Homework for Week 2

Part 1

1. For each of the following varieties, determine its dimension and whether it is irreducible. For each reducible variety, describe its irreducible components and their ideals.
   1. {(x,y) ∈ R²: x² + y²=0}
   2. {(x,y) ∈ C²: x² + y²=0}
   3. {(x,y,z) ∈ C³: x² + y²+ z²=0, x² - y²- z²+1=0}
2. Define X to be the zero locus of f(x,y)=x² + y² - 1 and g(x) = x-1. Find the ideal I(X). Is I(X)=(f,g)?

Part 2

Prove the following properties:

1. If S ⊆ T are subsets of k[x₁, ..., x_n], then V(S) ⊇ V(T).
2. If X ⊆ Y are subsets of Aⁿ, then I(X) ⊇ I(Y).
3. If S, T are subsets of k[x₁, ..., x_n], then V(S T) = V(S) ∪ V(T).
4. If T₁, T₂,... are subsets of k[x₁, ..., x_n], then V(∪ T_i) = ∩ V(T_i).

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