On the dimension of Voisin sets in the moduli space of abelian varieties
We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting in the collection of points $x \in A$ such that the zero-cycle $\{x\}-\left\{0_A\right\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/190458 |
| Acceso en línea: | https://hdl.handle.net/2445/190458 |
| Access Level: | acceso abierto |
| Palabra clave: | Varietats abelianes Geometria algebraica Cicles algebraics Abelian varieties Algebraic geometry Algebraic cycles |
| Sumario: | We study the subsets $V_k(A)$ of a complex abelian variety $A$ consisting in the collection of points $x \in A$ such that the zero-cycle $\{x\}-\left\{0_A\right\}$ is $k$-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $\operatorname{dim} V_k(A) \leq k-1$ and $\operatorname{dim} V_k(A)$ is countable for a very general abelian variety of dimension at least $2 k-1$. We study in particular the locus $\mathcal{V}_{g, 2}$ in the moduli space of abelian varieties of dimension $g$ with a fixed polarization, where $V_2(A)$ is positive dimensional. We prove that an irreducible subvariety $\mathcal{Y} \subset \mathcal{V}_{g, 2}$, $g \geq 3$, such that for a very general $y \in \mathcal{Y}$ there is a curve in $V_2\left(A_y\right)$ generating $A$ satisfies $\operatorname{dim} \mathcal{Y} \leq 2 g-1$. The hyperelliptic locus shows that this bound is sharp. |
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