On the Determination of Plane Curves by means of Assigned Singularities

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.