Math 176: Algebraic Geometry, Fall 2014- Assignment 1

1. Sketch the following varieties in R³.

(a) V(xz² - xy)

(b) V(x⁴ - zx, x³ - yx)

2. Let k be a field. For each of the following, determine, with justification, if the given set is an affine algebraic set.

(a) {(t, t², t³) ∈ A ³(k) | t ∈ k}.

(b) {(x, x) ∈ A² (l) | x ≠ 0}.

(c) The set of n × n non-invertible matrices whose entries are in k. (This is considered a subset of A^{n^2} (k).

3. Let C = V(f) for some f ∈ k[x, y] where deg(f) = n ≥ 2. Suppose L is a line in A²(k) with L ⊄ C. Show that L ∩ C is a finite set of no more than n points. (Hint: Suppose L = V(y - (ax + b)) and consider f(x, ax + b) ∈ k[x].)

4. Suppose V ⊂ Aⁿ(k) and W ⊂ A^m(k) are affine algebraic sets. Prove V × W, which is defined as

V × W := {(v₁, . . . , v_n, w₁, . . . , w_m) | (v₁, . . . , v_n) ∈ V, (w₁, . . . , w_m) ∈ W} is an affine algebraic set in A^m+n(k).