Purity (part 2)

Please read Purity (part 1) first.

Let f : X → Y be a dominant, finite type morphism of integral Noetherian schemes. We assume X is normal and Y regular. Let Sing(f) be the closed set of points of X where f isn’t smooth.

Question: Is codim Sing(f) ≤ 1 + dimension of generic fibre of f?

The discussion in the previous post shows that the answer is yes when f is a morphism between smooth varieties over a field all of whose fibres have the same dimension and that the bound given is best possible.

Dolgachev proved the answer to the question is yes in case X is a local complete intersection over Y (which happens for example if both X and Y are smooth over a common base scheme). There is a paper of Rolf k\”allstr\”om which has this result and much more.

As we’ve seen in the previous post, the answer to the question is “yes” when f is generically finite, i.e., when the dimension of the generic fibre is 0.

In the rest of this post we discuss the case where the generic fibre has dimension 1.

Let f : X → Y be as above with generic fibre of dimension 1. Let x be a generic point of an irreducible component of Sing(f) with image y in Y. We want to show that dim O_{X, x} is at most 2. To get a contradiction, assume this is not the case.

  1. dim O_{X, x} = dim O_{Y, y} + 1 – trdeg κ(x)/κ(y) by the dimension formula.
  2. If dim O_{X, x} > 3, then dim O_{Y, y} > 2 and we can find a regular divisor Y_0 in Y passing through y = f(x) and thereby reduce the dimension (lots of details missing, but I think this probably can be made to work).
  3. Assume dim O_{X, x} = 3 so dim O_{Y, y} = 2 + t where t is the transcendence degree of κ(x) over κ(y).

At the moment I have nothing intelligent to say in the case t > 0; suggestions are welcome. Assume t = 0.

In this case x is a closed point of the fibre. In particular, we see that, after shrinking X we may assume x is the only singular point of the fibre X_y. An argument similar to the one in miracle flatness shows that O_{Y, y} → O_{X, x} must be flat (here we use X is assumed to be normal).

Thus we see that we have to show something like this: given the formal germ C of an isolated reduced curve singularity over an algebraically closed field k there cannot be a flat deformation D/A of C over a 2-dimensional complete regular local ring A with residue field k which smooths out the singularity in all directions (no singular fibres except for the central one).

By an argument, which I think is due to Deligne, such a deformation can be globalized. In other words, given D/A we can find a proper flat family of geometrically connected curves X over Spec(A) and a closed point x of the closed fibre of X such that the completion of X at x is D and such that X is smooth over Spec(A) everywhere except at x. Essentially the way I think about this is that you first “attach” smooth projective curves to the germ C over k and then you use that deformations of curves always are smooth over the deformations of their singularities. In this construction we may and do assume the genera of the irreducible components of the closed fibre are > 1 and a fortiori that the genus of the generic fibre is > 1.

Thus we reduce to showing: there cannot be a flat proper family X of geometrically connected curves over a 2 dimensional complete local regular Noetherian ring A which is smooth except at finitely many points of the special fibre such that all irreducible components of the special fibre have genus > 1.

By a result of Moret-Bailly, given such an X there exists a different family X’ over A which agrees with X over the punctured spectrum and whose special fibre is proper smooth over k. A simple argument (using the genera of components being > 1) shows that X and X’ are isomorphic as schemes over A as desired.

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