# Is the set of surfaces over Spec Z with ample canonical sheaf empty

Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic fibre $X_{\mathbf Q}$ is ample?

Replacing "relative dimension two" by "relative dimension one", the answer is negative by a theorem of Abrashkin-Fontaine. I highly suspect the answer to be negative in this case too. Unfortunately, it is not known yet though as confirmed by Sándor.

Question 2. Does there exist a number field $K$ such that there are infinitely many $K$-isomorphism classes of smooth projective geometrically connected surfaces over $K$ with ample canonical sheaf and a smooth projective model over $O_K$?

The answer is positive if we replace "surfaces" by "curves". And as Will points out the answer is positive in the higher-dimensional case.

My main question is part of the arithmetic Shafarevich conjecture. As the terminology suggests, this conjecture is the arithmetic analogue of a conjecture for geometric objects. The latter (geometric) conjecture has been resolved by Arakelov, Bedulev, Kovács, Lieblich, Möller, Parshin, Viehweg, Zuo, et al. (Edit: Please see the references in Sándor's answer.) Its arithmetic analogue remains widely open for relative dimension $\geq 2$ to my knowledge, and was resolved in 1983 by Faltings for relative dimension 1.

With my second question I would like to assure myself of the non-triviality of a higher-dimensional arithmetic Shafarevich conjecture. It turns out to be trivial.

Let me state the results (due to the before-mentioned) in algebraic geometry relevant to this question. The base field is an algebraically closed field $k$ of characteristic zero.

Theorem 1. (Higher-dimensional geometric analogue of main question) There are no smooth projective (strongly?) non-isotrivial morphisms $X\to \mathbf P^1_k$ such that the canonical sheaf of the generic fibre of $X\to \mathbf P^1_k$ is ample.

Theorem 2. ("Folklore?" Higher-dimensional geometric analogue of second question) Fix $d\geq 0$. There exists a smooth projective connected curve $C$ such that there are infinitely many isomorphism classes of (strongly?) non-isotrivial smooth projective morphisms $X\to C$ of relative dimension $d$ whose generic fibre has ample canonical sheaf.

Now, Theorem 2 is one of the reasons that the following grand finiteness theorem is difficult.

Theorem 3. Let $C$ be a smooth projective connected curve and let $h$ be a polynomial. Then, there are only finitely many isomorphism classes of smooth projective (strongly?) non-isotrivial morphisms $X\to C$ whose generic fibre is canonically polarized with Hilbert polynomial $h$.

Let me note that I am considering function fields over a field of characteristic zero to be analogous to Spec $\mathbf O_K$. I know some of you prefer function fields over finite fields, but regarding these questions the analogy also "works" to a certain extent.

I might have stated Theorems 1-3 slightly incorrectly. In this case I apologize. (Also, I didn't state Theorem 3 in its full generality. The base curve doesn't need to be compact for instance.) Maybe, I should have only considered deformation types of families over $C$ in the statements.

Finally, let me point out some related MO questions:

What can be the dimension of a pointless smooth proper Z-scheme?

Smooth proper scheme over Z

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For question 2, if $C_1$ and $C_2$ are smooth curves over $\mathcal O_K$ with ample canonical bundle, then isn't $C_1\times C_2$ a smooth surface over $\mathcal O_K$ with ample canonical bundle? So you should be able to deduce the surfaces case from the curves case. –  Will Sawin Jan 13 '13 at 22:25
@Will. Ha! You're right. I don't know why I didn't think of that while typing the question. Anyway, I'll leave it in the question for now. In fact, I'm also interested in "other" examples. –  Ari Jan 13 '13 at 22:55
Dear Ariyan, could you give references for Th. 1-2-3 ? –  Damian Rössler Jan 14 '13 at 10:21
There is a phenomenon in fiber dimension $2$ that does not occur for fiber dimension $1$. Have you heard of "small resolutions" of threefold double point singularities? For instance, the hypersurface of $\mathbb{P}^3_{\mathbb{Z}_p}$ (with $p$ an odd prime) with defining equation $X_1^2 - p^2X_0^2 -X_2X_3$ is clearly singular at the in the closed fiber with homogeneous coordinates $[X_0,X_1,X_2,X_3] = [1,0,0,0]$. However, there is a birational modification (only in the closed fiber) that is smooth over $\text{Spec} \mathbb{Z}_p$. –  Jason Starr Jan 14 '13 at 13:49
Dear Damian, sorry for the late reply. The answer of Sándor Kovács contains a complete list of references. I can not add anything to that. For your convenience, let me just add that you will find easily all the precise references in Section 4 of the survey paper math.washington.edu/~kovacs/Seattle05/… –  Ari Jan 16 '13 at 18:23
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I don't think the answer to the first question is known.

