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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.17587 |
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| _version_ | 1866911119647440896 |
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| author | Burke, Andrew |
| author_facet | Burke, Andrew |
| contents | We realize a graded variant $K_0(Var_k^{dim})$ of the Grothendieck ring of varieties as a quadratic extension of the subring $K_0(Var_k^{sp})$ spanned by classes of smooth and proper varieties. As such, there exists a natural involution $\mathbb{D}$ on $K_0(Var_k^{dim})$. We show that $\mathbb{D}$ commutes with the symmetric power operations $Sym^m$ up to zero divisors. Moreover, we study varieties which are smooth up to cut-and-paste relations, which we call $\mathbb{D}$-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities Burke, Andrew Algebraic Geometry 14B05, 14G10 We realize a graded variant $K_0(Var_k^{dim})$ of the Grothendieck ring of varieties as a quadratic extension of the subring $K_0(Var_k^{sp})$ spanned by classes of smooth and proper varieties. As such, there exists a natural involution $\mathbb{D}$ on $K_0(Var_k^{dim})$. We show that $\mathbb{D}$ commutes with the symmetric power operations $Sym^m$ up to zero divisors. Moreover, we study varieties which are smooth up to cut-and-paste relations, which we call $\mathbb{D}$-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions. |
| title | Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities |
| topic | Algebraic Geometry 14B05, 14G10 |
| url | https://arxiv.org/abs/2508.17587 |