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A projective nonsingular variety $X$ over a number field $K$ has the notion of good reduction at places $p$ of $K$. Informally, $X$ has good reduction modulo $p$ if $X$ remains nonsingular when reduced modulo $p$. A theorem states that for all but finitely many primes we have good reduction (See Hindry and Silverman's "Diophantine Geometry: An Introduction", Proposition A.9.1.6, p.158).

Is there a similar notion of good reduction for singular varieties?

For example, suppose I have a surface with a single singularity, I expect that for almost all primes the surface will remain "nice" when reduced modulo $p$.

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Hutz in "Good reduction of periodic points on projective varieties" defines a more general good reduction for a proper scheme over a number field. However, the reduced scheme is still required to be smooth and proper. –  wishcow Oct 29 '12 at 15:50
    
I think in the curves setting one could consider stability or semi-stability as a property preserved by reduction. I don't know if this generalizes. –  Phillip Williams Oct 29 '12 at 16:14
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3 Answers

I think this is one of those instances when geometric intuition helps.

The (usual) geometric analogue of something defined over a number field is a variety defined over the function field of a curve (defined over a field of characteristic $0$). In this analogy good reduction corresponds to those fibers that are smooth and the theorem you're citing corresponds to the fact (in char $0$) that if the total space is smooth than there is an open subset of the base such that the fibers over that open subset are smooth.

This fact can be thought of depending on two key steps:

  • Deformation invariance: small deformations of smooth varieties are themselves smooth. (I.e., in a fibration the locus of smooth fibers is open).
  • Bertini: a general member of a basepoint-free linear system on a smooth variety is itself smooth. (I.e., in a fibration the locus of smooth fibers is non-empty).

So, to answer your question: The reasonable definition of "good reduction" for a class of singularities seems to be something like: If $X$ has singularities of type "blah", then for almost all primes $p$, the reduction mod $p$ of $X$ has also singularities of type "blah".

Now if the singularity class "blah" satisfies the two conditions above, then in the geometric setting the analogous statement will be true: For a family of varieties with singularities of type "blah", the locus of points in the base over which the fiber also (only) has singularities of type "blah" is a non-empty open set. If this is satisfied, you have a good chance that the arithmetic version will be true as well, although you might need a different proof.

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I think this definition is not the best one. I think if the general fiber has one singularity of type "blah", and a specific fiber has two singularities of type "blah", that is a fiber of bad reduction. I propose an alternate definition: Take the union of the projections to the base of all the irreducible components of the locus of non-smooth points that do not lie over the entire base. These are the fibers of bad reduction. To combine these two perspectives, one can do the same thing for the locus of points with especially bad reduction. –  Will Sawin Oct 29 '12 at 18:01
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One goal of such a definition is to replicate the result that the pushforward of a lisse sheaf is lisse on the fibers of good reduction, or in the number field case that the Galois representation is unramified there. Certainly there is some definition that agrees with this, since the pushforward of a constructible sheaf is constructible, so is lisse on some open set. However taking that open set is unsatisfying for other reasons. I don't know if this definition achieves that goal. –  Will Sawin Oct 29 '12 at 18:04
    
@Will: the number of singularities is not a good invariant. Singularities are local, their significant properties should be local. The global number of (isolated) singularities is not likely an interesting invariant. The right class for a "single" singularity is "isolated singularities". –  Sándor Kovács Oct 29 '12 at 20:16
    
@Will #2: Furthermore, even if you really want to go with the number of singularities, then yes, two singularities would be bad for the class of "single singularities". If the general fiber has a single singularity, the "double singularities" would still be limited to lie over a closed set, so no problem there. –  Sándor Kovács Oct 29 '12 at 20:16
    
@Will #3: Finally, your suggestion for an "alternate" definition (I suppose you mean "alternative") seems to produce the same notion as what I suggested (I suppose I mean "implied"). Cheers. –  Sándor Kovács Oct 29 '12 at 20:17
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Consider a family of elliptic curves with some nodal fibers, say $y^2=x(x-1)(x-\lambda)$. Base change to a base where there are two distinct sections that do not intersect the nodes, and glue the sections together. One now has a family of curves, most of which have a node singularity. Where the two sections intersect, there will be a cusp singularity, and where there was previously a node, there are now two nodes.

