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Tangent Spaces of a Normal Surface with Hyperelliptic Sections

Published online by Cambridge University Press:  20 November 2018

W. L. Edge
W. L. Edge*
Affiliation:
Inveresk House, Musselburgh, Scotland
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A rational normal curve C of order n in [n] has, at each point P, a nest of osculating spaces

as P moves on C the [n — 2] generates a primal Dn-1 of order 2n — 2.

Hilbert [3] found the multiplicities on Dn-1 not only of the vμ+1 generated by Dμ for each lesser value of μ but also those of all submanifolds common to these various Dμ.

A surface Φ in higher space has, as explained [4] by del Pezzo, a nest of tangent spaces

of respective dimensions

they raise the problem of finding the orders of manifolds generated by them and the multiplicity of each on the higher manifolds to which it belongs: the task does not seem to have been attempted, but it may well be eased if Φ is rational and normal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 1 , 01 February 1984 , pp. 131 - 143
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Castelnuovo, G., Sulle superficie algebriche le cui sezione piane sono curve iperellitiche, Rendiconti del circolo matematico di Palermo 4 (1980); Memorie Scelte, Bologna (Zanichelli, 1937), 189–202.Google Scholar
2. Edge, W. L., Algebraic surfaces with hyperelliptic sections, The Geometric Vein (Springer, New York, 1981).CrossRefGoogle Scholar
3. Hilbert, D., Über die Singularitäten der Diskriminantenfläche, Math. Annalen 30 (1887), 437–441: Gesammelte Abhandlungen 2 (Berlin, Springer, 1933), 117–120.Google Scholar
4. del Pezzo, P., Sugli spazi tangenti ad una superficie o ad una varieta immersa in uno spazio di pui dimensioni, Rendiconti Ace. Napoli 25 (1886), 176–180.Google Scholar
5. Segre, C., Su una classe di superficie delg' iperspazii legate colle equazioni lineari alle derivate parziali di 2° ordine, Atti. Ace. Torino 42 (1906-07), 559–591. Opere 2 (Edizioni Cremonese, Rome, 1958).Google Scholar