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On the Geometrical Genus of Certain Surfaces

Published online by Cambridge University Press:  24 October 2008

Edwin A. Maxwell
Edwin A. Maxwell
Affiliation:
Queens' College
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Extract

In spite of a large amount of general theory, it remains a surprising fact that the number of known irregular surfaces in space of three dimensions is quite small. Moreover, although the evaluation of the arithmetical genus pα is comparatively direct, it is often a matter of considerable difficulty to calculate the geometrical genus pg. This paper deals with a number of surfaces for which the equations of the canonical system can be written down and their number counted. An extension to primals in higher space is also suggested for the simplest case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 185 - 193
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

* See Bertini, , Introduzione alla geometria proiettiva degli iperspazi (1923), p. 321.Google Scholar

† Rend. Ist. Lombardo, (2), 24, (1891), 127–37,Google Scholar nota I. For a convenient account see Baker, , Principles of geometry, 6 (1933), 203.Google Scholar

* Proc. Camb. Phil. Soc. 31 (1935), 48.Google Scholar

† This same difficulty is mentioned by Baker, , J. Lond. Math. Soc. 10 (1935), 61.Google Scholar

* Noether, , Ann. di Math. (2), 5 (1897), 163–77.Google Scholar The number of linearly independent surfaces of order n, having an s−fold curve of order ε and genus p is

if n is sufficiently high.

* Op. cit. p. 181.Google Scholar

* See Appendix.

† I am indebted for the method employed in this proof to Prof. Baker, who has greatly simplified my own work.