Math 176: Algebraic Geometry, Fall 2014- Assignment 8

1. Compute each of the following, where P = (0, 0).

(a) I(P, C ∩ D) where C = V(y - x²) and D = V(y² + x³ - x²).

(b) I(P, C ∩ D) where C = V(y² - x³) and D = V(y³ - x⁴).

2. Let C = V(f) where f ∈ k[x, y, z] be an irreducible curve on P².

(a) If f has degree 3, prove that C has at most one singularity.

(b) If f has degree 4, prove that C cannot have three collinear double points.

3. Let P₁, . . . , P₅ ∈ P² and assume no four of them are collinear. Show that there is a unique conic through P₁, . . . , P₅.

4. Let R be a commutative ring with a multiplicative identity 1. Let Spec(R), called the prime spectrum of R, be the set of prime ideals in R. For each ideal I of R, define

V(I) := {p ∈ Spec(R) : I ⊂ p}.

Prove the following:

(a) V({0}) = Spec(R); V(R) = ∅;

(b) V(I) ∪ V(J) = V(I ∩ J) for any ideals I, J of R

(c) V(I) ∩ V(J) = V(I + J) for any ideals I, J of R

What is this problem about?