Colloquium publications.

LINEAR SYSTEMS OF CURVES ON ALGEBRAIC
SURFACES.
BY HENRY S. WVITE.
CHAPTER 1.
Transition from Plane Carves to Suifac(s.
THE notion of equivalence as formulated in projective geometry
has simplified greatly the study of algebraic curves and surfaces,
particularly those of low order. The next step toward a wider
survey is the admission of all birational transformations of the
plane, or of space of three or more dimensions. In the plane, the
theory of Cremona transformations is no longer new, and the
elements are familiar to all students of geometry. Not so, however, in space of more than two dimensions; probably for the
reason that nothing is known analogous to the theorem that a
plane Cremona transformation is resolvable into a succession of
quadric transformations and collineations. And even in plane
geometry the intricacies of the transformations themselves have
kept most students from the matter of higher importance, the
properties of figures that remain invariant under all transformations of the group. Yet there does exist a body of doctrine under
the accepted title of "Geometry on an algebraic curve," and a
fair beginning has been made upon a similar theory, the " Geometry
on an algebraic surface." These titles are intended to cover
*Consult, for an outline of the geometry upon an algebraic curve, Pascal's
Repertorium der hoheren Mathematik, Part II, Chapter V, i 4; or the more
extended articles: C. Segre, "Introduzione alla geometria sopra un ente
algebrico semplicemente infinite"; E. Bertini, "La geoinetria delle serie
lineari sopra una curva piana secondo il metodo algebrico," —both in Annali
di Matematica, ser. 2, vol. 22 (1894). For the corresponding theories regarding surfaces, the best reference is to the comprehensive summary by Castelnuovo and Enriques: "Sur quelques recents resultats dans la theorie des
surfaces algebriques," Math. Annalen, vol. 48 (1896). Supplementary results
are summarized in a later paper by the same authors: "Sopra alcune questioni
fondamentali nella teoria delle superficie algebriche," Annali di matematic: pura
ed applicata, ser. 3, vol. 6 (1901).

1

1