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On abelian subvarieties generated by symmetric correspondences. (English) Zbl 0685.14018

Let C be a non rational algebraic projective smooth curve defined or \({\mathbb{C}}\). For any \(\alpha\in Pic^ 2(C)\) we consider the map \(\Phi_{\alpha}\) between the symmetric product \(C^{(2)}\) of the curve and its Jacobian, \(J(C)\), such that \(\Phi_{\alpha}(P,Q)= {\mathcal O}_{P+Q}\otimes \alpha^{-1}\) for all \((P,Q)\in C^{(2)}\). Let A be a proper Abelian subvariety of \(J(C)\).
We prove that \(\dim[\Phi_{\alpha}^{-1}(A)]=1\) if and only if there exist two curves K and \(K'\) and two non constant maps \(f: C\to K\) and \(h: K\to K'\) such that \(\deg(h)=2\), and \[ A\supseteq (h\circ f)^*[J(K')] + \{\text{connected component of }\ker(f_*)\}; \] \((f_*\), \((h\circ f)^*\) are the induced maps between Jacobians). We also prove that if \(\dim[\Phi_{\alpha}^{-1}(A)]<1\) for every \(\alpha\) and A, then \(h^ 1(-D)=0\) for every effective connected reduced divisor D of \(C^{(2)}\). Finally, we use our results to show that there are no genus 9 curves in the generic Abelian variety of dimension 5; this improves all bounds we know about this subject.
Reviewer: A.Alzati

MSC:

14H40 Jacobians, Prym varieties
14K05 Algebraic theory of abelian varieties
14F17 Vanishing theorems in algebraic geometry
32L20 Vanishing theorems
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References:

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