Colloquium publications.

LINEAR SYSTEMS OF CURVES.

of curves is still susceptible of precise definition, and that in two
ways whose equivalence constitutes an important theorem.
If on any surface, rational or not, there exists a system of
curves doubly infinite, such that two arbitrary points determine
one and only one curve containing them, that may be termed a
linear net upon the surface in question; and Enriques proves that
the o02 curves of such a system can be projectively related to the
straight lines of a plane. If the series is oo3, and if three arbitrary points determine uniquely a curve of the system which shall
contain them, then its curves are referable projectively to the
planes of three-space, etc. Only simply infinite systems escape
this far-reaching theorem, and thus give rise to a most interesting
unsettled question, indicated by Castelnuovo.*
Definitions of residual and corresidual curves upon a surface
are those which any one could formulate at once from the use of
these terms for sets of points upon a curve; their significance
upon a twisted curve is the same as upon its plane projection.
So of complete systems, both of curves and of surfaces, the latter
admitting of course multiple curves as well as base points. For
a surface of order m, the adjoints invariantively related are of
order n - 4, containing as (s - 1)-fold curve every s-fold curve
of the given surface. If these first adjoint surfaces form a k-fold
infinite linear system, the number k is an invariant of the surface
and is termed its geometric deficiency (pg). Attempting to express
this number in terms of the order m of the surface, the order d
and deficiency tr of its double curve (if any), and of the number
t of triple points on this double curve, one would find a second
number
Pn = i(m - 1) (m - 2) (m - 3) - d (m - 4) + 2t + w - 1,
called the numerical deficiency of the surface. This number also
is an invariant of the surface, as Noether first proved, and may
* Castelnuovo: " Alcuni risultati sui sistemi lineari di curve appartenenti ad
una superficie algebrica." Memorie di matematica e di fisica della Societd Italiana
delle Scienze, ser. 3, vol. 10 (1896), pp. 82-102. See especially the close of his
preface.