Mixed Ladder Determinantal Varieties
Mixed Ladder Determinantal Varieties
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混合阶梯行列式品种
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
C. Miller
中科院分区：
文献类型：
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作者：
N. Gonciulea;C. Miller
摘要本文研究了由单侧梯子L的不同区域（阶梯）中可能不同大小的子式的理想所定义的梯子行列式簇。这些簇是经典阶梯行列式簇的重要推广（即，与等大小的子式），因为他们是非常密切相关的舒伯特品种，这是本文的第一个主要结果。我们表明，它们对应于相反的细胞在舒伯特品种在旗品种的类型A n。由此，我们可以推导出这些具有混合大小子式理想的单边阶梯行列式簇的正规性和Cohen-Macaulay性，以及它们具有有理奇点的事实。接下来，我们证明，直到仿射空间的乘积，这些簇中的每一个都是经典阶梯行列式簇中的基本开集（即，与相等大小的未成年人），它包含作为一个基本的开放集另一个经典的阶梯行列式品种。这个结果，沿着与一般的本地化引理用来显示它，使我们能够计算的除数类群和奇异轨迹的坐标环的这些品种，以及确定当他们Gorenstein。
Abstract We investigate ladder determinantal varieties defined by ideals of minors of possibly different sizes in the different regions (the steps) of one-sided ladders L . These varieties are an important generalization of the classical ladder determinantal varieties (i.e., with equal-size minors) since they are very closely related to Schubert varieties, this being the first main result of this paper. We show that they correspond to opposite cells in Schubert varieties in flag varieties of type A n . In consequence, one deduces the normality and the Cohen–Macaulayness of these one-sided ladder determinantal varieties with ideals of mixed-size minors, as well as the fact that they have rational singularities. Next we show that, up to products by affine spaces, each of these varieties is a basic open set in a classical ladder determinantal variety (i.e., with equal-size minors) and that it contains as a basic open set another classical ladder determinantal variety. This result, along with a general localisation lemma used to show it, enables us to compute the divisor class group and singular locus of the coordinate rings of these varieties, as well as to determine when they are Gorenstein.