Homework for Week 8

Part 1

  1. Find the projective closure of the affine variety given by y3 + xy + x2=0.
  2. Let Pn be a hyperplane in Pn+1 and let p be a point in Pn+1 - Pn. Define a map f: Pn+1 - p → Pn by letting

    f(q) = the intersection of Pn with the line containing p and q.

    Explain why this map is a well-defined morphism.

Part 2

  1. Compute the degree of the projective twisted cubic, namely the image of the curve
    [x0, x1] → [x03, x02x1, x0x12, x13] in P4.
  2. The affine twisted cubic is the image of the map t → (t,t2, t3), which is defined in A3 by the equations y-x2 and xy-z. Show that the projective twisted cubic is not defined by the equations yw-x2 and xy-zw. (Hint: ...Bezout...) Compare this to the problem on projective closures from Part 1: any thoughts?
  3. Let Y be an algebraic variety of pure dimension r (namely, each irreducible component of Y has dimension r). Show that the following are equivalent:
    1. The degree of Y is 1.
    2. Y is a linear variety (namely Y is the zero locus of linear polynomials).