Homework for Week 8
Part 1
- Find the projective closure of the affine variety given by y3 + xy + x2=0.
- 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
- Compute the degree of the projective twisted cubic, namely the image of the curve
[x0, x1] → [x03, x02x1, x0x12, x13] in P4.
- 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?
- 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:
- The degree of Y is 1.
- Y is a linear variety (namely Y is the zero locus of linear polynomials).
- Part 1: Due Tuesday
- Part 2: Due Thursday