The author proves the following theorem: Let X be an n-dimensional abelian variety over an algebraically closed field k of any characteristic. Suppose that G is an effective 1-cycle generating X and that D is an ample divisor on X with the intersection number (D·G) equal to n = dimX. Then (X,D,G) is a Jacobian triple, i.e., (1)G = G1+· · ·+Gr is a sum of nonsingular irreducible curves G1, · · · ,Gr; (2) X is isomorphic to the product of the Jacobian varieties J1, · · · , Jr of G1, · · · ,Gr, respectively; (3) the inclusion G \to X coincides, up to translation, with the one naturally induced by inclusions Gi to Ji (1 i r); and (4) D coincides with the sum D1 + · · ·+Dr of the pullbacks Di to X of the theta divisors i of Ji (1 .. r).
A new proof of the Ran-Matsusaka criterion for Jacobians.Proc. Amer. Math. Soc. 92 (1984), no. 3, 329–331.1088-6826
COLLINO, Alberto
1984-01-01
Abstract
The author proves the following theorem: Let X be an n-dimensional abelian variety over an algebraically closed field k of any characteristic. Suppose that G is an effective 1-cycle generating X and that D is an ample divisor on X with the intersection number (D·G) equal to n = dimX. Then (X,D,G) is a Jacobian triple, i.e., (1)G = G1+· · ·+Gr is a sum of nonsingular irreducible curves G1, · · · ,Gr; (2) X is isomorphic to the product of the Jacobian varieties J1, · · · , Jr of G1, · · · ,Gr, respectively; (3) the inclusion G \to X coincides, up to translation, with the one naturally induced by inclusions Gi to Ji (1 i r); and (4) D coincides with the sum D1 + · · ·+Dr of the pullbacks Di to X of the theta divisors i of Ji (1 .. r).
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