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Zbl 0864.14027
Bardelli, Fabio; Ciliberto, Ciro; Verra, Alessandro
Curves of minimal genus on a general abelian variety. (English)
[J] Compos. Math. 96, No.2, 115-147 (1995). ISSN 0010-437X; ISSN 1570-5846/e

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.
[L.Chiantini (Siena)]

MSC 2000:: *14K30 Picard schemes, higher Jacobians
14H45 Special curves and curves of low genus

Keywords: abelian variety; jacobian dimension

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