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.
Monthly Archives: April 2020
Algebraic Geometry Jottings 6
The tome Commutative Algebra by Zariski and Samuel opens with the memorable sentence, “This book is the child of an unborn parent.” As Zariski explains,
Filed under Algebraic Geometry
Algebraic Geometry Jottings 5
We’ve been looking at Kendig‘s two definitions of intersection multiplicity; now let’s look at Fulton‘s.
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:
Filed under Algebraic Geometry
Algebraic Geometry Jottings 4
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.
Filed under Algebraic Geometry
Algebraic Geometry Jottings 3
Bézout’s theorem requires us to count intersection points according to their multiplicity. OK, what’s multiplicity? (Fulton uses the phrase intersection number.)
Filed under Algebraic Geometry
Algebraic Geometry Jottings 2
As I said last time, I’m learning some algebraic geometry, starting with Bézout’s theorem, and using Fulton’s Algebraic Curves and Kendig’s A Guide to Plane Algebraic Curves as the texts. Right now we’re looking at this example from Fulton:
Filed under Algebraic Geometry
Diagonal Argument 