Néron models of jacobians over base schemes of dimension greater than 1

Abstract

We investigate to what extent the theory of Néron models of jacobians and of abel-jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a Néron model. We use this to show that, in general, Néron models do not exist even after making a modification or even alteration of the base. On the other hand, we show that Néron models do exist outside some codimension-2 locus. Idea of the proof: Néron models via the relative Picard functorOur existence and non-existence results for the Néron model proceed via an auxiliary object, the 'total-degree-zero relative Picard functor' PicC/S to the locus U ⊆ S where C is smooth coincides with the Jacobian of C U /U. It is also rather easy to show thatC/S satisfies the 'existence' part of the Néron mapping property whenever C is regular. However, if C/S has non-irreducible fibres then Pic [0]C/S is in general highly non-separated, and for this reason it fails to satisfy the 'uniqueness' part of the Néron mapping property. As such, we wish to construct some kind of 'separated quotient' of Pic [0]C/S , in the hope that this will satisfy the whole Néron mapping property.Now the failure of separatedness in a group-space is measured by the failure of the unit section to be a closed immersion; as such, a natural way to construct a 'separated quotient' of Pic [0] C/S is to quotient by the closure of the unit section. Quotients by flat subgroup-spaces always exist in the category of algebraic spaces, but the problem is that the closure of the unit section in Pic [0]C/S (which we shall refer to asē) is not in general flat over S, and so this quotient does not exist 3 . By a slightly more delicate argument, one can even show that the existence of a Néron model is equivalent to the flatness ofē. The technical heart of this paper is 3 One might be tempted to apply [RG71] to flattenē by blowing up S, but note that Pic [0] C/Sis not quasi-compact over S, and so their results do not apply; indeed, one consequence of our results is that, if C/S is not aligned, thenē does not become flat after any modification of S.

Torsion points and height jumping in higher-dimensional families of abelian varieties

“…Proof. The equivalence of (a), (d) and (e) follows from the main results of [Hol14b]. It is clear that (b) implies (c).…”

“…For the sake of those readers not familiar with the theory of Néron models over higher dimensional bases developed in [Hol14b] we note that the assumption that J admit a Néron model over S can be replaced by the assumption that S \ U (with reduced induced scheme structure) is smooth over Q, or that the fibres of C → S have tree-like dual graphs.…”

Moduli of stable maps in genus one and logarithmic geometry, I

“…Note that the notion of radial alignments, as well as variants which follow later in the paper, are distinct from the alignment condition introduced by Holmes in work on the Néron models [13]. It is related to the notion of aligned logarithmic structure introduced by Abramovich, Cadman, Fantechi, and the third author [1].…”

Néron models and the height jump divisor

“…Proof. Part (a) is [11,Corollary 1.3]. For part (2), by a limiting argument we find that formation of the Néron model commutes with pullback to the spectrum of the local ring at the generic point of a boundary divisor of U in S. Such a local ring is Dedekind, so the Néron model over it is of finite type ([1, 1.2]).…”

“…If A is a family of jacobian varieties with semistable reduction and S is a quasicompact regular separated base scheme, then by results of the second author [11] there exists a largest open subscheme V ⊂ S containing U such that A does have a Néron model N(A, V ) over V . Moreover, the complement of V has codimension at least two in S. Now given two sections P, Q ∈ A(U ), there exist m, n ∈ Z >0 such that the multiples mP, nQ extend as sections of the fiberwise connected component N 0 (A, V ) of N(A, V ) over V (perhaps after slightly shrinking V ).…”