Reference no: EM131083962
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, . . . , xn] 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) = mP1 ∩ mP2 ∩ mP3, where I = (x2 - 2xy4 + y8, y3 - y) and mPi = I({Pi}).
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