The authors study the nontriviality of the normal function on Mg given by the Abel-Jacobi invariant of the cycle C+ − C− associated to a curve C in its Jacobian. This study was first carried out by G. Ceresa [Ann. of Math. (2) 117 (1983)] by the degeneration method; this led him to conclude that for g \geq 3 and general C, C+ is not algebraically equivalent to C− in JC, following a path of reasoning initiated by Griffiths. Here the approach is infinitesimal: the authors study in detail the infinitesimal invariant , which is at the point C an element of the dual of Ker( P^{g−2,g−1} \otimes H^1 (T_C) \to P^{g−2,g−1}, where P denotes the primitive part of H^{2g−3}(JC) and the map is “IVHS”. (This \delta_{\nu} is constructed using the differential of \nu , and was first introduced by Griffiths.) This study allows the authors to conclude that does not vanish on subloci of Mg of codimension less than (g +2)/3, g > 3, not contained in the hyperelliptic locus, which shows that on such a subvariety C+ is not algebraically equivalent to C− at the general point. It is also shown that, suitably interpreted, for g = 3, C non-hyperelliptic, gives exactly the equation of C in its canonical embedding. (Review by Claire Voisin)
The Griffiths infinitesimal invariant for a curve in its Jacobian.
COLLINO, Alberto;
1995-01-01
Abstract
The authors study the nontriviality of the normal function on Mg given by the Abel-Jacobi invariant of the cycle C+ − C− associated to a curve C in its Jacobian. This study was first carried out by G. Ceresa [Ann. of Math. (2) 117 (1983)] by the degeneration method; this led him to conclude that for g \geq 3 and general C, C+ is not algebraically equivalent to C− in JC, following a path of reasoning initiated by Griffiths. Here the approach is infinitesimal: the authors study in detail the infinitesimal invariant , which is at the point C an element of the dual of Ker( P^{g−2,g−1} \otimes H^1 (T_C) \to P^{g−2,g−1}, where P denotes the primitive part of H^{2g−3}(JC) and the map is “IVHS”. (This \delta_{\nu} is constructed using the differential of \nu , and was first introduced by Griffiths.) This study allows the authors to conclude that does not vanish on subloci of Mg of codimension less than (g +2)/3, g > 3, not contained in the hyperelliptic locus, which shows that on such a subvariety C+ is not algebraically equivalent to C− at the general point. It is also shown that, suitably interpreted, for g = 3, C non-hyperelliptic, gives exactly the equation of C in its canonical embedding. (Review by Claire Voisin)
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