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Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?

If not, under what additional conditions?

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Do you mean Néron models defined for abelian varieties over higher local fields? –  Clark Barwick Jan 25 '10 at 16:38
I think he means: if $S$ is an integral scheme with function field $K$ and $A_{/K}$ is an abelian variety, does there exist a smooth group scheme $A_{/S}$ satisfying the Neron mapping property? –  Pete L. Clark Jan 25 '10 at 17:02
Sorry, we also want some kind of nonsingularity hypothesis on $S$: to fix ideas, perhaps we suppose that $S$ is smooth? –  Pete L. Clark Jan 25 '10 at 17:06
Either way, great question: +1. My guess is that the answer is "no" (meaning, not always), but I'm looking forward to hearing why. –  Pete L. Clark Jan 25 '10 at 17:23
Yes, I mean what Pete wrote. –  Timo Keller Jan 25 '10 at 17:26
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2 Answers

This is not answer to the OP, but to the remark of Pete on singular base scheme $S$. Here is an example explainning why one should assume regularity even in dimension $1$.

Let $T$ be a smooth curve over a field $k$ and identfity two rational points $t_1\ne t_2$ on $T$. We get a finite birational morphism $T\to S$ of integral curves over $k$. Let $E\to T$ be an elliptic curve with non isomorphic fibers at $t_1, t_2$. Let $\eta$ be the common generic point of $T$ and $S$. Then I claim that $E_\eta$ has no Néron model over $S$.

Suppose that we have a Néron model $E'$ over $S$. As $E$ is the Néron model of $E_\eta$ over $T$, we get a birational morphism f: $E'\times_S T\to E$ which is an isomorphism away from $\{ t_1, t_2\}$. Denote by $E_i$ the fiber of $E$ over $t_i$. We have a morphism $f_i : E'_{s}\to E_i$ where $s$ is the image of $t_i$ in $S$.

  • We can not have $f_1, f_2$ both quasi-finite, because otherwise $f$ would be quasi-finite, hence an open immersion (Zariski's Main Theorem), so $f_1, f_2$ would be open immersions and $E_1$ would be birational, hence isomorphic to $E_2$.

  • Suppose $f_1$ is not quasi-finite, then it is constant. Now take two rational points of $E_\eta$ which specialize to two distinct rational points $a, b$ in $E_1$. By Néron property, they also have specializations in $E_s$. But the latter are mapped by $f_1$ respectively to $a$ and $b$. Contradiction.

An idea to construct an example in higher dimension smooth basis: let $S$ be the affine plane containning a nodal curve passing through the origin $o$. Blow-up $o$ to get a smooth surface $T$ and consider an elliptic curve $E$ over $T$ with non isomorphic fibers at $t_1, t_2$ (intersection points of the exceptional divisor with the strict transform of the nodal curve). If any abelian scheme is the Néron model of its generic fiber (which seems reasonable if Néron model exists over regular base schemes), then similarily to the above we could prove that the generic fiber of $E$ has no Néron model over $S$.

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Take a family of elliptic curves over the punctured plane whose $j$-invariant at $(x,y)$ is $y/x$.

Perhaps you should ask if there exists a blow-up of $S$ on which a model exists.

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You beat me to it! Nice. –  Clark Barwick Jan 25 '10 at 20:39
James: I wonder how to prove the Néron model does not exit in your example. The idea of blowing-up the base is nice. There is a paper by J. de Jong and Oort on extending families of stable curves. However it not clear whether Néron model will exists even with alteration (à la de Jong) of $S$. –  Qing Liu Jan 25 '10 at 22:15
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