Continuous Non-linear Systems

In II, 1 we have accepted for temporary purposes a definition of an algebraic system Σ of curves C on a surface F, as a system cut out on F by an algebraic system Σ′ of hypersurfaces. It will serve as a preliminary clarifying remark if we point out immediately why that definition is not sufficiently general. The base loci of the given system Σ′ of hypersurfaces determine a com­plete linear system of hypersurfaces of the same order as the hyper­surfaces of Σ′. This linear system cuts out on F a linear system of curves of the same order as the curves C and containing Σ. Hence Σ is totally contained in a linear system of curves. However—and this is a fundamental point of the theory, which will be discussed in section 3 of this chap­ter—there exist surfaces (notably, irregular surfaces) which carry algebraic systems of curves not contained in linear systems. The simplest example is given by surfaces carrying an irrational pencil Σ of curves. Obviously, such a pencil (supposing for simplicity that the curves of the pencil are irreducible) cannot be contained in a linear system of dimension r, where r is necessarily ≧2, because the curves of the pencil are of virtual degree zero (II, 1). If the pencil is reducible, it is seen immediately that its curves are composed of the curves of another irrational pencil, and the statement that Σ is not contained in a linear system is essentially equivalent to the statement to the statement that an irrational involution of sets of points on an algebraic curve cannot be contained in a linear series.