How does one intersect non-transverse divisors on Mg-bar.

Question

Let M_g-bar be the Deligne-Mumford compactification of genus g curves, and let δ₁ be the divisor of degenerate curves of the form `genus 1 meeting a genus g-1 transversely".

Question 1: What is the n-fold self intersection of δ₁?

I've seen mentioned in the literature that the answer for n = g is rational curves with g many genus 1 tails. This certainly seems reasonable -- this cycle has the right dimension and isvisually' obtained by g-1 many degenerations.

Question 2: Let C be a curve in M_g-bar contained in δ₁. How do I calculate the intersection of C and δ₁?

I'm interested in the case where C is a family of rational curves with g many fixed elliptic tails (so that each elliptic tail is the same elliptic curve), each with at most two rational components. The answer is in the literature, but without proof. Does anyone know a reference or a quick way to do this calculation?

Answer 1

Both questions reduce to showing the 2-fold intersection of D_1 is a the closure of the locus of genus g-2 curve with two elliptic tails.

The first question follows from this claim by induction and using the map D_1 -> M(g-1)^bar x M(1,1)^bar.

The second is a direct application of the claim.

Which brings us to the claim: Instead of working on Mg^bar - which is very hard, you can rigidify the question, and work on some compactified Hurwitz scheme. The Hurwitz scheme H(g,d) is the scheme of degree d covers of P^1 by genus g curves, with simple ramficiation only. The crucial facts here are:

• Given branch points on P^1, there are only finitely many curves with these branch points.

• You can compactify this space: when branch points colide you add a "buble": a copy of P^1, connected to the original P^1 where these points colided, and marked ramification points on the buble. You now have a node connecting the two P^1s; the rule is that the over this nodes, the ramification pattern is identical for the curves lying over the buble and the original P^1. Needless to say, you can add as many bubles as you want.

The pullback of D1 to H(g,d) is the closure of the locius of a pair of P^1's, such that: there are 2(g-2+d)-1 marked branch points on one of them, and 2d-1 marked branch points on the other. This is a purely combinatorial condition. Intersecting the condition with itself you get the closure of the loci of a chain of 3 P^1s, with 2d-1, 2(g-3+d) -2, 2d-1 points. Which is a pullback to H(g,d) of the expected class.

Note that we worked with pullbacks, so we have an equality of classes, not of the underlying reduced schenes.