An investigation of the tangent splash of a subplane of $$\mathrm{PG}(2,q^3)$$PG(2,q3)

In $$\mathrm{PG}(2,q^3)$$PG(2,q3), let $$\pi $$π be a subplane of order $$q$$q that is tangent to $$\ell _\infty $$ℓ∞. The tangent splash of $$\pi $$π is defined to be the set of $$q^2+1$$q2+1 points on $$\ell _\infty $$ℓ∞ that lie on a line of $$\pi $$π. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck–Bose representation of $$\mathrm{PG}(2,{q}^3)$$PG(2,q3) in $$\mathrm{PG}(6,{q})$$PG(6,q). We show that a tangent splash of $$\mathrm{PG}(1,q^3)$$PG(1,q3) is a $$\mathrm{\text{ GF }}(q)$$GF(q)-linear set of rank 3 and size $$q^2+1$$q2+1; this allows us to use results about linear sets from Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89–104, 2010) to obtain properties of tangent splashes.

[1]  Wen-Ai Jackson,et al.  A characterisation of tangent subplanes of PG(2, q3) , 2014, Des. Codes Cryptogr..

[2]  R. C. Bose,et al.  The construction of translation planes from projective spaces , 1964 .

[3]  Norman L. Johnson,et al.  Handbook of Finite Translation Planes , 2007 .

[4]  R. C. Bose,et al.  Linear representations of projective planes in projective spaces , 1966 .

[5]  The Geometry of GF(q 3) , 1986, Canadian Journal of Mathematics.

[6]  Susan G. Barwick,et al.  The tangent splash in PG(6, q) , 2015, Discret. Math..

[7]  J. Thas,et al.  General Galois geometries , 1992 .

[8]  Susan G. Barwick,et al.  Sublines and subplanes of PG(2, q3) in the Bruck-Bose representation in PG(6, q) , 2012, Finite Fields Their Appl..

[9]  Susan G. Barwick,et al.  Exterior splashes and linear sets of rank 3 , 2014, Discret. Math..

[10]  Michel Lavrauw,et al.  On linear sets on a projective line , 2010, Des. Codes Cryptogr..

[11]  Johannes André,et al.  Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe , 1954 .

[12]  Robert Winter,et al.  Dimensional crossover in Sr2RuO4 within a slave-boson mean-field theory , 2008, 0812.3731.

[13]  Guglielmo Lunardon,et al.  Translation ovoids of orthogonal polar spaces , 2004 .

[14]  J. Hirschfeld Finite projective spaces of three dimensions , 1986 .

[15]  Olga Polverino,et al.  Linear sets in finite projective spaces , 2010, Discret. Math..

[16]  Leo Storme,et al.  Results on Maximal Partial Spreads in PG(3, p3) and on Related Minihypers , 2003, Des. Codes Cryptogr..