Colloquium publications.

LINEAR SYSTEMS OF CURVES.

3

systems are carried over into linear systems. A complete linear
system is defined most easily by specifying the multiplicity that a
curve of the system must have in each point of a fundamental set
and by prescribing the order of the curves. Thus (as... ) can indicate that in at every curve is to have a multiple point of order
at least sl, etc. If the base points alone, with their respective
multiplicities, determine a system under consideration, that system
is termed complete. If the base points actually impose, for curves
of order m, fewer conditions than would be expected from their
several multiplicities, the system is special; otherwise it is reyular. It is an important theorem that no set of r base points can
be so located as to produce an (r + 1)th variable multiple point
on the curves of the system; i. e., the multiple points of the
generic curve of a plane linear system lie all in the base points of
the system.
Adjoint curves of a linear system are familiar to the student
of function theory; they have in every multiple point of order s
for the given system a multiplicity of order at least s - 1. The
adjoints of order lower by 3 than the original system are important
from the fact that they transform always into the corresponding
system of adjoints to the transformed curves. On this account
the term adjoint, as used ordinarily, implies a curve of order
- 3 unless differently specified. Second adjoints are adjoint to
adjoints of the system, etc. The employment of successive adjoint
systems as a means of investigation is due to S. Kantor and to G.
Castelnuovo, the latter acknowledging the priority of the former.*
On every curve its adjoints cut out a unique complete series gp2,
called the canonical series. The deficiency of the first or second
adjoints of a linear system is denoted by P1 or P2, and may be
termed first, or second, canonical deficiency. Aside from the
canonical series upon curves of a system, the most important are
the characteristic series of the system, that is the totality of sets of
points in which two curves of the system intersect. If a plane
linear system is complete, then the characteristic series on each
*See JMath. Annalen, vol. 44 (1894), p. 127.