## 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:

 [1] Bardelli, F., Pirola, G.P.: Curves of genusg lying on ag-dimensional Jacobian variety, Invent. Math.95, 263–276 (1989) · Zbl 0638.14025 [2] Ciliberto, C., Van der Geer, G., Teixidor, M.: On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms (preprint) · Zbl 0806.14020 [3] Harary, F.: Graph theory, Reading, Mass. Addison-Wesley 1969 · Zbl 0182.57702 [4] Pirola, G.P.: Base number theorem for Abelian varieties. Math. Ann.282, 361–368 (1988) · Zbl 0625.14024 [5] Serre J.P.: Groupes algébriques et corps de classes. Paris: Hermann 1959 · Zbl 0097.35604
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