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On the Determination of Plane Curves by means of Assigned Singularities

Published online by Cambridge University Press:  24 October 2008

J. A. Todd
J. A. Todd
Affiliation:
Trinity College
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Extract

The principal object of the present paper is to give a set of sufficient conditions, of considerable generality, under which a plane curve assigned to pass through certain points with given multiplicities shall be irreducible. The case considered is that in which the assigned multiple points of the curve are of such generality that the conditions presented to the curve at these points are independent. We shall shew that subject to certain explicit restrictions on the points, which ensure, for example, that no three points shall be collinear whose prescribed multiplicities in aggregate exceed the order of the curve, the curve determined by the points will be irreducible if its freedom is not negative, save in the case when it degenerates into a repeated elliptic curve. A familiar example of this last case arises in the problem of constructing a sextic with nine assigned nodes; in general the curve in question consists of the unique cubic through the nine points, counted twice.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 3 , July 1931 , pp. 291 - 305
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Severi, , Vorlesungen über algebraische Geometrie (Leipzig, 1921), pp. 349 ff., which may be consulted for the details.CrossRefGoogle Scholar

* Noether, , Acta Mathematica, 8 (1886), 161–192CrossRefGoogle Scholar

† Caporali, , Mem. di Geom. 171.Google Scholar

* In this case we clearly have a pencil of rational curves.

* Mem. di Torino (2) 42 (1892), 1.Google Scholar

* Op. cit. ante.