Colloquium publications.

2

THE BOSTON COLLOQUIUM.

only such properties of a curve or surface as appertain to the
entire class of curves or surfaces that can be related birationally
to the fundamental form.
A plane algebraic curve may have its order changed by a Cremona transformation, but not its deficiency (genre, Geschlecht).
As to sets of points on the curve, two sets which together make
up a complete intersection of a second curve with the first do not
lose that property by birational transformation, if we exclude
from consideration fundamental points introduced by the transformation itself.* Mutually residual sets of points, and corresidual
sets, preserve their relation. Hence the group of sets of points
corresidual with any given set becomes of importance. If a given
set of D points lies on a curve of deficiency p, and if a corresidual
set can be found containing k arbitrary points, then these numbers
are connected by the relation constituting the Rfemann-Roch
theorem
k-= D -p + p,
where, is zero if D > 2p - 2.
The totality of all sets of D points corresidual to any one set is
termed a group or series, and is denoted by a symbol gr. Such a
series is called complete. If by any algebraic restrictions a series
is separated out from it, of course that would be called incomplete
or partial. For example, on a plane nodal cubic a series Cg2 is
cut out by all straight lines, incomplete because any three arbitrary points of the curve are corresidual to any other three.
Every series YgD can be cut out upon the fundamental curve by a
linear system of auxiliary curves whose equation may be written,
with k parameters:
F0+ IFl + F2+ +.+ 1,F, =0.
As on a single curve sets of points, so in a plane, linear systems
of curves are studied. By every birational transformation, linear
*Or if we employ no auxiliary curves except such as are adjoint to that containing the point sets.