## 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 Poincaré’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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