Let O be a point on a smooth surface S. As in the preceding chapter, denote by O = OS, O the local ring of S at O and by M its maximal ideal. We say that a family of conditions imposed on germs of curve at O is linear if and only if the germs satisfying these conditions describe a linear system, that is (section 2.7) if and only if their equations describe the set of non-zero elements of an ideal of O. Given any positive integer ν, it is clear that the condition of having multiplicity at least v at O is a linear one, the corresponding ideal of O being just Mν. One may ask the germs to go through given infinitely near points with prescribed multiplicities: if K is a cluster with origin at O and for each p ∈ K there is given an integer νp, we may consider the family of conditions on ξ,

In general such a family is not linear. Indeed, for an easy example, take O and the point p in its first neighbourhood on the x-axis as the points of K; then it is clear that all germs ξλ: x2 − y2 + λy = 0, λ ≠ 0, have at both O and p multiplicity non-less than (in fact equal to) one. Nevertheless, the germ ∈0 does not go through p. In this chapter we introduce families of linear conditions that, although weaker, are to a certain extent similar to the conditions of going through the points of a cluster with prescribed multiplicities.