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.