT. Penttila, Ovoids in PG(3,16) are elliptic quadrics,J. Geom. 38 (1990), 95–106.
C.M. O'keefe andT. Penttila, Ovoids in PG(3,16) are elliptic quadrics, II,J. Geom. 44 (1992), 140–159.
C.M. O'Keefe, T. Penttila andG.F. Royle, Classification of ovoids in PG(3, 32),J. Geom. 50 (1994), 143–150.
T. Penttila andG.F. Royle, Sets of type (m, n) in projective and affine planes of order 9,Des. Codes Cryptogr. 6 (1995), 229–245.
C.T.Quinn,Baer Structures, Unitals and Associated Finite Geometries. PhD Thesis, University of Adelaide, 1997.
C.T. Quinn andL.R.A. Casse, Concerning a characterisation of Buekenhout-Metz unitals,J. Geom. 52 (1995), 159–167.
R. Vincenti, A survey on varieties of PG(4,q) and Baer subplanes of translation planes,Ann. Discrete Math. 18 (1983), 775–780.
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Barwick, S.G., O'Keefe, C.M. & Storme, L. Unitals which meet Baer subplanes in 1 moduloq points. J Geom 68, 16–22 (2000). https://doi.org/10.1007/BF01221057
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DOI: https://doi.org/10.1007/BF01221057