Fulton characterizes the intersection number, I(P, E∩F), with seven properties. Let me just repeat the last one:
- I(E∩F) = I(E∩(F+AE)) for any A.
Fulton characterizes the multiplicity I(E∩F) with seven properties (§3.3). (Fulton calls it the intersection number. Also, he writes I(P,E∩F) for the intersection number at P. I’ll usually assume P is the origin O, and omit writing it.) The last three properties stand out:
Last time we looked at Kendig‘s first definition of multiplicity. A branch of E, parametrized by (xE(t),yE(t)), passes through the origin O, as does the curve F. Assume xE(t) and yE(t) are power series in t. Plug them into the polynomial F(x,y), getting a power series F(t). The order of F(t) (the degree of the first nonzero term) is the multiplicity of that intersection—that is, of the branch of E with the entire curve F at O.