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摘要翻译：
设$k$为字段，$v$和$w$为$k$-维数为$m$和$n$的向量空间。设$\phi$为从$hom(V,W)$到$hom(\wedge^t V,\wedge^t W)$的规范映射。我们研究$\phi$的映像$Y_T$的Zariski闭包$X_T$。在$t=\min(m,n)$的情况下，$y_t=x_t$是Grassmannian上的锥体，但是$x_t$大于$y_t$（对于$1<t<\min(m,n)$）。通过相应的$G$-稳定素理想，分析了$X_t$中的$G=\gl(V)\times\gl(W)$-轨道。结果表明，它们是由两个数值不变量分类的，其中一个是秩，另一个是相关的不变量，我们称之为小秩。令人惊讶的是，$x_t\setminus y_t$中的轨道来自于$u<t$的镜像$y_u$和简单的代数运算。在最后一节中，我们确定了$x_t$的奇异轨迹。除了很好理解的例外情况之外，它是由$Y_T$中的rank$\le1$元素组成的。
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英文标题：
《The variety of exterior powers of linear maps》
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作者：
Winfried Bruns, Aldo Conca
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Commutative Algebra 交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $\phi$ be the canonical map from $Hom(V,W)$ to $Hom(\wedge^t V,\wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $\phi$. In the case $t=\min(m,n)$, $Y_t=X_t$ is the cone over a Grassmannian, but $X_t$ is larger than $Y_t$ for $1<t<\min(m,n)$. We analyze the $G=\GL(V)\times\GL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime ideals. It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in $X_t\setminus Y_t$ arise from the images $Y_u$ for $u<t$ and simple algebraic operations. In the last section we determine the singular locus of $X_t$. Apart from well-understood exceptional cases, it is formed by the elements of rank $\le 1$ in $Y_t$.
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PDF链接：
https://arxiv.org/pdf/0705.3399
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