(a) Let S ⊂ ℙ3 be a smooth cubic surface. How many lines L ⊂ ℙ3 are contained in S? (Answer on page 253.)
(b) Let F and G be general homogeneous polynomials of degree 4 in four variables, and consider the corresponding family of quartic surfaces in ℙ3. How many members St of the family contain a line? (Answer on page 233.)
(c) Let F and G be general homogeneous polynomials of degree d in three variables, and let be the corresponding family of plane curves of degree d. How many of the curves Ct will be singular? (Answer on page 268.)
In this chapter we will introduce the machinery for answering these questions; the answers themselves will be found in Chapters 6 and 7.
Introduction: Chern classes and the lines on a cubic surface
Cartier divisors—defined through the vanishing loci of sections of line bundles— are of enormous importance in algebraic geometry. More generally, it turns out that many interesting varieties of higher codimension may be described as the loci where sections of vector bundles vanish, or where collections of sections become dependent; this reduces some difficult problems about varieties to easier, linear problems.
Chern classes provide a systematic way of treating the classes of such loci, and are a central topic in intersection theory. They will play a major role in the rest of this book. We begin with an example of how they are used, and then proceed to a systematic discussion. To illustrate, we explain the Chern class approach to a famous classical result:
Theorem 5.1.Each smooth cubic surface in ℙ3 contains exactly 27 distinct lines.
Sketch: Given a smooth cubic surface X ⊂ ℙ3 determined by the vanishing of a cubic form F in four variables, we wish to determine the degree of the locus in G(1, 3) of lines contained in X.
We linearize the problem using the observation that, if we fix a particular line L in ℙ3, then the condition that L lie on X can be expressed as four linear conditions on the coefficients of F.