Abstract
Let be a complete discrete valuation ring with algebraically closed residue field and fraction field . Let be a proper smooth and geometrically connected scheme over . Néron defined a canonical pairing on between -cycles of degree zero and divisors which are algebraically equivalent to zero. When is an abelian variety, and if one restricts to those -cycles supported on -rational points, Néron gave an expression of his pairing involving intersection multiplicities on the Néron model of over . When is a curve, Gross and Hriljac gave independently an analogous description of Néron’s pairing, but for arbitrary -cycles of degree zero, by means of intersection theory on a proper flat regular -model of .
We show that these intersection computations are valid for an arbitrary scheme as above and arbitrary -cycles of degree zero, by using a proper flat normal and semifactorial model of over . When is an abelian variety, and is a semifactorial compactification of its Néron model , these computations can be used to study the relative algebraic equivalence on . We then obtain an interpretation of Grothendieck’s duality for the Néron model , in terms of the Picard functor of over . Finally, we give an explicit description of Grothendieck’s duality pairing when is the Jacobian of a curve of index one.
Citation
Cédric Pépin. "Néron's pairing and relative algebraic equivalence." Algebra Number Theory 6 (7) 1315 - 1348, 2012. https://doi.org/10.2140/ant.2012.6.1315
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