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Note on curves in a jacobian. (English) Zbl 0802.14002
For an abelian variety \(A\) over \(\mathbb{C}\) and a cycle \(\alpha \in CH_ d (A)_ \mathbb{Q}\) we define a subspace \(Z_ \alpha\) of \(CH_ d (A)_ \mathbb{Q}\) by: \(Z_ \alpha : = \langle n_ * \alpha : n \in \mathbb{Z} \rangle \subset CH_ d (A)_ \mathbb{Q}\). Results of Beauville imply that \(Z_ \alpha\) is finite dimensional. In case \(A = J(C)\), the jacobian of a curve \(C\), Ceresa has shown that the cycle \(C-C^ - : = C - (-1)_ *C \in Z_ C\) is not algebraically equivalent to zero for generic \(C\) of genus \(g \geq 3\), which implies that for such a curve \(\dim_ \mathbb{Q} Z_ C \geq 2\). – In this note we investigate the subspace \(Z_{W_ m}\) of \(CH_ m (J(C))_ \mathbb{Q}\), with \(W_ m\) the image of the \(m\)-th symmetric power of \(C\) in \(J(C)\) (so \(W_ 1 = C)\). To simplify matters we will actually work modulo algebraic equivalence.
Let \(Z_ \alpha/ \approx_{\text{alg}}\) be the image of \(Z_ \alpha \subset CH_ d (A)_ \mathbb{Q}\) in \(CH_ d (A)_ \mathbb{Q}/ \approx_{\text{alg}}\). A \(d\)-cycle \(\alpha\) is Abel-Jacobi equivalent to zero, \(\alpha \approx_{\text{AJ}}0\), if \(\alpha\) is homologically equivalent to zero and its image in \(J_ d (A)\), the \(d\)-th primitive intermediate Jacobian of \(A\), is zero. Recall that any curve of genus \(g\) is a cover of \(\mathbb{P}^ 1\) for some \(d \leq (g + 3)/2\).
Theorem. (1) For any abelian variety \(A\) and any \(\alpha \in CH_ d (A)\) we have: \(\dim_ \mathbb{Q} (Z_ \alpha/ \approx_{\text{AJ}}) \leq 2\).
(2) For any curve of genus \(g\) and \(1 \leq n \leq g - 1\) we have: \(\dim_ \mathbb{Q} (Z_{W_{g-n}}/ \approx_{\text{alg}}) \leq n\).
(3) For a curve \(C\) which is a \(d:1\)-cover of \(\mathbb{P}^ 1\) we have: \(\dim_ \mathbb{Q} (Z_ C/ \approx_{\text{alg}}) \leq d - 1\).
Recall that Ceresa showed that the image of \(W_ m-W_ m^ -\) in \(J_ m (J(C))\) is nonzero for generic \(C\) of genus \(g \geq 3\) and \(1 \leq m \leq g - 2\). Therefore (1) and (2) for \(n = 2\) are sharp. In case \(C\) is hyperelliptic, so \(C\) is a 2:1 cover of \(\mathbb{P}^ 1\), the cycles \(W_ m\) and \(W^ -_ m\) are however algebraically equivalent. Therefore (3) is sharp for hyperelliptic curves \((d = 2)\) and generic trigonal curves \((d = 3)\).

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14H40 Jacobians, Prym varieties
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References:
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