Keynote Questions

1. (a) If {Ct = V(t0F + t1G) ⊂ ℙ2}tϵℙ1 is a general pencil of plane curves of degree d, how many of the curves Ct are singular? (Answer on page 253.)

2. (b) Let {Ct ⊂ ℙ2}tϵℙ2 be a general net of plane curves. What is the degree and geometric genus of the curve Г ⊂ ℙ2 traced out by the singular points of members of the net? What is the degree and geometric genus of the discriminant curve = {t ϵ ℙ2|Ct is singularg? (Answer in Section 7.6.2.)

3. (c) Let {C ⊂ ℙr be a smooth nondegenerate curve of degree d and genus g. How many hyperplanes H ⊂ ℙr have contact of order at least r + 1 with C at some point? (Answer on page 268.)

4. (d) If {Ct ⊂ ℙ2}tϵℙ1 is a general pencil of plane curves of degree d, how many of the curves Ct have hyperflexes (that is, lines having contact of order 4 with Ct)? (Answer on page 405.)

5. (e) If {Ct ⊂ ℙ2}tϵℙ4 is a general four-dimensional linear system of plane curves of degree d, how many of the curves Ct have a triple point? (Answer on page 257.)

In this chapter we introduce the bundle of principal parts associated with a line bundle L on a smooth variety X. This is a vector bundle on X whose fiber at a point p ϵ X is the space of Taylor series expansions around p of sections of the line bundle, up to a given order. We will use the techniques we have developed to compute the Chern classes of this bundle, and this computation will enable us to answer many questions about singular points and other special points of varieties in families. We will start out by discussing hypersurfaces in projective space, but the techniques we develop are much more broadly applicable to families of hypersurfaces in any smooth projective variety X, and in Section 7.4.2 we will see how to generalize our formulas to that case.

In the last section (Section 7.7) we introduce a different approach to such questions, the “topological Hurwitz formula.”.