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# Is the set of two-dimensional surfaces over Spec Z whosegenericfibrehaswith ample canonical sheaf empty

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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 of course I highly suspect as Will points out the answer to be is positive in this the higher-dimensional casetoo.

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. (I explain what I mean more precisely below.)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:

http://mathoverflow.net/questions/111708/what-can-be-the-dimension-of-a-pointless-smooth-proper-z-scheme

http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z

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# Is the set of two-dimensional surfaces over Spec Z whose generic fibre has 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.

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 of course I highly suspect the answer to be positive in this case too.

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. 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. (I explain what I mean more precisely below.)

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:

http://mathoverflow.net/questions/111708/what-can-be-the-dimension-of-a-pointless-smooth-proper-z-scheme

http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z