By Serge Lang (auth.)
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Additional info for Abelian Varieties
Finally, we note that the genus of Ct and Ct , (or rather of their normalization) is the same, again because of the isomorphism (1 . We may therefore speak of the genus of the generic curve. As usual, we may choose t generic over any given field K containing k. For example, if f : V -+ B is a rational map of V into an abelian variety, defined over K, we shall take Ct generic over K . The dimension of an abelian variety generated by V is bounded by the genus of the generic curve. This will come from the universal mapping property of the Jacobian, and the following lemma.
X and we have therefore q;(u + x) = q;(u) - d· x. We know that q; is everywhere defined (Theorem 2). Putting u = 0, we obtain [II, § 1J 29 MAPS OF VARIETIES INTO ABELIAN VARIETIES d . x, which shows that the restriction of
For example, if f : V -+ B is a rational map of V into an abelian variety, defined over K, we shall take Ct generic over K . The dimension of an abelian variety generated by V is bounded by the genus of the generic curve. This will come from the universal mapping property of the Jacobian, and the following lemma. LEMMA 6. If (V, f) generates an abelian variety A, then the restriction of f to the generic curve of V also generates A. Proof: Let k be a field of definition for f : V -+ A. The generic curve Ct being defined over k(t), let ft : Ct -+ A be the induced rational map of C t into A, defined over k(t).
Abelian Varieties by Serge Lang (auth.)