There are many proofs of the stable reduction theorem for curves. A good overview is given in Abbes : Réduction semi-stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, …. I personally found the introduction of Temkin’s paper Stable modification of relative curves quite helpful. Yet another proof (not discussed in the references just given) can be found in a preprint by Kai Arzdorf and Stefan Wewers entitled “Another proof of the Semistable Reduction Theorem”. In this blog post I’d like to discuss their argument (up to a point).
General remark: The goal on this blog is not new or original research, but rather the goal is to understand material in a way that is easy to explain with what is currently available in the Stacks project.
Let’s start with a complete discrete valuation ring R with fraction field K whose residue field k is algebraically closed. Assume we have a smooth projective geometrically irreducible curve C over K. Then we can choose a flat projective sheme X over R whose generic fibre is C. We may normalize X and assume X is a normal scheme. Denote X_0 the special fibre. This is a projective connected scheme over k which satisfies (S_1).
During the proof we will finitely many times
- replace R by the integral closure R’ of R in a finite K’/K and X by the normalized base change X’, and
- replace X by a normalized blowup of X.
The assertion of the stable reduction theorem is that in doing so we can get to a situation where X_0 has at worst nodes as singularities.
Step 1. A theorem of Epp guarantees that we can do an operation of type 1 to get reduced special fibre X_0. See for example Tag 09IJ. We observe that any further base change of X (by an extension of dvrs) is normal, so that in particular the isomorphism type of the special fibre is preserved under this operation.
Step 2. If X_0 is reduced, then for a closed point p \in X_0 we have two local invariants: the number of formal branches m_p of X_0 at p and the δ-invariant δp.
Step 3. Let p be a singular point of X_0 with δp > 1. Choose a normalized blow up Y —> X at p such that each of the formal branches B_i at p lifts to a nonsingular branch at some point q_i of Y lying over p and all the q_i are distinct. (Of course you have to show such a blow up exists, but I think this is completely standard. Moreover we have results like this available in the Stacks project; for example Tag 080P can be used to separate the branches and the proof of Tag 00P8 gives a sequence of blowups that normalize the branches.) Of course now Y no longer needs to have reduced special fibre. Thus we replace Y and X by normalized base changes so that both X and Y have reduce special fibre.
Step 4. Consider the map Y —> X we produced in the previous step. To finish the proof it suffices to show that the maximal local δ-invariant of Y at points mapping to p is < δp.
To do this my first guess was to try and prove following criterion (this is probably wrong, although I have no counter example):
If R1f*OY0 has nonzero stalk at p, then Step 4 works.
It is easy to see that the vanishing of R1f*OY0 has all kinds of pleasurable consequences (such as the rationality of the irreducible components of Y0 lying over p — kind of like having a rational singularity at p), so there is hope we can prove that this vanishing isn’t possible if δp > 1.
To prove the statement quoted in italics above, we try to use the relationship between local invariants and the genus of the curve plus the fact that X_0 and Y_0 have the same (arithmetic) genus. However it won’t work. Let’s take the example suggested by Anand where X_0 is a cuspidal rational curve. Then what might happen is that Y_0 is a smooth rational curve (the normalization of X_0) attached transversally to a cuspidal rational curve (the exceptional fibre). Here we could end up doing an infinite sequence of normalized blow-ups over and over again. Moreover, somehow at each step there is a unique point where you blow up.
This kind of problem is solved in the paper by a (more) careful choice of the ideal to blow up in (I haven’t yet looked enough at the paper to understand how). In resolution of singularities of surfaces, a similar problem comes up. For example, look at the discussion of resolution of rational double points in Artin’s paper “Lipman’s proof of ….”. The problem is dealt with by showing that an infinite sequence of infinitely near but not satellite points gives a nonsingular arc passing through the singular point and showing that along a nonsingular arc the procedure of normalized blowups always works. I think the references mentioned at the beginning of this post, tell us a similar method will work here, but the question is how difficult it will be.
I’ll add updates here if I think about it more.