Relative Mather discrepancy on arc spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912579349118976 |
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| author | de Fernex, Tommaso Mere, Zach |
| author_facet | de Fernex, Tommaso Mere, Zach |
| contents | Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative Mather discrepancy on arc spaces de Fernex, Tommaso Mere, Zach Algebraic Geometry Primary 14E18, Secondary 14B05 Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring. |
| title | Relative Mather discrepancy on arc spaces |
| topic | Algebraic Geometry Primary 14E18, Secondary 14B05 |
| url | https://arxiv.org/abs/2508.12420 |