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C is not algebraically equivalent to \(C^-\) in its Jacobian. (English) Zbl 0538.14024
Let \(W_ r\), 1\(\leq r\leq g-1\), be the image of the Abel map for the r- fold symmetric product of a non-singular algebraic curve C of genus g; and \(W^-_ r\) be the ”inverse set” of \(W_ r\), i.e. \(W^-_ r\) is the image of \(W_ r\) under the involution \(w\mapsto -w.\) As cycles, \(W_ r\) and \(W^-_ r\) are homologically equivalent on the Jacobian J(C). Moreover, it is well known that \(W_ r\) and \(W^-_ r\) are algebraically equivalent on J(C) when \(r=g-1\) (the Riemann symmetry of the \(W_{g-1}=\Theta\)-divisor!) or, for all r, when C is hyperelliptic. The paper under review shows that for 1\(\leq r\leq g-2\) on a generic Jacobian variety the cycles \(W_ r\) and \(W^-_ r\) are algebraically independent. The proof uses an ”inversion theorem” for Abelian varieties and is done by reduction to a singular Abelian case. The crucial step is the case \(g=3\), \(r=1\). This result implies that PoincarĂ©’s formula is not valid for the algebraic equivalence ring of J(C) with C generic.
Reviewer: V.V.Shokurov

MSC:
14H40 Jacobians, Prym varieties
14C99 Cycles and subschemes
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