Abstract
Ak-nucleus of a normal rational curve inPG(n, F) is the intersection over allk-dimensional osculating subspaces of the curve (k ε {−1,0,...,n− 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristicp}>0 and #F≥n the number of non-zero digits in the representation ofn + 1 in basep equals the number of distinct nuclei. An explicit formula for the dimensions ofk-nuclei is given for #F=F≥k + 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brauner, H.,Geometrie projektiver Räume II, BI-Wissenschaftsverlag, Mannheim Wien Zürich, 1976.
Brouwer, A.E., andWilbrink, H.A.,Block Designs, in Handbook of Incidence Geometry, Buekenhout, F., ed., Elsevier, Amsterdam, 1995, ch. 8, pp. 349–382.
Glynn, D.G.,The non-classical 10-arcof PG(4,9), Discrete Math., 59 (1986), pp. 43–51.
Hasse, H.,Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten, J. reine angew. Math., 177 (1937), pp. 215–237.
Havlicek, H.,Normisomorphismen und Normkurven endlichdimensionaler projektiver Desargues-Räume, Monatsh. Math., 95 (1983), pp. 203–218.
Havlicek, H.,Die automorphen Kollineationen nicht entarteter Normkurven, Geom. Dedicata, 16 (1984), pp. 85–91.
Havlicek, H.,Applications of Results on Generalized Polynomial Identities in Desarguesian Projective Spaces, in Rings and Geometry, Kaya, R., Plaumann, P., and Strambach, K., eds., D. Reidel, Dordrecht, 1985, pp. 39–77.
Herzer, A.,Die Schmieghyperebenen an die Veronese-Mannigfaltigkeit bei beliebiger Charakteristik, J. Geometry, 18 (1982), pp. 140–154.
Hexel, E., andSachs, H.,Counting residues modulo a prime in Pascal's triangle, Indian J. Math., 20 (1978), pp. 91–105.
Hirschfeld, J.W.P.,Projective Geometry over Finite Fields, Clarendon Press, Oxford, second ed., 1998.
Hirschfeld, J.W.P., andThas, J.A.,General Galois Geometries, Oxford University Press, Oxford, 1991.
Karzel, H.,Über einen Fundamentalsatz der synthetischen algebraischen Geometrie von W. Burau und H. Timmermann, J. Geometry, 28 (1987), pp. 86–101.
Long, C.T.,Pascal's triangle modulo p, Fibonacci Q., 19 (1981), pp. 458–463.
Thas, J.A.,Normal rational curves and (q+ 2)-arcs in a Galois space S q−2,q (q = 2h), Atti Accad. Naz. Lincei Rend., 47 (1969), pp. 249–252.
Timmermann, H.,Descrizioni geometriche sintetiche di geometrie proiettive con caratteristica p>0, Ann. mat. pura appl., IV. Ser. 114, (1977), pp. 121–139.
Timmermann, H.,Zur Geometrie der Veronesemannigfaltigkeit bei endlicher Charakteristik, Habilitationsschrift, Univ. Hamburg, 1978.
Author information
Authors and Affiliations
Additional information
Research supported by the Austrian National Science Fund (FWF), project P-12353-MAT.
Rights and permissions
About this article
Cite this article
Gmainer, J., Havlicek, H. Nuclei of normal rational curves. J Geom 69, 117–130 (2000). https://doi.org/10.1007/BF01237480
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01237480