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Néron models and the height jump divisor. (English) Zbl 1374.14023

A curve \(C\) over a separably algebraically closed field \(k\) is called semistable if \(C\) is connected, reduced and projective, and each singular point of \(C\) is an ordinary double point. A morphism of schemes \(p\, : C \to S\) is a semistable curve if \(p\) is proper, flat and finitely presented, and if all geometric fibers of \(p\) are semistable curves. For a semistable curve \(C \to S\), we write \(Sm(C/S)\) the locus where \(C\to S\) i s smooth and \(\mathrm{Sing}(C/S)\) for the locus where \(C \to S\) is not smooth.
The definition of height jump divisor requires too much space to be described here (it appears in Section 3 of the paper, to which we refer for a precise definition). This is however an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced by R. Hain. The authors observe that one can compute the multiplicity of the height jump divisor J along a prime divisor \(Z\) in \(T\) using the ‘test curve’, which is the canonical map from the spectrum of the local ring of \(T\) at the generic point of \(Z\) to \(T\) itself. It follows that in order to study height jumping one can mostly restrict to the case where \(T\) is the spectrum of a discrete valuation ring (i.e., a trait). In this case, the height jump divisor is a rational multiple \(J = j \cdot[t]\) of the closed point \(t\) of \(T\).
We quote two of the main results of the paper.
Theorem A: Let \(D\) be a divisor of relative degree zero on \(C \to S\), with support contained in the smooth locus \(Sm(C/S)\) of \(C \to S\). Let \(s\in S\) be a point. Then for all non-degenerate morphisms of pointed schemes \(f : (T, t) \to (S, s)\) with \((T, t)\) a trait, the height jump divisor \(J(f ; D, D)\) is an effective divisor on \(T\) .
Theorem B: Assume that \(S\) is of finite type over a field \(k\). Let \(D\) be a divisor of relative degree zero on \(C \to S\) with support contained in \(Sm(C/S)\). Let \(T\) be a smooth projective geometrically connected curve over \(k\), and let \(f\,:\, T \to S\) be a non-degenerate \(k\)-morphism. Then the \(Q\)-line bundle \(f^\ast\langle D, D\rangle_a^{\otimes-1}\) has non-negative degree on \(T\).

MSC:

14H10 Families, moduli of curves (algebraic)
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14K15 Arithmetic ground fields for abelian varieties
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