Definition of intersection multiplicity of a curve with some hyperplanes

Question

I'm studying the chapter 2 of this paper and I have the following doubt:

What is the definition of intersection multiplicity of a curve $C$ with some hyperplanes at a point $P$?

Remark: My only background is Fulton's Algebraic curves book.

I really need help.

Thanks

Answer 1

@user64687 gives an adequate description for many purposes.

Historically, the definition of intersection multiplicity has not been easy/straightforward. Some of this history can be followed through André Weil's "Foundations of Algebraic Geometry" (1946) and William Fulton's "Intersection Theory" (1984). I recall (but perhaps recall fuzzily) that Jean-Pierre Serre's "Algèbre locale, Multiplicités" (1965) definition via "Serre's Tor-formula" finally captured what was generally meant by the term entirely rigorously. (Aha. The Wikipedia article Intersection Theory: Intersection multiplicities seems to agree, so I may be correctly remembering. This section of this article has a version of this definition. This section also has a diagram demonstrating a place where user64687's description is perhaps vastly more enlightening.)