Quotient singularities in the Grothendieck ring of varieties
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929288093106176 |
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| author | Esser, Louis Scavia, Federico |
| author_facet | Esser, Louis Scavia, Federico |
| contents | Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the Grothendieck ring of varieties. We show that the answer is negative if $BG$ is not stably rational and affirmative if $G$ is abelian. The case when $X=Z^n$ for some smooth projective variety $Z$ and $G=S_n$ acts by permutation of the factors is of particular interest. We make progress on it by showing that $[Z^n/S_n]-[Z\langle n\rangle / S_n]$ is divisible by $[\mathbb{A}^1]$, where $Z\langle n\rangle$ is Ulyanov's polydiagonal compactification of the $n$-th configuration space of $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14313 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quotient singularities in the Grothendieck ring of varieties Esser, Louis Scavia, Federico Algebraic Geometry 14L30, 14E05, 14M20 Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the Grothendieck ring of varieties. We show that the answer is negative if $BG$ is not stably rational and affirmative if $G$ is abelian. The case when $X=Z^n$ for some smooth projective variety $Z$ and $G=S_n$ acts by permutation of the factors is of particular interest. We make progress on it by showing that $[Z^n/S_n]-[Z\langle n\rangle / S_n]$ is divisible by $[\mathbb{A}^1]$, where $Z\langle n\rangle$ is Ulyanov's polydiagonal compactification of the $n$-th configuration space of $Z$. |
| title | Quotient singularities in the Grothendieck ring of varieties |
| topic | Algebraic Geometry 14L30, 14E05, 14M20 |
| url | https://arxiv.org/abs/2208.14313 |