No CrossRef data available.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 24 October 2008
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.
* 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.