Isolated singular points in the theory of algebraic surfaces

Author:

Babbage D. W.

Abstract

If F(x0, x1, x2, x3) = 0 is the equation of a surface in space of three dimensions which has an ordinary isolated s-ple point O, then by means of the substitutionswhere Φ0 = 0, Φ1 = 0, …, Φr = 0 are the equations of r + 1 linearly independent surfaces passing simply through O, F is transformed into a surface F′ in [r], on which to the point O of F there corresponds a simple curve γ. The points of γ arise from the points of F in the first neighbourhood of O, and in this simple case the genus of γ is ½ (s − 1) (s − 2). In the study of properties which are common to all members of an infinite family of birationally equivalent surfaces no distinction is made between O and γ, O being regarded as a curve which has become infinitesimal on the particular surface of the family in question.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference4 articles.

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1. Geometry at Cambridge, 1863–1940;Historia Mathematica;2006-08

2. Base conditions and covariant systems in an algebraic threefold;Mathematical Proceedings of the Cambridge Philosophical Society;1939-04

3. On the regularity of surfaces, II;Mathematical Proceedings of the Cambridge Philosophical Society;1934-07

4. On the transformation of certain singular surfaces;Mathematical Proceedings of the Cambridge Philosophical Society;1933-07-30

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