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Curves of minimal genus on a general abelian variety. (English) Zbl 0864.14027

Let \(C\) be a smooth projective curve and let \(\varphi:C\to A\) be a morphism, with \(A\) abelian variety. If the image \(\varphi(C)\) generates \(A\), then \(A\) is isomorphic to a quotient of the Jacobian of \(C\); in general, one finds infinitely many curves \(C\) which map to \(A\) as above and the minimal genus of these curves is called the jacobian dimension of \(C\). Several lower bounds are known for the jacobian dimension of a general abelian variety \(A\). The authors are mainly concerned here with the case \(\dim(A)=4,5\); they show that a general abelian variety of dimension 5 has jacobian dimension 11; for general abelian fourfolds \(A\), they prove that the jacobian dimension is 7, furthermore they characterize curves of genus 7 mapping to \(A\) as above and compute their number.

MSC:

14K30 Picard schemes, higher Jacobians
14H45 Special algebraic curves and curves of low genus
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References:

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