Colombo, Elisabetta; van Geemen, Bert Note on curves in a jacobian. (English) Zbl 0802.14002 Compos. Math. 88, No. 3, 333-353 (1993). 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 \((d = 3)\). Cited in 2 ReviewsCited in 13 Documents MSC: 14C05 Parametrization (Chow and Hilbert schemes) 14H40 Jacobians, Prym varieties Keywords:Abel-Jacobi equivalent cycles; Chow group; curves in a jacobian PDFBibTeX XMLCite \textit{E. Colombo} and \textit{B. van Geemen}, Compos. Math. 88, No. 3, 333--353 (1993; Zbl 0802.14002) Full Text: arXiv Numdam EuDML References: [1] E. Arbarello , M. Cornalba , P. Griffiths and J. Harris , Geometry of Algebraic Curves I, Grundlehren 267, Springer Verlag (1985). · Zbl 0559.14017 [2] A. Beauville , Quelques Remarques sur la Transformation de Fourier dans I’Anneau de Chow d’une Variété Abélienne, Algebraic Geometry (Tokyo/Koyto 1982) . Notes Math. 1016, 238-260, Springer 1983. · Zbl 0526.14001 [3] A. Beauville , Sur l’anneau de Chow d’une variété abéliene , Math. Ann. 273 (1986) pp. 647-651. · Zbl 0566.14003 [4] G. Ceresa , C is not algebraically equivalent to C- in its Jacobian , Annals of Math. 117 (1983) 285-291. · Zbl 0538.14024 [5] C. Deninger and J. Murre , Motivic decomposition of abelian schemes and the fourier transform , J. reine angew. Math. 422 (1991) 201-219. · Zbl 0745.14003 [6] B.H. Gross and C. Schoen , The canonical cycle on the triple product of a pointed curve , to appear. · Zbl 0822.14015 [7] K. Künneman , On the Chow motive of an abelian scheme , in: Proceedings of the Conference on Motives , Seattle 1991, (eds.), U. Jannsen, S. Kleiman and J. P. Serre (AMS Proceedings in Pure Mathematics). · Zbl 0823.14032 [8] J.P. Murre , On a conjectural filtration on the Chow groups of an algebraic variety , in: Proceedings of the Conference on Motives , Seattle 1991, (eds.), U. Jannsen, S. Kleinman and J. P. Serre (AMS Proceedings in Pure Mathematics). · Zbl 0805.14001 [9] M.V. Nori , Algebraic Cycles and Hodge Theoretic Connectivity , preprint University of Chicago (1991). · Zbl 0822.14008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.