Math 176: Algebraic Geometry, Fall 2014- Assignment 4
1. Use polynomial maps to prove V(xz - y2, yz - x3, z2 - x2y) ⊂ A3 is a variety.
2. (a) Let V = V(xy - z) ⊂ A3. Prove V is isomorphic to A2 and provide an explicit isomorphism φ and associated k-algebra isomorphism φ∗ from Γ[V] to Γ[A2], along with their inverses.
(b) Is V(xy - z2) isomorphic to A2? Prove or disprove.
3. Let f ∈ k(V), P ∈ V, and f = f1/f2 where f1, f2 ∈ Γ(V).
(a) Suppose the fraction f1/f2 is a witness that f is defined at P. Prove that if f is written as f'1/f'2 where f'I ∈ Γ(V) for each i, then f'2(P) ≠ 0. (In other words, show that being defined at P is well-defined.)
(b) If f2(P) = 0 and f1(P) ≠ 0, prove P is a pole of f .
4. Let V = V(y2 - x3 - x2) ⊂ A2, and let z = y/x ∈ k(V). Find the pole set of z and z2.