Ambient Spaces of Dimensional Dual Arcs
Satoshi Yoshiara
DOI: 10.1023/B:JACO.0000022564.51008.63
Abstract
A d-dimensional dual arc in PG( n, q) is a higher dimensional analogue of a dual arc in a projective plane. For every prime power q other than 2, the existence of a d-dimensional dual arc ( d 
2) in PG( n, q) of a certain size implies n 
d( d + 3)/2 (Theorem 1). This is best possible, because of the recent construction of d-dimensional dual arcs in PG( d( d + 3)/2, q) of size 
d-1 i=0 q i, using the Veronesean, observed first by Thas and van Maldeghem (Proposition 7). Another construction using caps is given as well (Proposition 10).
2) in PG( n, q) of a certain size implies n d( d + 3)/2 (Theorem 1). This is best possible, because of the recent construction of d-dimensional dual arcs in PG( d( d + 3)/2, q) of size d-1 i=0 q i, using the Veronesean, observed first by Thas and van Maldeghem (Proposition 7). Another construction using caps is given as well (Proposition 10).Pages: 5–23
Keywords: dual arc; dual hyperoval; Veronesean
Full Text: PDF
References
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2. B. Cooperstein and J.A. Thas, “On generalized k-arcs in P G(2n, q),” Ann. Combin. 5 (2001), 141-152.
3. A. Del Fra, “On d-dimensional dual hyperovals,” Geometriae Dedicata 79 (2000), 157-178.
4. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, 1993.
5. C. Huybrechts, “Dimensional dual hyperovals in projective spaces and c.AG* geometries,” Discrete Math. 255 (2002), 193-223.
6. C. Huybrechts and A. Pasini, “Flag-transitive extensions of dual affine spaces,” Contrib. Algebra Geom. 40 (1999), 503-532.
7. A. Pasini and S. Yoshiara, “On a new family of flag-transitive semibiplanes,” European J. Combin. 22 (2001), 529-545.
8. A. Pasini and S. Yoshiara, “New distance regular graphs arising from dimensional dual hyperovals,” European J. Combin. 22 (2001), 547-560.
9. J.A. Thas and H. van Maldeghem, “Characterizations of the finite quadric Veroneseans V2n n ,” Quart. J. Math. Oxford Ser. 2.
10. J.A. Thas and H. van Maldeghem, “Characterizations of the finite quadric and Hermitian Veroneseans over finite fields,” J. Geom. 76 (2003), 282-293.
11. S. Yoshiara, “A new family of d-dimensional dual hyperovals in P G(2d + 1, 2),” European J. Combin. 20 (1999), 489-503.
12. S. Yoshiara, “On a family of planes of a polar space,” European J. Combin. 22 (2001), 107-118.