Colloquium publications.

6

THE BOSTON COLLOQUIUM.

be either equal to or less than p9, but never greater. Rational
surfaces have pg =p - 0; ruled surfaces have pl negative. If
Pg = p., then the above-mentioned theorem of Enriques concerning linearity holds true also for systems which are only simply
infinite. Surfaces of the first adjoint system cut out upon a given
surface a system of curves, each of deficiency p(') or less. This
invariant number p(') we may call the canonical deficiency of the
surface; the curves form an unique complete linear system, just
as do the point sets of the canonical series on a plane curve.
The definitions here given are but a part of those found useful
in this fascinating branch of geometry. The true way to learn
something of the subject is not to master first all its definitions
and distinctions, but to study the proofs of some few leading
theorems. Such are Enriques's proof of the equivalence of two
geometrical definitions of the linearity of a system (mentioned
above), and the following less elementary propositions:
1. Surfaces whose plane or hyperplane sections are irreducible
unicursal curves are either ruled or rational (Noether).*
2. So also surfaces whose plane or hyperplane sections are irreducible elliptic curves (Castelnuovo),t or hyperelliptic of any deficiency r (Enriques).t For plane sections, not hyperelliptic, of
deficiency z > 2, the corresponding theorem is not yet fully
known.~ The proof of this theorem I shall give in full.
3. Upon any algebraic surface f (x, y, z, zt)= 0 a linear differential of first kind is said to exist (Picard), if an expression involving four rational functions P,, P2, P3, P4, of the coordinates:
f [P1 (, y, z, t) l dx + P, l dy + P3, ' + PI cdt]
is finite and determinate, independent of the path of integration,
*Noether's theorem is more general. See lMath. Anrnalen, vol. 3 (1871):
"Ueber Flachen, welche Schaaren rationaler Curven besitzen."
t "Sulle superficie algebriche," etc., Lincei Rendiconti, January, 1894.: "Sui sistemi lineari," etc., Math. Annalen, vol. 46 (1895), pp. 179-199.
Q For full information, see the second paper, cited above, of Castelnuovo and
Enriques. I regret that this paper had not come to my notice before giving these
lectures.