Will has already pointed out the trivial answer to the second question. However this is not the right question. I mean this is kind of trivial. The interesting question is if you fix the genus and require that the curve over $K$ has good reduction everywhere (outside a fixed set of primes). If you ask it in that way, then the answer for curves is negative (by Faltings) and so the easy fix to do it in higher dimensions does not work.

Here are some comments and references to Theorems 1,2,3:

Theorem 1 is known in more general context.

It does not need "strong", non-isotrivial is enough.

Relevant references are:

Kovács, Sándor J.(1-UT) Smooth families over rational and elliptic curves. J. Algebraic Geom. 5 (1996), no. 2, 369–385.

Kovács, Sándor J.(1-MIT) On the minimal number of singular fibres in a family of surfaces of general type. J. Reine Angew. Math. 487 (1997), 171–177.

Kovács, Sándor J.(1-CHI) Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom. 9 (2000), no. 1, 165–174.

Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10 (2001), no. 4, 781–799.

Kovács, Sándor J.(1-WA) Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math. 131 (2002), no. 3, 291–317.

There are also generalizations for families over higher dimensional bases. See for instance:

Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) Base spaces of non-isotrivial families of smooth minimal models. Complex geometry (Göttingen, 2000), 279–328, Springer, Berlin, 2002.

Kebekus, Stefan(D-KOLN); Kovács, Sándor J.(1-WA) Families of canonically polarized varieties over surfaces. (English summary) Invent. Math. 172 (2008), no. 3, 657–682.

Kebekus, Stefan(D-FRBG); Kovács, Sándor J.(1-WA) The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1–33.

Patakfalvi, Zsolt(1-PRIN) Viehweg's hyperbolicity conjecture is true over compact bases. (English summary) Adv. Math. 229 (2012), no. 3, 1640–1642.

Theorem 2:

This is a triviality unless you fix some invariants. On the other hand for relative dimension $1$ and fixed genus this is not true. This is the geometric version of Shavarevich's conjecture and was first proved by Parshin:

Paršin, A. N. Algebraic curves over function fields. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 1191–1219,

and then in a more general case by Arakelov:

Arakelov, S. Ju. Families of algebraic curves with fixed degeneracies. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293.

In higher dimensions, the statement is true indeed by taking the product of an arbitrary family of curves and an arbitrary curve (each of the appropriate genus). The second curve can be moved in moduli which gives even a continuous family of families.

In fact, this was what led to the notion of strong isotriviality.

Some relevant references are:

Kovács, Sándor J.(1-WA) Strong non-isotriviality and rigidity. Recent progress in arithmetic and algebraic geometry, 47–55, Contemp. Math., 386, Amer. Math. Soc., Providence, RI, 2005.

Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.

Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.

Zsolt Patakfalvi Arakelov-Parshin rigidity of towers of curve fibrations, connections to the infinitesimal Torelli problem http://arxiv.org/abs/1010.3069

Theorem 3:

as I explained above, even this is not true without the "strong" assumption.

For strongly non-isomorphic families it is proven in

Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.

I would expect it to be true for a somewhat larger class of families, but the actual class still needs to be defined. The key modulo this paper is rigidity.

For more details see

Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.

or

Chapter III of Hacon, Christopher D.(1-UT); Kovács, Sándor J.(1-WA) Classification of higher dimensional algebraic varieties. Oberwolfach Seminars, 41. Birkhäuser Verlag, Basel, 2010. x+208 pp. ISBN: 978-3-0346-0289-1

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Thank you very much. This affirms my expectation that we don't know the answer to Q1. I do expect the answer to be negative (because I "believe" in the analogy.) Do you know of any references where the arithmetic Shafarevich conjecture is studied except for the papers by Faltings and André concerning abelian varieties, curves and K3 surfaces and the survey papers by Zarhin, Parshin, Szpiro, etc. ? For instance, do you know whether the arithmetic Shafarevich conjecture is known for surfaces of Kodaira dimension one (=elliptic fibrations)? –  Ari Jan 15 '13 at 17:05
Other than what you mention, you could look at Vojta's papers. His conjectures on various height estimates are considered higher dimensional analogous of Mordell, which, I am sure you know, is closely related to Shafarevich. There is a chance that one can get a sort of boundedness statement from some height estimates (this is what happens for curves) and then you need rigidity to get finiteness. As the example in my answer shows, in the geometric case you need more than just non-isotriviality for rigidity, but this could be a place where the number field case is different. –  Sándor Kovács Jan 15 '13 at 18:54
Also, besides Vojta, there is the paper by Levin in Annals. I think his higher-dimensional Siegel conjectures should be "easier" than Vojta's conjectures and they probably also imply results towards Shafarevich. I didn't read them that carefully yet though. –  Ari Jan 18 '13 at 22:23