Vanishing criteria for Ceresa cycles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914252650971136 |
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| author | Laga, Jef Shnidman, Ari |
| author_facet | Laga, Jef Shnidman, Ari |
| contents | Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove vanishing criteria for the Ceresa cycle $κ(C) \in \mathrm{CH}_1(J)\otimes \mathbb{Q}$ in the Chow group of 1-cycles on $J$. Namely,
$(A)$ If $\mathrm{H}_{\mathrm{prim}}^3(J)^{\mathrm{Aut}(C)} = 0$, then $κ(C)$ vanishes;
$(B)$ If $\mathrm{H}^0(J, Ω_J^3)^{\mathrm{Aut}(C)} = 0$ and the Hodge conjecture holds, then $κ(C)$ 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 $\mathcal{M}_3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_03891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vanishing criteria for Ceresa cycles Laga, Jef Shnidman, Ari Algebraic Geometry Number Theory 14C25 (Primary) 14H45, 14K12 (Secondary) Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove vanishing criteria for the Ceresa cycle $κ(C) \in \mathrm{CH}_1(J)\otimes \mathbb{Q}$ in the Chow group of 1-cycles on $J$. Namely, $(A)$ If $\mathrm{H}_{\mathrm{prim}}^3(J)^{\mathrm{Aut}(C)} = 0$, then $κ(C)$ vanishes; $(B)$ If $\mathrm{H}^0(J, Ω_J^3)^{\mathrm{Aut}(C)} = 0$ and the Hodge conjecture holds, then $κ(C)$ 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 $\mathcal{M}_3$. |
| title | Vanishing criteria for Ceresa cycles |
| topic | Algebraic Geometry Number Theory 14C25 (Primary) 14H45, 14K12 (Secondary) |
| url | https://arxiv.org/abs/2406.03891 |