Algebraic curves and maximal arcs
A. Aguglia
, L. Giuzzi2
and G. Korchmáros3
2A. Aguglia
3Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy
DOI: 10.1007/s10801-008-0122-7
Abstract
A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set K \mathcal{K} of the Desarguesian plane PG(2, q) is obtained. The case where K \mathcal{K} is a maximal ( k, n)-arc is considered in greater depth.
Pages: 531–544
Keywords: keywords algebraic curves; maximal arcs
Full Text: PDF
References
1. Abatangelo, V., Korchmáros, G.: A generalization of a theorem of B. Segre on regular points with respect to an ellipse of an affine Galois plane. Ann. Mat. Pure Appl. 72, 87-102 (1997)
2. Abatangelo, V., Larato, B.: A characterization of Denniston's maximal arcs. Geom. Dedicata 30, 197- 203 (1989)
3. Aguglia, A., Giuzzi, L.: An algorithm for constructing some maximal arcs in PG(2, q2). Results Math., to appear.
4. Aguglia, A., Korchmáros, G.: Blocking sets of external lines to a conic in PG(2, q), q odd. Combinatorica 26(4), 379-394 (2006)
5. Ball, S., Blokhuis, A.: On the size of a double blocking set in PG(2, q). Finite Fields Appl. 2(2), 125-137 (1996)
6. Ball, S., Blokhuis, A.: An easier proof of the maximal arcs conjecture. Proc. Am. Math. Soc. 126(11), 3377-3380 (1998)
7. Ball, S., Blokhuis, A., Mazzocca, F.: Maximal arcs in Desarguesian planes of odd order do not exist. Combinatorica 17(1), 31-41 (1997)
8. Barlotti, A.: Sui (k, n)-archi di un piano lineare finito. Boll. Un. Mat. Ital. 11(3), 553-556 (1956)
9. Denniston, R.H.F.: Some maximal arcs in finite projective planes. J. Comb. Theory 6, 317-319 (1969)
10. Giulietti, M.: Blocking sets of external lines to a conic in PG(2, q), q even. Eur. J. Comb. 28(1), 36-42 (2007)
11. Goppa, V.D.: Geometry and Codes. Kluwer, Amsterdam (1988) J Algebr Comb (2008) 28: 531-544
12. Hamilton, N., Mathon, R.: More maximal arcs in Desarguesian projective planes and their geometric structure. Adv. Geom. 3, 251-261 (2003)
13. Hamilton, N., Penttila, T.: A Characterisation of Thas maximal arcs in translation planes of square order. J. Geom. 51(1-2), 60-66 (1994)
14. Hamilton, N., Penttila, T.: Groups of maximal arcs. J. Comb. Theory Ser. A 94(1), 63-86 (2001)
15. Hirschfeld, J.W.P.: Projective Geometries over Finite Fields, 2nd edn. OUP, Oxford (1998)
16. Hirschfeld, J.W.P., Korchmáros, G.: Arcs and curves over finite fields. Finite Fields Appl. 5(4), 393- 408 (1999)
17. Leep, D.B., Yeomans, C.C.: The number of points on a singular curve over a finite field. Arch. Math.
2. Abatangelo, V., Larato, B.: A characterization of Denniston's maximal arcs. Geom. Dedicata 30, 197- 203 (1989)
3. Aguglia, A., Giuzzi, L.: An algorithm for constructing some maximal arcs in PG(2, q2). Results Math., to appear.
4. Aguglia, A., Korchmáros, G.: Blocking sets of external lines to a conic in PG(2, q), q odd. Combinatorica 26(4), 379-394 (2006)
5. Ball, S., Blokhuis, A.: On the size of a double blocking set in PG(2, q). Finite Fields Appl. 2(2), 125-137 (1996)
6. Ball, S., Blokhuis, A.: An easier proof of the maximal arcs conjecture. Proc. Am. Math. Soc. 126(11), 3377-3380 (1998)
7. Ball, S., Blokhuis, A., Mazzocca, F.: Maximal arcs in Desarguesian planes of odd order do not exist. Combinatorica 17(1), 31-41 (1997)
8. Barlotti, A.: Sui (k, n)-archi di un piano lineare finito. Boll. Un. Mat. Ital. 11(3), 553-556 (1956)
9. Denniston, R.H.F.: Some maximal arcs in finite projective planes. J. Comb. Theory 6, 317-319 (1969)
10. Giulietti, M.: Blocking sets of external lines to a conic in PG(2, q), q even. Eur. J. Comb. 28(1), 36-42 (2007)
11. Goppa, V.D.: Geometry and Codes. Kluwer, Amsterdam (1988) J Algebr Comb (2008) 28: 531-544
12. Hamilton, N., Mathon, R.: More maximal arcs in Desarguesian projective planes and their geometric structure. Adv. Geom. 3, 251-261 (2003)
13. Hamilton, N., Penttila, T.: A Characterisation of Thas maximal arcs in translation planes of square order. J. Geom. 51(1-2), 60-66 (1994)
14. Hamilton, N., Penttila, T.: Groups of maximal arcs. J. Comb. Theory Ser. A 94(1), 63-86 (2001)
15. Hirschfeld, J.W.P.: Projective Geometries over Finite Fields, 2nd edn. OUP, Oxford (1998)
16. Hirschfeld, J.W.P., Korchmáros, G.: Arcs and curves over finite fields. Finite Fields Appl. 5(4), 393- 408 (1999)
17. Leep, D.B., Yeomans, C.C.: The number of points on a singular curve over a finite field. Arch. Math.