Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T01:37:30.222Z Has data issue: false hasContentIssue false
  • English
  • Français

The fixed part of the canonical system on an algebraic surface

Published online by Cambridge University Press:  24 October 2008

Patrick Du Val
Patrick Du Val
Affiliation:
Trinity College
Get access Rights & Permissions [Opens in a new window]

Extract

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 1 , January 1938 , pp. 1 - 5
Copyright
Copyright © Cambridge Philosophical Society 1938

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Hodge, W. V. D., Journ. London Math. Soc. 12 (1937), p. 58.CrossRefGoogle Scholar
(2)Bronowski, J., in the press. [Since writing the above I find that Bronowski's proof was substantially anticipated by Segre, B., Annali di Mat. (4), 16, (1937), 157.]Google Scholar
(3)Picard, E. and Simart, G., Théorie des fonctions algébriques de deux variables indépendantes, 1 (Paris, 1897), p. 223.Google Scholar
(4)Pedoe, D., Proc. Cambridge Phil. Soc. 31 (1935), p. 536.CrossRefGoogle Scholar
(5)Bronowski, J., Journ. London Math. Soc. 12 (1937), p. 140.CrossRefGoogle Scholar
(6)Segre, B., Rend. dell' Ist. di Bologna, 01 1936.Google Scholar
(7)Campedelli, L., Atti della R. Acc. di Torino, 71 (1936), p. 370.Google Scholar
(8)Enriques, F. and Campedelli, L., Lezioni sulla teoria delle superficie algebriche (Padua, 1932), p. 372.Google Scholar
(9) Cf. Enriques, F. and Campedelli, L., Lezioni sulla teoria delle superficie algebriche (Padua, 1932), p. 299.Google Scholar