Homework for Week 10

Part 1

1. Let X be a smooth projective curve and f ∈ k[X]. Consider f as a regular function f: X → P¹. Prove that (f) = f^*(0-∞ ), where 0 - ∞ is the divisor on P¹.
2. Suppose f: X → Pⁿ is a rational map. Can you identify the points where f fails to be regular in terms of divisors?

Part 2

1. Compute the "Chow ring" of the torus S¹ x S¹ (which looks like the surface of a doughnut).
2. Compute the Chow ring of Pⁿ.

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