Math 176: Algebraic Geometry, Fall 2014- Assignment 7
1. Consider V(yz - x2) in P2. What type of conic does this algebraic set look like in the 3 natural affine sections of P2 (i.e. in U0, U1, U2)? What are the points at infinity (i.e. the points where z = 0)?
2. Let V = V(x0x2 - x3x4, x20x3 - x1x22) ⊂ P4. Find equations for the affine algebraic sets V ∩ U0 and V ∩ U3. Are these varieties?
3. Determine, with justification, if the following statements are true or false. Again we assume k is an algebraically closed field with characteristic 0.
(a) A homogeneous prime ideal is radical.
(b) Ia(C(V)) = I(V) for any projective variety V ⊂ Pn.
4. If Y ⊂ An is an algebraic set, recall the Zariski closure is the smallest algebraic set in An containing Υ. We saw on the midterm that this is V(I(Υ)). What is a natural candidate for the projective closure of Υ? Justify why your choice of definition for projective closure is reasonable.