论文标题
Grothendieck品种环中的商奇异性
Quotient singularities in the Grothendieck ring of varieties
论文作者
论文摘要
让$ g $为有限的群体，$ x $是一种忠实的$ g $ action的平滑复杂的投影品种，而$ y $是$ x/g $的奇异点的分辨率。 Larsen和Lunts询问$ [x/g] - [y] $是否可以在变种的Grothendieck环中被$ [\ Mathbb {a}^1] $排除。我们表明，如果$ bg $不是$ g $是Abelian，则答案是负面的。对于某些平滑的投射品种$ z $和$ g = s_n $作为因素置换的情况时，$ x = z^n $特别值得一提。我们通过证明$ [z^n / s_n] - [z \ langle n \ rangle / s_n] $可以由$ [\ mathbb {a}^1] $对，其中$ z \ langle n \ rangle $是ulyanov的polydiagonal compactification ulyanov $ n $ n $ n $ z $ z $ z $ z $
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 $Z$.
