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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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