Vanishing criteria for Ceresa cycles

Abstract

Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove vanishing criteria for the Ceresa cycle $\kappa (C)\in {\mathrm{CH}}_1(J)\otimes \mathbb{Q}$ in the Chow group of 1-cycles on $J$. Namely, $(A)$ If ${\mathrm{H}}_{\mathrm{prim}}^3(J{)}^{\mathrm{Aut}(C)}=0$, then $\kappa (C)$ vanishes; $(B)$ If ${\mathrm{H}}^0(J,{\Omega }_J^3{)}^{\mathrm{Aut}(C)}=0$ and the Hodge conjecture holds, then $\kappa (C)$ vanishes modulo algebraic equivalence. We then study the first interesting case where $(B)$ holds but $(A)$ does not, namely the case of Picard curves $C:y^3=x^4+a{x}^2+bx+c$. Using work of Schoen on the Hodge conjecture, we show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. Moreover, we determine exactly when it is torsion in the Chow group. As a byproduct, we show that there are infinitely many plane quartic curves over $\mathbb{Q}$ with torsion Ceresa cycle (in fact, there is a one parameter family of such curves). Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over ${\mathcal{M}}_3$.

arXiv comment24 pages, added more examples, comments welcome

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