Chow stability of curves

This post is part of my effort to understand what is currently the cheapest way to prove semi-stable reduction of curves. One possible approach, championed by Gieseker, is to prove it using GIT stability. To explain the strategy, fix a genus g > 1. Fix an integer d >> g. Set P(t) = dt + 1 – g ∈ Z[t]. Suppose we can prove

  1. the Hilbert point of any smooth projective curve C of genus g embedded by the full linear system of a degree d line bundle is GIT stable, and
  2. every closed subscheme of P^{d – g} with Hilbert polynomial P whose Hilbert point is GIT semi-stable, is a nodal curve.

Then semi-stable reduction pops out. Namely, it is a basic fact of GIT that a specialization of a stable point can be replaced, modulo the group action, to a specialization whose limit is semi-stable (one may in addition assume the limit point has a closed orbit in the semi-stable points). It always seemed to me that proving 1 is more difficult than proving 2, so we’ll focus on that in this post.

I’d like to observe here that if one is only interested in proving semi-stable reduction, then one can play around with the quantifiers above in a fun way. For example, to use the argument above it suffices to prove for a given curve C that there exists some very ample line bundle of large degree such that the corresponding Hilbert point is stable. Next, it suffices to prove this after specializing the curve, hence we may assume that the ground field is finite. In that case there are only finitely many flags to consider in the Hilbert-Mumford criterion, hence this makes some of the uniformity questions that arise trivial. I find this a psychologically useful thing to observe, but I am not sure it actually make things easier to prove.

In a paper by Ross and Thomas it is mentioned that one can get a geometric proof of Chow stability of curves. In particular they claim that one can avoid doing some of the combinatorics that often arise when proving stability. I think I have now understood a simple way to do this (which is, if not equal, very close to what Ross and Thomas say in that article), which I am going to explain here. I haven’t worked out all the details, so caveat emptor.

Let C be a smooth projective curve sitting in P^n. We are going to check the Hilbert-Mumford criterion for stability of its Chow point. To do this we consider closed subschemes X of P^n x A^1, flat over A^1, and invariant under the action of G_m acting by a 1-parameter subgroup of SL_{n + 1} on P^n and by the standard action on A^1. We normalize weights of the action of G_m such that the coordinate t on A^1 has weight 1 (this is, I think, the opposite of the normalization in the paper of Ross-Thomas). Let w_l be the weight of the G_m-action on H^0(X, O_X(l))/tH^0(X, O_X(l)). It is shown in Mumford, Stability of projective varieties, that

w_l = a_0 l^2 + a_1 l + a_2

for l >> 0 and some rational numbers a_i. Mumford also shows that a_0 (up to a factor or 2, 1, or 1/2) is the weight of G_m on the ample line bundle of the Chow variety corresponding to the Chow point of the G_m fixed point corresponding to the special fibre X_0. Thus the Hilbert-Mumford criterion tells us we have to show

for every X as above we have a_0 < 0

Now, we are going to apply Lemma 5.1 of the paper of Ross-Thomas. It says that we can compute a_0 on any modification of X. I claim that with some simple algebraic geometry we can find (after possibly taking an nth root of t and replacing G_m accordingly) a G_m-equivariant map

f : Y —> X

where Y is flat over A^1, where f induces an isomorphism of generic fibres, and where the special fibre Y_0 is a nodal curve. Because the generic fibre is constant with value C, this implies that Y_0 is equal to C union a bunch of chains of P^1’s. Moreover, the G_m-action on these chains is easy to describe: for t -> 0 it pushes points towards the component C. The pullback of O(1) to Y_0 is an invertible sheaf L, equipped with a G_m-action, with the following properties

  1. L is globally generated, and
  2. the weight on H^0(Y_0, L) – H^1(Y, L) is nonpositive.

It takes a bit of work to prove this. Now I claim that this, together with the description of how G_m acts on the chains of P^1’s I mentioned above, implies that the weights on H^0(Y_0, L^l) grows asymptotically as (neg) l^2 unless the scheme X is the trivial degeneration of C. To prove this involves a little bit of combinatorics of the chains of P^1’s but not much. (I’ve gone over this computation twice, and both times it came out correctly. If you’re interested, stop by my office and I’ll explain it to you.)

Although beautiful, this method of proving semi-stable reduction for curves takes too many twists and turns to make it suitable for inclusion into the Stacks project, at least for the moment…

PS: I strongly encourage anybody trying to understand Kempf’s result on uniqueness of destabilizing flags, to look at Sections 2 and 3 of Burt Totaro’s paper on tensor products in p-adic Hodge theory. It is a marvelous piece of mathematical exposition if I ever saw one.

Graded direct sums

A graded (preadditive) category is a preadditive category such that the hom groups have a Z-grading compatible with composition. In Heller’s paper of 1958 he talks about direct sums in graded categories: one requires the projections and the coprojections to be homogeneous (I would also require them to have degree 0 but Heller doesn’t require this).

Today seems to be the day for silly questions, because I was wondering if a graded category which has direct sums as an additive category (i.e., ignoring the grading) necessarily has direct sums as a graded category.

The answer is no (please stop reading here; it won’t get any clearer from here on out). For example, start with a semi-simple abelian category A generated by two non-isomorphic simple objects X and Y. Then consider the graded category Grgr(A) of graded objects of A (see Tag 09MM). Let’s denote [n] the shift functors on graded objects. Then consider the subcategory B of Grgr(A) containing 0, containing arbitrary finite direct sums of shifts of copies of K = X ⊕ Y and containing arbitrary shifts of L = X ⊕ Y[1] and M = X ⊕ Y[2]. Then, forgetting the grading, we see that K ⊕ K is the direct sum of L and M. But, even with the definition in Heller, K ⊕ K is not the graded direct sum of L and M in this category. In fact, the direct sum L ⊕ M in Grgr(A) is not isomorphic to any object of B, but B, viewed as an preadditive category has direct sums.

Graded idempotents

Today, I was on and off wondering about idempotents in Z-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology graded idempotents which refers to idempotents which are homogeneous of degree 1. This suggests that there exist others. And indeed, it is easy to make examples of non-homogeneous idempotents by conjugation with units. But we can ask for more.

  1. Is there an example of an idempotent which is not conjugate to a graded idempotent?
  2. Is there an example of a Z-graded associative algebra with a nontrivial idempotent but no nontrivial graded idempotents?


Some more searching and google finally turned up the paper Idempotents in ring extensions by Kanwar, Leroy, and Matczuk which provides the answer to 1. There’s probably tons of papers that make this observation. Namely, suppose that R is a (commutative) domain such that R[x, x^{-1}] and R don’t have the same Picard group. For example R = k[t^2, t^3] with k a field (details omitted). Let L be an invertible module over R[x, x^{-1}] which is not isomorphic to the pullback of an invertible module from R. Pick a surjection

R[x, x^{-1}]⊕ n —> L

As L is a projective R[x, x^{-1}]-module we obtain an idempotent e in the Z-graded ring M_n(R[x, x^{-1}]) = M_n(R)[x, x^{-1}]. And this idempotent is not conjugate to an element of M_n(R) as that would mean L does come from R.

So this answers 1. I do not know the answer to 2.

4000 pages

Well, actually 4001 pages at this very moment. Also

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