### Regular Linear Systems of Curves with the Singularities of a Given Curve as Base Points

American Journal of Mathematics, Vol. 54, No. 3
(July 1932), 471-488.

Summary

An irreducible algebraic curve of order m, f(x,y) = 0, in the plane π(z=0) is given with singularities of a specified kind. The surface F, z^{n} = f(x,y) where n is a positive integer, gives rise to adjoint surfaces φ_{ν}, and to complete linear systems of surfaces φ_{ν}—dπ which together with the plane π taken d times form adjoints φ_{ν}. These systems of surfaces cut the plane π in systems of Curves | C^{d} |, shown to be complete. The irregularity of the surface F can be expressed in terms of the superabundances of some of the systems | C^{d} |, hence application of the theorem that F is regular if n is a power of a prime shows the regularity of the systems of plane curves involved. For f(x,y) = 0 having only nodes and cusps, Zariski obtains in this manner regular systems of curves with the cusps of f as base points. In this paper we consider f possessing either of two types of singularities:—ordinary k-fold points and k-fold points each of which is the origin of one branch of order k. The result theorems are analogous to the well-known theorem: Curves of order m–3 with (k–1)-fold points at the k-fold points of f form a regular system; we prove that curves of order m–3–j) with (k–i)-fold points (i = 1, 2, ..., k–1) at the k-fold points of f form regular systems for j having a certain range of values. Thus restrictions are imposed on the number and position of specified singularities for f. Further, since the surface F is regular for all positive integral values of n when f(x,y) has a cyclic fundamental group with respect to its carrying complex plane, the connection between the irregularity of F and the super-abundances of the linear systems described enables us to state sufficient conditions that f(x,y) = 0 have a non-cyclic fundamental group; these conditions show that there exists a connection between the position of the singular points other than nodes and the structure of the fundamental group.

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