An invariant detecting rational singularities via the log canonical threshold
An invariant detecting rational singularities via the log canonical threshold
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通过对数正则阈值检测有理奇点的不变量
DOI：
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发表时间：
2019
期刊：
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通讯作者：
M. Mustaţă
中科院分区：
文献类型：
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作者：
R. Cluckers;M. Mustaţă
我们证明，如果 f 是光滑复变 X 上的非零、不可逆函数，并且 J_f 是 f 的雅可比理想，则当且仅当由 f 定义的超曲面具有有理奇点时，lct(f, J_f^2)>1。而且，如果不是这种情况，则lct(f, J_f^2)=lct(f)。我们给出了两个证明，一个依赖于弧空间，另一个证明 f 的最小指数至少与 lct(f, J_f^2) 一样大。在 Q 代数闭包上的多项式的情况下，我们还证明了后一个不等式的类似物，其中最小指数被动机振荡指数 moi(f) 代替。
We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, J_f^2)=lct(f). We give two proofs, one relying on arc spaces and one that shows that the minimal exponent of f is at least as large as lct(f, J_f^2). In the case of a polynomial over the algebraic closure of Q, we also prove an analogue of this latter inequality, with the minimal exponent replaced by the motivic oscillation index moi(f).