This will be a “chewing” post.
Category Archives: Algebraic Geometry
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.
I’ve decided to learn algebraic geometry. Or at least more algebraic geometry—I’m not starting from zero. But I’m still sampling the appetizers; I’m using Fulton’s Algebraic Curves and Kendig’s A Guide to Plane Algebraic Curves as my initial texts. Eventually I’d like to understand schemes, but that’s dessert; I plan on making a long, leisurely meal of it, with plenty of time savoring examples and history, and chewing proofs to extract all the flavor. (Can you tell I’m writing this before lunch?)