Vanishing criteria for Ceresa cycles

TL;DR

This paper establishes criteria for the vanishing of Ceresa cycles on Jacobians of smooth projective curves, explores specific cases like Picard curves, and identifies conditions under which these cycles are torsion or vanish.

Contribution

It provides new vanishing criteria for Ceresa cycles based on cohomological conditions and analyzes the torsion properties of these cycles in specific curve families.

Findings

Vanishing of Ceresa cycle under certain cohomological conditions.

Picard curves have Ceresa cycles that are torsion in the Griffiths group.

Existence of infinitely many plane quartic curves over  with torsion Ceresa cycle.

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 $C \colon y^3 = x^4 + ax^2 + bx + c$. 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 $\mathcal{M}_3$.