I think both kinds of bad fibers count as bad reduction.

My argument for the second kind is as follows: Looking at things locally in a neighborhood of the node, the other singularity is invisible, so this might as well be the original, pre-gluing family, in which the nodal singularities were of course bad fibers, as the generic fiber was smooth and they weren't. I do not think that creating a singularity elsewhere on the variety should take a bad fiber and make it good.

Furthermore, note that both kinds of fibers have vanishing cycles.

(Alternately, take an elliptic curve over a number field with positive rank and one semistable prime. Glue a power of an element of infinite order to the zero section, giving the variety a node singularity and creating a prime with two node singularities. This prime should remain a prime of bad reduction.)

Thus I propose the following notion of a fiber of bad reduction: For each singularity on the generic fiber (or on the variety over $K$), its closure is an irreducible closed subset of the total space (or the variety extended to $\mathcal O_K$). Since the singular locus is closed, these closed subsets are entirely singular - they are the "unavoidable" singularities. If a special fiber has an "avoidable" singularity that does not lie on any of these irreducible closed subsets, or it has a singularity on one of these subsets that has a worse singularity type in some meaningful sense than the general one, then it is a fiber of bad reduction. All other fibers are fibers of good reduction. (Similarly, for each singularity in the variety over a number field, we may get as a consequence one or more singularities in its reduction mod $p$. Singularities that are not a consequence of a characteristic zero singularity, or singularities that are worse than the characteristic zero one necessitates, make the prime bad reduction. If that does not occur, it's good reduction.)

As long as the set of points with a singularity type worse than a given singularity type is closed, which I believe it is for all reasonable measurements of singularity badness, then the fibers of good reduction are a nonempty open set. This is clear because the set of bad fibers is a finite union of closed sets that do not contain the generic point.

I do not know if there is a definition of singularity type badness that will work here so that all points with vanishing cycles / ramified Galois representations are fibers of bad reduction.

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@Will: I'll say it again: I think your definition is a special case of mine. I don't see why you are saying that by my definition that bad prime would become good when I did not give an explicit definition. I just expressed a philosophy to approach the problem. My definition depends on the definition of "blah" which could be taken to get what you think my definition is or what your explicit definition is or something completely different. –  Sándor Kovács Nov 1 '12 at 20:00
    
Then perhaps my definition is just a further clarification of yours, making it more explicit? –  Will Sawin Nov 1 '12 at 20:26
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My original intepretation of what you said is that "singularities of type "blah"", means "all singularities are of type "blah", where "blah"" is a type of singularity." Clearly this would not count "only one singularity" as a valid "type" of singularity. I now see that you mean something much more general, where "blah" can be almost anything. But clearly most versions of what "blah" means are silly. I am trying to characterize in more detail what sorts of things "blah" can be. Alternately I am showing how one could actually define a class of varieties that has your two properties. –  Will Sawin Nov 1 '12 at 20:31
    
I apologize for misrepresenting your views and mistyping your name. –  Will Sawin Nov 1 '12 at 20:31
    
@Will: yes, I agree with your comment (voted it up). I think we're on the same page now. I didn't perceive it as "misrepresentation", just "misunderstanding" or "misinterpretation" and I am probably the one to blame to not make it more clear what I meant. Unfortunately written media does not reflect tone, so I imagine you could see my first remark as being offended, but I wasn't and it was not my intention to give that impression. The remark about the name was just a joke. No worries. Cheers. –  Sándor Kovács Nov 1 '12 at 21:08
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How about the condition: $X/S$ has "good reduction" if the scheme-theoretic singular locus is flat over $S$?

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Take the family $S=\operatorname{Spec}k[t]$, $X=\operatorname{Spec}k[x,y,t]/(y^2-x^2(x-t))$. Is the scheme-theoretic singular locus $\operatorname{Spec}k[x,y,t]/(x,y)$, thus making t=0$ a fiber of good reduction, or is it defined more subtly? –  Will Sawin Nov 2 '12 at 19:27
    
Thanks. But isn't there an embedded point at $t=0$? I will check this later. –  Piotr Achinger Nov 3 '12 at 2:16
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