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?



Are there Néron models over higher dimensional base schemes?
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. 


