This chapter is devoted to studying pencils and linear systems of germs of curve. We introduce their clusters of base points and prove that all but finitely many germs in a pencil and generic germs in a linear system go sharply through the corresponding clusters of base points. This allows us to prove restricted versions (for pencils and linear systems) of the µ-constant theorem. Linear systems are then used for determining which monomials in the equation of a reduced germ ξ are irrelevant to either the equisingularity type (E-sufficiency) or the analytic type (A-sufficiency) of the germ.

Linear series on a projective line

In this section we recall some elementary facts about linear series on a complex projective line. They will be applied to the linear series cut out by pencils and linear systems on exceptional divisors.

Let ℙ1 be a one-dimensional complex projective space, and assume that X0, X1 are projective coordinates on ℙ1. An effective divisor or group of points of ℙ1 is a formal sum of finitely many points in ℙ1 or, equivalently, an almost zero map from ℙ1 into the set of the non-negative integers. A group of points D is thus uniquely written

where the coefficients np are non-negative integers, all but finitely many equal to zero. The integer np is usually called the multiplicity of p in D. Points of positive multiplicity are said to belong to D. Points p with np = 1 (resp. np > 1) are called simple (resp. multiple) points of D and ∑pnp is the degree or the number of points of D.