Find the pole set of z, Math 176: Algebraic Geometry, Fall

Find the pole set of z

Reference no: EM131083988

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/f₂where f₁, f₂ ∈ Γ(V).

(a) Suppose the fraction f₁/f₂is 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.

Reference no: EM131083988

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