Néron's pairing and relative algebraic equivalence

Abstract

Let $R$ be a complete discrete valuation ring with algebraically closed residue field $k$ and fraction field $K$. Let $X_K$ be a proper smooth and geometrically connected scheme over $K$. Néron defined a canonical pairing on $X_K$ between $0$-cycles of degree zero and divisors which are algebraically equivalent to zero. When $X_K$ is an abelian variety, and if one restricts to those $0$-cycles supported on $K$-rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model $A$ of $A_K$ over $R$. When $X_K$ is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary $0$-cycles of degree zero, by means of intersection theory on a proper flat regular $R$-model $X$ of $X_K$.

We show that these intersection computations are valid for an arbitrary scheme $X_K$ as above and arbitrary $0$-cycles of degree zero, by using a proper flat normal and semifactorial model $X$ of $X_K$ over $R$. When $X_K=A_K$ is an abelian variety, and $X=\overline{A}$ is a semifactorial compactification of its Néron model $A$, these computations can be used to study the relative algebraic equivalence on $\overline{A}∕R$. We then obtain an interpretation of Grothendieck’s duality for the Néron model $A$, in terms of the Picard functor of $\overline{A}$ over $R$. Finally, we give an explicit description of Grothendieck’s duality pairing when $A_K$ is the Jacobian of a curve of index one.