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Hi everyone,

let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic fiber of X over S is isomorphic to X.

  • Are there general conditions on X, S, k, such that X exists?
  • If X is smooth and projective, what are the conditions, such that there is a smooth model X?
  • Any good references that go into models and reduction in general, and not only in the case of curves?
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Childish joke about the title. –  Harry Gindi Jan 1 '10 at 6:19
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5 Answers 5

up vote 8 down vote accepted

I would also be very interested if someone could point to good references for this question, but I am not yet allowed to vote for the question.

Nekovar's survey article on the Beilinson conjectures from the early 90s mentions some results for varieties over Q. He says in section 5.3 that given a smooth projective variety over Q, there always exists a proper flat model over Z, but that a regular such model is rarely known to exist. However, in the published version of the same survey, there is an added note at the very end of the article saying that "Spivakovsky recently announced a general result on resolution of singularities, which implies that a regular proper flat model of X mentioned in 5.3 always exist". However, I have never seen this result of Spivakovsky mentioned anywhere else, so I doubt that it is true. Does anyone else know more about this?

The survey is available here: http://people.math.jussieu.fr/~nekovar/pu/mot.pdf

For the published version, google "Serre Jannsen Motives", click at the Google Books link, and then search for "Spivakovsky" within the book.

In general for the case when k is a number field and S is the ring of integers in k, it seems unreasonable to ask for general theorems about the existence of smooth models, although one could maybe hope for something about regular models. For example, I think there are no nontrivial smooth and proper curves over Spec(Z) at all (although there are smooth and proper such schemes of higher dimension, and there are also smooth and proper such schemes in any dimension, including 1, over rings of integers in other number fields).

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I'm happy to present my example of a smooth projective surface $X$ over $K=\mathbb{Q}_p$ ($p$ prime) such that $X(K)\neq\emptyset$, whose $l$-adic cohomology groups are unramified (for all primes $l$) and which still has bad reduction : there is no smooth $\mathbb{Z}_p$-scheme whose generic fibre is $X$. (The method works for any finite extension of $\mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $\mathbb{P}_1$ with four degenerate fibres, so it is a rational surface in the sense of being $\bar K$-birational to $\mathbb{P}_2$. It will be clear that the example is not isolated.

If $p$ is odd, let $d\in\mathbb{Z}_p^\times$ be a unit which is not a square, and take $d=5$ if $p=2$, so that $K(\sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in $\mathbb{P}({\cal O}(2)\oplus{\cal O}(2)\oplus{\cal O})$ (coordinates $y:z:t$) over $\mathbb{P}_1$ (coordinates $x:x'$) defined by the equation $$ y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2. $$ I claim that this $X$ has all the properties stated above, if $v_p(e_1-e_2)>0$.

First, $X(K)\neq\emptyset$ because each degenerare fibre is a pair of intersecting lines conjugated by $\mathrm{Gal}(\bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of $\mathrm{Gal}(\bar K|K)$ on the Picard group $\mathrm{Pic}(\bar{X})$ of $\bar X=X\times_K\bar K$ factors via the quotient $\mathrm{Gal}(K(\sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group $A_0(X)_0$ of $0$-cycles of degree $0$ is $\mathbb{Z}/2\mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156), and a theorem of Bloch (th. 0.4, On the Chow groups of certain rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 14 no. 1 (1981), p. 41-59, available at Numdam) asserts that if a conic bundle has good reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

Addendum (in response to a question in an email I received). One can show moreover that no smooth projective surface $Y$ over $\mathbf{Q}_p$ which is $\mathbf{Q}_p$-birational to $X$ can have good reduction. This follows from the facts recalled above and the theorem of Colliot-Thélène and Coray (which can be found in Fulton's Intersection theory) : $A_0(Y)_0$ is isomorphic to $A_0(X)_0$.

