# Are there Néron models over higher dimensional base schemes?

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

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

This is really a comment in response to JBorger and Qing Liu's questions about existence of Néron models after blowing up or altering the base, but is too long for the comment box.

In general, Néron models do not exist over bases of dimension greater than 1, even allowing alterations of the base. This non-existence seems quite robust - it does not help if you allow Néron lft models, or allow your Néron model to be an algebraic space, or…

The simplest example is probably to take $S = \operatorname{Spec} \mathbb{C}[[u,v]]$ (complete, regular, local,...), and to let $C/S$ be the nodal curve in weighted projective space $\mathbb{P}(1,1,2)$ over $S$ given by the affine equation $$y^2 = (x-1)(x-1-u)(x+1)(x+1+v).$$ If you let $J$ be the jacobian of the generic fibre of $C/S$, then $J$ does not admit a Néron model over $S$, or even over $S’$ where $S’ \rightarrow S$ is proper surjective (for example, an alteration). I do not know a very short proof of this latter fact; it can be found in http://arxiv.org/abs/1402.0647

More generally, given a nodal curve over a regular separated base, the jacobian will usually not admit a Néron model. There are two cases where a Néron model clearly does exist: if the curve is of compact type, or if it arises as pullback along a smooth morphism from a curve over a DVR. It turns out that these two situations, together with `combinations of the two’, are in some sense the only situations where Néron models do exist. Moreover, altering the base will usually make no difference to the existence of a Néron model. More precise statements can be found in the above reference.

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See my preprint http://arxiv.org/abs/1410.5293 Theorem 3.2 on page 7ff.:

Let $S$ be a regular, Noetherian, integral, separated scheme, and $g: \{\eta\} \hookrightarrow S$ the inclusion of the generic point. Let $\mathcal{A}/S$ be an Abelian scheme. Then $$\mathcal{A} = g_*g^*\mathcal{A}$$ as sheaves on $S_{\mathrm{sm}}$. (This is the "Néron mapping property" for $\mathcal{A}/S$.)

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Am I correct when I paraphrase that as "every abelian scheme over $S$ is the Néron model of its general fibre"? However, this doesn't say anything about which $A/\eta$ have a Néron model. But it does say that $g_{*}A$ is the sheaf that is the candidate. Nice to see you returning to your own question after 4 years! – jmc Jan 13 at 14:06
Yes, you are right, exactly! – Timo Keller Jan 13 at 14:30
I think it would be nice if the accepted answer would expand a little bit. Currently the term “Néron model” isn't even mentioned. – jmc Jan 13 at 15:19
There is no need to reference the linked arXiv paper: the nontrivial part (i.e., the surjectivity of the generic fiber pullback) of the result is already a special case of 8.4/6 of "Neron models" by Bosch, Lutkebohmert, Raynaud (this reference should be mentioned in the proof of Thm. 3.2 of the linked paper). – Question Mark Jan 13 at 17:13