Homework for Week 4

Part 1

For each of the following, what is the dimension and what are the singular points?

1. x² = x⁴ + y⁴
2. xy = x⁶ +y⁶

Part 2

1. Define the map f: A → A² by f(t)=(t², t³). Show that f is a bijective continuous map with continuous inverse from A onto the curve y²=x³. Nonetheless, show that f is not an isomorphism.
2. Let X be the curve defined by y²=x²+x³. At which points of X is the rational function y/x regular? Prove that y/x is not in k[X].

• Part 1: Due Tuesday
• Part 2: Due Thursday