In my class, the lecturer told me that in order to calculate the intersection multiplicities for multiple space curves, sometimes I may have to calculate the Gröbner Basis. I just can't see how calculating the Gröbner Basis helps to find the intersection multiplicities.
How do I guarantee that if I have two algebraic curves $F$ and $G$ then the Gröbner Basis of $(F,G)$ still consists of two curves? Because if not, then how do I calculate the intersection multiplicities?
Also, can you give an example of algebraic curves such that calculating the Gröbner Basis is much more easier than calculating the intersection multiplicities?