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PMATH 764
Algebraic Curves
Instructor: Rahim Moosa
Term: Winter 2012 (1121)
University of Waterloo
Finalized: April 2012
Disclaimer: These notes are provided as-is, and may be incomplete or contain errors.
Contents
1 Introduct

PMATH 764, Algebraic Curves
Assignment 7, due April 2
Problems
Do the following problems from Fultons book:
5.18. Show that there is only one conic passing through the ve points [0 : 0 : 1], [0 : 1 : 0], [1 : 0 : 0],
[1 : 1 : 1], and [1 : 2 : 3], show tha

PMATH 764, Algebraic Curves
Assignment 3, due February 10
Problems
Do the following problems from Fultons book:
2.17. Let V = V (Y 2 X 2 (X + 1) A2 , and X, Y the residues of X, Y in (V ); let z = Y /X k(V ). Find
2
the pole sets of z and of z .
2.18.

PMATH 764, Algebraic Geometry
Assignment 2, due February 3
Problems
Do the following problems from Fultons book:
1.25. (a) Show that V (Y X 2 ) A2 (C) is irreducible; in fact, I(V (Y X 2 ) = (Y X 2 ). (b) Decompose
V (Y 4 X 2 , Y 4 X 2 Y 2 + XY 2 X 3 ) A2

PMATH 764, Algebraic Curves
Assignment 4, due March 2
Problems
1. Do the following problems from Fultons book:
2.14. A set V An (k) is called a linear subvariety of An (k) if V = V (F1 , . . . , Fr ) for some polynomials
Fi of degree 1. (a) Show that if T

PMATH 764, Algebraic Curves
Assignment 6, due March 23
Problems
Do the following problems from Fultons book:
4.4. Let I be a homogeneous ideal in k[X1 , . . . , Xn+1 ]. Show that I is prime if and only if the following
condition is satised; for any forms

PMATH 764, Algebraic Curves
Assignment 5, due March 9
Problems
Do the following problems from Fultons book:
3.19. A line L is tangent to a curve F at a point P if and only if I(P, F L) > mP (F ).
3.20. If P is a simple point on F , then I(P, F (G + H) min

PMATH 764, Algebraic Geometry
Assignment 1, due January 23
Problems
1. Do the following problems from Fultons book:
1.10. Give an example of a countable collection of algebraic sets whose union is not algebraic.
1.11. Show that the following are algebraic