Colloquium publications.

LINEAR SYSTEMS OF CURVES.

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i2 = 0. Notice that 01, 02, 03 are adjoint O's of all sextics of the
system, so that Q is the conjugate point of P on every sextic that
contains them.
We mention systems of this second kind, only in order to exclude them from further discussion here. Let (H) be an oc3 system
of the first kind of hyperelliptic plane curves H1, HI2, etc., of order
n, and let (0) be the system of adjoint curves of order n - 3, i. e.,
let the curves q5 4, * - * have as (i - 1)-fold points the i-fold base
points of the system (EH). Consider any point P of the plane.
Its conjugate Q on any curve H of the system must lie, by definition, upon every +-curve containing P. Since Q is a variable
point, its locus must needs form a part of every f-curve through
P, and these cf-curves accordingly must be degenerate. By parity
of argument every +-curve must consist of (p - 1) distinct parts
where p is the common deficiency of curve H, and each part must
intersect every curve H in only two points, a conjugate pair, outside the multiple base points of the system (H). For an example
of this, let the system (H) consist of all curves of order n having
in a fixed point 0 a multiple point of order n - 2. Any O-curve
must have in 0 an (n - 3)-fold point, and is itself of order n - 3,
therefore it will consist of n - 3 right lines through 0. Every
constituent right line has with any curve H it - 2 intersections
in 0, and two outside that point; the latter two are conjugate points
on the curve, which is consequently hyperelliptic. Another example, with the O's compounded of conies, is the system of curves
of order 2mn + 3 with four fixed multiple points of order m + 1.
The fact that for these plane systems the points conjugate to a
given point fill out a definite locus is the thing to which we shall
wish to recur.
In space of three dimensions, let a surface F have all its plane
sections hyperelliptic curves (C) of deficiency p. Can these be
represented by a system of curves all in one plane? Is the surface F rational, i. e., transformable into a plane, point-for-point,
rationally? This question again may be approached by the aid
of conjugate pairs of points. We should expect of course that