Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
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2009-03
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通讯作者：
M. Kawakita
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作者：
M. Kawakita
我们引入了黎曼-罗赫定理的一种方法来解决固定维度上最小对数差异的有界问题。将其简化为 Gorenstein 终端奇点的情况后，首先我们证明，如果多重性或嵌入维数有界，则其最小对数差异是有界的。其次，我们在没有明确分类的情况下恢复了 Reid 对 Gorenstein 终端三重奇点的表征，以及 Markushevich 的最小对数差异的精确边界。最后我们提供了一个特殊的四重奇点的精确边界，其一般超平面截面有一个终端部分。
We introduce an approach of Riemann--Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that its minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded. Secondly we recover the characterisation of a Gorenstein terminal three-fold singularity by Reid, and the precise boundary of its minimal log discrepancy by Markushevich, without explicit classification. Finally we provide the precise boundary for a special four-fold singularity, whose general hyperplane section has a terminal piece.