Keynote Questions

1. (a) Let F0, F1, F2∈ k[X, Y, Z] be three general homogeneous cubic polynomials in three variables. Up to scalars, how many linear combinations t0F0 + t1F1 + t2F2 factor as a product of a linear and a quadratic polynomial? (Answer on page 65.)

2. (b) Let F0, F1, F2, F3∈ k[X, Y, Z] be four general homogeneous cubic polynomials in three variables. How many linear combinations t0F0 + t1F1 + t2F2 + t3F3 factor as a product of three linear polynomials? (Answer on page 65.)

3. (c) Let A,B,C be general homogeneous polynomials of degree d in three variables. Up to scalars, for how many triples t = (t0, t1, t2) ≠ (0, 0, 0) is (A(t), B(t), C(t)) a scalar multiple of (t0, t1, t2)? (Answer on page 55.)

4. (d) Let Sd denote the space of homogeneous polynomials of degree d in two variables. If V ⊂ Sd and W ⊂ Se are general linear spaces of dimensions a and b with a + b = d + 2, how many pairs (f, g)∈ V × W are there (up to multiplication of each of f and g by scalars) such that f | g? (Answer on page 56.)

5. (e) Let S ⊂ ℙ3 be a smooth cubic surface and L ⊂ ℙ3 a general line. How many planes containing L are tangent to S? (Answer on page 50.)

6. (f) Let L? ℙ3 be a line, and let S and T? ℙ3 be surfaces of degrees s and t containing L. Suppose that the intersection S n Tis the union of L and a smooth curve C. What are the degree and genus of C? (Answer on page 71.)

In this chapter we illustrate the general theory introduced in the preceding chapter with a series of examples and applications.

The first section is a series of progressively more interesting computations of Chow rings of familiar varieties, with easy applications. Following this, in Section 2.2 we see an example of a different kind: We use facts about the Chow ring to describe some geometrically interesting loci in the projective space of cubic plane curves.

Finally, in Section 2.4 we briefly describe intersection theory on surfaces, a setting in which the theory takes a particularly simple and useful form.