Homework 7
Due Wednesday, March 12
1. For each of our examples of DVRs in class, identify a uniformizing parameter.
2. (a) Prove that in a general DVR, the uniformizing parameter is unique up to multiplication
by a unit. (This means that if s and t are bo
Homework 9
Due Wednesday, April 9
1. (Fultons 3.17, partial) Find (with proof, of course) the intersection number at P := (0, 0) of
each curve in cfw_A, B, C, D below with E and F . Try I (E, F ) too. (Its done in the book, but
its fun to play with.)
2. C
Homework 10
Due Wednesday, April 16
1. Let F := Y 2 X 3 + X, G := X 2 Y 2 1.
(a) Find all points of intersection of F and G in A2 (C) and the intersection numbers at each
point.
(b) Find all innite points on F and G, that is, points in P2 (C) that are not
Homework 8
Due Wednesday, March 26
d
1. Your formulae and proofs for dimk k [X, Y, Z] / X, Y, Z last week were generally correct,
but they lacked vision. To encourage you to obtain such vision, this week you have to nd
d
and prove a (closed form) formula
1
Algebraic geometry notes c 2014 by Will Murray and California State University. Handout 4, Wednesday, March 19, 2014.
Background for proofs of intersection properties
Correspondence theorem: Let V An be a variety, let I := I(V ), and let P V . Recall
th