论文标题
在最大Cohen-MaCaulay模块上的正常奇异性的爆炸时
On the blow-up of a normal singularity at maximal Cohen-Macaulay modules
论文作者
论文摘要
雷诺（Raynaud）和格鲁森（Gruson）开发了一种理论，即沿着连贯的捆绑$ m $用代代数品种$ x $ x $，因为存在$ x $的爆炸$ x'$ x'$ x'$ x'$ x $，因此，$ m $的“严格转换”是$ x'$的“严格转换”，并且可以满足blight up的通用（最小值）财产。但是，对爆炸的奇异性知之甚少。在本文中，我们证明，如果$ x $是正常的表面奇异性，而$ m $是一种反射$ \ Mathcal {o} _ {x} $ - 模块，那么这种爆炸就自然而然地源于McKay对应的理论。我们表明，通过$ x $的分辨率获得了与$ m $相关的完整的捆绑$ \ nathcal {m} $（即，$ m $的背式反射性船体）获得全球范围的全球套头，然后是$ nikers clients $ nikers $ nikers $ nikers $ nikers $ nights $。此外，我们证明，如果$ x $是戈伦斯坦（Gorenstein），而$ m $在Wunram和Riemenschneider的意义上是特殊的（Bobadilla和作者先前的作品中概括），那么Raynaud和Gruson的爆炸是正常的。最后，我们使用Eisenbud开发的基质分解理论来提供此类爆炸的具体例子。
Raynaud and Gruson developed the theory of blowing-up an algebraic variety $X$ along a coherent sheaf $M$ in the sense that there exists a blow-up $X'$ of $X$ such that the "strict transform" of $M$ is flat over $X'$ and the blow-up satisfies an universal (minimality) property. However, not much is known about the singularities of the blow-up. In this article, we prove that if $X$ is a normal surface singularity and $M$ is a reflexive $\mathcal{O}_{X}$-module, then such a blow-up arises naturally from the theory of McKay correspondence. We show that the normalization of the blow-up of Raynaud and Gruson is obtained by a resolution of $X$ such that the full sheaf $\mathcal{M}$ associated to $M$ (i.e., the reflexive hull of the pull-back of $M$) is globally generated and then contracting all the components of the exceptional divisor not intersecting the first Chern class of $\mathcal{M}$. Moreover, we prove that if $X$ is Gorenstein and $M$ is special in the sense of Wunram and Riemenschneider (generalized in a previous work by Bobadilla and the author), then the blow-up of Raynaud and Gruson is normal. Finally, we use the theory of matrix factorization developed by Eisenbud, to give concrete examples of such blow-ups.
