Infinitely near points of algebraic surfaces were introduced by Max Noether (1876).
Infinitely near points can also be defined for higher-dimensional varieties: there are several inequivalent ways to do this, depending on what one is allowed to blow up.
The points of C have the geometric interpretation as the tangent directions at P to S. They can be called infinitely near to P as way of visualizing them on S, rather than S*.
More generally this construction can be iterated by blowing up a point on the new curve C, and so on.
The genus of C is given by where N is the normalization of C and mx is the multiplicity of the infinitely near point x on C.