Are there Néron models over higher dimensional base schemes?

Question

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?

Answer 1

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.

Answer 2

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 $S$.

Answer 3

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.

Answer 4

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 $\mathcal{A}/S$.)