Suppose we have a pencil of plane curves of degree – how many are singular? In this note, we discuss a purely topological approach to this enumerative problem, which is juxtaposed against the “usual” technique involving Chern classes. Throughout, for simplicity, we assume our base field is (by the Lefschetz Principle, our discussion carries over to all algebraically closed fields of characteristic zero.)
As a warmup, we start with the Riemann-Hurwitz formula. Let be a smooth curve of degree , and let be a branched cover of degree . We want to determine a relationship between the Euler characteristics of the curves; to do this, we start with the simplest case of an unramified cover. Purely topologically, if is a -fold covering space, then their Euler characteristics satisfy as the Euler characteristic is additive for disjoint unions. When ramification enters the picture, we modify the above formula with an error term: where is the ramification index at the point .
To generalize Riemann-Hurwitz, we can think of it as a relationship between the topological Euler characteristics of a covering space and a base, up to some contribution from ramification behavior. More generally, though, for a fiber bundle with fiber , we have the following relation: and we recall that covering spaces are simply fiber bundles with discrete fibers. Thus, we may approach a form of Riemann-Hurwitz that applies to morphisms that are fiber bundles away from finitely many points.
Let be a smooth, projective variety, and let be a map to a smooth curve of genus . Since we are in characteristic , there exist finitely many points such that the fiber is singular. Let be the divisor of given by the (disjoint) union of these fibers, i.e. Then, , and since the open set is a fiber bundle over , we have that where is a general point of .
By additivity of the Euler characteristic on disjoint unions, we have that which by the previous calculations, we can rewrite as Rephrasing the first right-hand summand and extending the latter over all points of , we get the generalized Riemann-Hurwitz formula: That is, we get the expected relation of if were a true fiber bundle over , with an error term that counts the deviation of the Euler characteristic of each singular fiber from that of a generic one.
Let us apply the generalized Riemann-Hurwitz formula to our enumerative problem from the Introduction. Recall the setup: is a general pencil of plane curves of degree . Because we assumed that the polynomials and were general, the base locus of the pencil is reduced points, and the total space of the pencil, expressed as the graph of the rational map , is the blowup of along . Now, is smooth, so is simply projection onto the first factor.
Since is the blow-up of at points, we have Next, we know that a general fiber of the map is a smooth plane curve of degree ; by the degree-genus formula, it has genus , so Now, each singular fiber has a single node as singularity – no more, no worse. Therefore, the normalization of each nodal curve has genus , so it has Euler characteristic . We can view as coming from by identifying two points that form the node, which bumps down the Euler characteristic by one, that is, which is greater than the Euler characteristic of a general fiber. Thus, the contribution of each singular fiber to the generalized Riemann-Hurwitz formula is , and we can count the number of singular fibers by the following calculation: which is the number we want!
[EH16] Eisenbud, D., & Harris, J. (2016). 3264 and all that: A second course in algebraic geometry.
[Ha77] Hartshorne, R. (1977). Algebraic geometry (Vol. 52). Springer-Verlag.
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