Abelian Varieties by Serge Lang

Author:Serge Lang
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2019-03-11T16:00:00+00:00

with points on an abelian variety J, and a divisor T on the product J × V × W. Let k be a field of definition for J, V, W over which T is rational. We may suppose u0 and u1 generic over J (but not necessarily independent) after adding to them a generic point of J. The theorem of the cube asserts that on J × J × V × V × W × W, the notations being those of Chapter III, § 2. Let v1 w1 be independent generic points of V, W over k(u0, u1), and take the intersection of with We then obtain the following cycles.

For j = k = 1, we find 0, since (ui v1 w1) cannot be in any component of T.

For j = k = 0, we obtain the two terms occurring in the expression for X, that is to say, if we project on V × W on the third and fifth factors, we find T(u0) — T(u1).

The other terms give divisors of type Y × W and V × Z. One sees directly from the definition of algebraic equivalence that

REMARK. If V, W are complete and non-singular in codimension 1, then we can give an alternate proof for the preceding proposition, by taking X(v) for v generic on V, and using the seesaw principle. We leave the details to the reader, observing merely that there are fewer indices in this procedure than in the one using the theorem of the cube.

PROPOSITION 7. Let V, W be two varieties, complete and nonsingular in codimension 1, both defined over k. Let (A, D) and (B, E) be Picard varieties of V and W respectively defined over k. Let F be the divisor on V × W × A × B obtained from the divisor

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