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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.17033 |
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| _version_ | 1866909654293938176 |
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| author | Achter, Jeff Casalaina-Martin, Sebastian Vial, Charles |
| author_facet | Achter, Jeff Casalaina-Martin, Sebastian Vial, Charles |
| contents | Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times X)$, a morphism $T \to A$ of varieties over $K$. In this note we show that, if $T$ admits no $K$-point, the data $(T,Z)$ determines a torsor $A^{(T,Z)}$ over $K$ under $A$ and a $K$-morphism $T \to A^{(T,Z)}$. This can be used to provide an obstruction to the existence of algebraic cycles defined over $K$. We then connect this obstruction to some recent results of Hassett--Tschinkel and Benoist--Wittenberg on rationality of threefolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular homomorphisms, with a twist Achter, Jeff Casalaina-Martin, Sebastian Vial, Charles Algebraic Geometry Let $X/K$ be a variety over a field, and $A/K$ an abelian variety. A regular homomorphism to $A$ (in codimension $i$) induces, for every smooth geometrically connected pointed $K$-scheme $(T,t_0)$ and every cycle class $Z \in CH^i(T\times X)$, a morphism $T \to A$ of varieties over $K$. In this note we show that, if $T$ admits no $K$-point, the data $(T,Z)$ determines a torsor $A^{(T,Z)}$ over $K$ under $A$ and a $K$-morphism $T \to A^{(T,Z)}$. This can be used to provide an obstruction to the existence of algebraic cycles defined over $K$. We then connect this obstruction to some recent results of Hassett--Tschinkel and Benoist--Wittenberg on rationality of threefolds. |
| title | Regular homomorphisms, with a twist |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.17033 |