\overline{M}_{g} IS IRREDUCIBLE

RAVI VAKIL

April 1998: The original note was from some time in 1996. I’ve edited it slightly, and removed excessively naive (or wrong) statements.

We will show that \overline{M}_{g} is irreducible (in characteristic 0) using semistable reduction and minimal facts about \overline{M}_{g}. This idea was provoked by a comment Joe Harris once made about the power of tightly-controlled codimension 1 degenerations. I have since looked at Fulton’s two-page note “On the Irreducibility of the Moduli Space of Curves”, and I realize that this note is only the additional observation that we don’t even need to invoke anything like the compactification of the Hurwitz scheme.

0.1. A degeneration question. Fix a point ∞ ∈ ℙ^{1}, and consider a regular curve C with a degree d map π to ℙ^{1}, with r simple ramifications away from ∞, and π^{−1}(∞) a union of p distinct points. Call these p points “∞-sections”. Then by Riemann-Hurwitz, we have

r = d + 2g + p − 2  (1)

ramification points away from ∞. Move one of the ramification points to ∞, keeping the others fixed. We can use the (characteristic 0) recipe for semistable reduction to find a limit map from a nodal curve, after an appropriate base change.^{1} (Essentially, take the limit stable map.) By base-changing at the start, we may assume that the ∞-sections are distinguishable.

The limit curve has two parts, one part C′ consisting of components mapping dominantly to ℙ^{1}, and the other C^{∞} of components mapping to ∞. By blowing up further, we may assume C′ is regular, and that the total space of the family is regular (so the ∞-sections have limits that are regular points of the central fiber). Let g′ and g^{∞} be the arithmetic genera of C′, C^{∞} respectively, n the number of nodes where C′ and C^{∞} meet, p′ the number of preimages of ∞ on C′, β the number of the p ∞-sections whose limit is in C^{∞}. Of the p′ pre-images of ∞ in C′, n of them are points of intersection with C^{∞}, and the rest are limits of the remaining p − β ∞-sections of the general curve (which could theoretically come together). Hence

Date: April 27, 2000.

1In a nutshell, take any limit map — where the total family of curves if flat, and there is a family of maps, but the central fiber may have non-reduced components and various singularities. Blow up the surface until it is regular and the central fiber is set-theoretically nodal, make a base change of order the lcm of the multiplicities of the components of the central fiber, normalize, and then blow down (-1)-curves (of the family) on the central fiber that don’t map dominantly to ℙ^{1}.

p′ ≤ n + (p − β).  (2)

By Riemann-Hurwitz for the curve C′ (similar to (1)), r −1 = d+2g′ + p′ − 2, so (comparing with (1))

2g′ + p′ = 2g + p − 1.  (3)

As the arithmetic genus of the central fiber is g,

g′ + g^{∞} + n − 1 = g.  (4)

By combining (2)-(4) ((2) - (3)+ 2 (4)),

2g^{∞} − 2 + n + β ≤ 1.  (5)

For the j^{th} of the (say, k) connected components of C^{∞}, let g_{j}^{∞} be the arithmetic genus, let n_{j} be the number of intersections with C′, and let β_{j} be the number of (limits of) ∞-sections on it. Then (5) can be restated as ∑_{j=1}^{k}(2g_{j}^{∞} − 2 + n_{j} + β_{j}) ≤ 1. As the central fiber is connected, n_{j} ≥ 1. Also, for each connected component of C^{∞}, at least one of the p ∞-sections must lie on it.^{2} Hence ∑_{j=1}^{k} 2g_{j}^{∞} ≤ 1, so the arithmetic genus of each connected component of C^{k} must be 0. Thus all collapsed components of the central fiber are rational. (We can conclude more, but we won’t need to for our purposes.)

0.2. Brief sketch of irreducibility argument. By the usual arguments, we need only show that any regular genus g curve C can be degenerated to a nodal stable curve. (By this, I mean that the component of \overline{M}_{g} containing [C] also contains a nodal curve.) Map C to ℙ^{1} so that all ramification is simple. Fix a point ∞ ∈ ℙ^{1}. Specialize the ramifications to lie over ∞ one at a time. If it breaks into pieces where the stable model has a genus g component, this component must map dominantly to ℙ^{1}, so we’ll throw away the “rational tails” and continue. (This may decrease d.) This process can’t continue forever, as no dominant morphism from a genus g curve to ℙ^{1} can be ramified over only one point.

2This is intuitively clear to me, but I haven’t thought of a two-line argument, although I’m sure one exists. Here’s a longer argument. Assume otherwise that C_{j}^{∞} is a connected component of C^{∞} not meeting any ∞-section. The pullback of O_{ℙ^{1}}(∞) to the universal family has degree 0 when restricted to C_{j}^{∞}. The pullback of the divisor ∞ is a positive linear combination of irreducible components of C_{j}^{∞} (plus other components not meeting C_{j}^{∞}). But each irreducible component of C_{j}^{∞} has non-positive intersection number with C_{j}^{∞}, and at least one has strictly negative intersection (as (C_{j}^{∞})^{2} = −n_{j}). Thus we have a contradiction.