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Thanks for the addendum! I had previously worked through the entire example, and was pretty sure that the reason was that any birational surface must have the same Chow group property, but wasn't sure where to look. –  Matt Nov 5 '12 at 2:26
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There is a very nice answer for abelian varieties.

Theorem 1.1 of Conrad and Brinon's notes on p-adic hodge theory gives a nice theorem in the case of Abelian varieties: let A be an abelian variety (for simplicity over Q), l a prime, and $p \neq l$ a second prime. Then A has good reduction (e.g. there exists a smooth model) iff the l-adic Tate module (equivalently the l-adic etale cohomology) is unramified at p. For A an elliptic curve an elementary proof of this is in Silverman's Arithmetic of Elliptic Curves, Theorem 7.1 (Criterion of Neron–Ogg–Shafarevich).

The case l = p is in a sense the beginning of p-adic hodge theory. Grothendieck gave a nice criterion in terms of p-divisible groups: A extends to a smooth model iff each torsion subscheme A[n] extends to an integral model (in a compatible way). This is the same as saying the p-divisible group associated to A extends to an integral model. This is all explained very well in Conrad and Brinon's notes (Section 7).

Later it was proved that this is equivalent to the p-adic Tate module being Crystalline. This is also in Brinon and Conrad's notes.

Finally, this type of theorem fails for surfaces, even when $l \neq p$! I think Shenghao Sun knows an example where the l-adic etale cohomology is unramified at p, but the surface still had bad reduction.

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ur...I guess Shenghao doesn't remember this example any more. Could you remind him? Thanks. –  shenghao Oct 16 '09 at 21:38
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For Theorem 1.1 alluded to above, see Serre, Jean-Pierre; Tate, John Good reduction of abelian varieties. Ann. of Math. (2) 88 1968 492--517. –  Chandan Singh Dalawat Mar 13 '10 at 3:27
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If you don't require X to be flat over S, then you can use X = X whenever K=k. I don't think that's what you wanted.

Assuming flatness, I don't know any sufficient conditions for existence or smoothness, aside from tautological ones. Example: say X arises as a k-point in a moduli space of smooth varieties. This defines a rational map from S_k. If it can be promoted to a morphism from S, then the answer is yes.

There are some necessary conditions, involving good cohomology behavior. If X is smooth and proper over the complex numbers, you get a variation of Hodge structure and a map to the corresponding clasifying space. To extend smoothly, the Hodge structure can't degenerate and the Gauss-Manin connection needs to be regular. I think there are similar conditions with p-adic cohomology ("potentially semistable" and "DeRham" are phrases I've heard here) and l-adic cohomology ("mixed" versus "pure"). In general, I don't know how close you are to getting a smooth model once these obstructions are overcome.

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Finding a global integral model is too hard, so let's assume S is local, e.g. a complete DVR. I don't know if there is any general result on the existence of local integral models, and even it exists, it might not be unique.

But for Shimura varieties of abelian type, there is a general result. Milne posed an extra condition on local integral models, and called a model satisfying this condition a "canonical integral model". With this condition one can show that the canonical model, if exists, is unique.

Let G be a reductive group over rationals and $K$ a compact open subgroup of G(A_f)=finite adelic points of G, and let $X$ be a set of G(\mathbb R)-conjugacy classes of homomorphisms S --> G_{\mathbb R}. Assume (G,X) is a Shimura datum of abelian type, and assume G and K are unramified at a rational prime p, then the Shimura variety S_K(G,X) defined by (G,X,K) has canonical good reduction at every finite place v of the reflex field lying above p, i.e., canonical models exists, and it has good reduction. The condition that G and K are unramified at p is equivalent to the existence of a hyperspecial subgroup K_p of G(\mathbb Q_p) and the p-component of K being K_p. See some papers/notes of Milne, e.g. introduction to Shimura varieties, p.130-131.

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"...even it exists, it might not be unique." Models can certainly fail to be unique, even if you put a minimality condition on them! –  David Zureick-Brown Oct 15 '09 at 6:03
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