The classical algebro-geometric way of studying the singularity of a germ of plane curve is to analyze its behaviour when intersected with other germs. In this chapter we begin the study of the intersection behaviour of (singular) plane germs: this performs a deep geometrical analysis of singularities which will lead to their classification. Our main tool in this study will be the points infinitely near to a point O on a smooth surface S. Infinitely near points were first introduced by M. Noether [63] and extensively studied by Enriques ([35], book IV). In spite of the fact that infinitely near points lie on different surfaces, the whole set of them provides a sort of infinitesimal space which displays the local geometry at O of the curves on S.

Classical authors used to define infinitely near points by means of different kinds of birational transformations dependent on many arbitrary choices, which nevertheless led them to basically correct and consistent results. The modern definition we shall present in this chapter follows the same idea but uses a single type of transformation, named blowing-up. This makes a definition free of arbitrary choices and avoids some undesired effects of the classical transformations.

Blowing up

Once a point O on a (smooth) surface S has been fixed, choose local coordinates x, y at O and assume that they are analytic in an open neighbourhood U of O.