Math 176 algebraic geometry fall 2014- assignment 3

Question

Math 176: Algebraic Geometry, Fall 2014- Assignment 3

1. Decompose each of the following algebraic sets into irreducible components. Justify that each of your irreducible component is indeed irreducible.

(a) V(y⁴ - x², y⁴ - x²y² + xy² - x³) ⊂ A²(C).

(b) V(x² - yz, xz - x) ⊂ A³().

2 Let I = (x² - y³, y² - z³) ⊂ C[x, y, z]. In this problem we will prove V(I) is a variety.

(a) Show that every element of C[x, y, z]/I can be represented as a + xb + yc + xyd for some polynomials a, b, c, d ∈ C[z].

(b) Define φ : C[x, y, z] → C[t] by φ(x) = t⁹, φ(y) = t⁶, φ(z) = t⁴.

If f = a + xb + yc + xyd for some a, b, c, d ∈ C[z], and φ(f) = 0, conclude f = 0.

(c) Deduce that ker(φ) = I and V(I) is irreducible.

3. Let f ∈ k[x₁, . . . , x_n] and let V = V(f). Suppose V is a variety. Prove there is no variety W with V W  Aⁿ.

4. Let V, W ⊂ Aⁿ be algebraic sets and suppose V ⊂ W. Show that each irreducible component of V is contained in some irreducible component of W.

5. Prove that there are three points P₁, P₂, P₃ ∈ A²(C) such that, rad(I) = m_P1 ∩ m_P2 ∩ m_P3, where I = (x² - 2xy⁴ + y⁸, y³ - y) and m_Pi = I({Pi}).