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On the transformation of certain singular surfaces
Published online by Cambridge University Press: 24 October 2008
Extract
Nöther in his classical paper Zur Theorie des eindeutigen Ent-sprechens algebraischer Gebilde has given a set of sixteen examples of the computation of his invariants pn, p(2), p(1) for algebraic surfaces in ordinary space. In the following I discuss, as a matter of some interest, the birational transformation of his surfaces into surfaces which are non-singular.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 3 , 30 July 1933 , pp. 319 - 330
- Copyright
- Copyright © Cambridge Philosophical Society 1933
References
* Nöther, , Math. Annalen, 8 (1875), 495.CrossRefGoogle Scholar
† These formulae were essentially given in a lecture by Professor H. F. Baker in the Lent term, 1932, for the cases occurring here, when every s is 2 or 3. I have since obtained them in other ways. See Babbage, , “Isolated singular points in the theory of algebraic surfaces”, Proc. Camb. Phil. Soc. 29 (1933), 212.CrossRefGoogle Scholar
* The numerical genus p n is calculated from Nöther's formula I + ω = 12p n + 9. Cf. Nöther, loc. cit.
† There are ten linearly independent quadrics through 
and therefore thirteen through 
. The freedom of quadric sections is thus thirty-one and it can be shewn by means of a formula of Nöther's on the postulation of a multiple curve to a surface that this is also the freedom of surfaces of the type F 6. Cf. Nöther, , “Sulle curve multipli di superficie algebriche”, Annali di Mat. (2), 5 (1871), 163.CrossRefGoogle Scholar A similar argument should be supplied in several of the cases which follow, but I have omitted it for brevity.
* Babbage, , “A series of rational loci with one apparent double point”, Proc. Camb. Phil. Soc. 27 (1931), 399CrossRefGoogle Scholar: Edge, , “The number of apparent double points of certain loci”, Proc. Camb. Phil. Soc. 28 (1932), 285.CrossRefGoogle Scholar
* Enriques, , “Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere lineare p (1) = 1”, Rend. Lincei (5), 23 (1914), 206, 291.Google Scholar
* The cusps of 
are winding-places of three sheets of the triple plane.
* The general cubic section does not, however, contain an elliptic sextic of 
.
* Cf. Babbage, , “Rational normal octavic surfaces with a double line, in space of five dimensions”, Proc. Camb. Phil. Soc. 29 (1933), 95–102 (100).CrossRefGoogle Scholar
† In the representative [5], corresponds to a general solid, the points of C to the planes through p in the solid II, and the points of a line g to the neighbourhoods of points of p in a given prime through II; the points of a generator of the opposite system on Q correspond to the neighbourhoods of a given point of p in the ∞1 primes through II.