论文标题
在光滑品种上的局部局部结构上的本地结构
On the local structure of the Brill-Noether locus of locally free sheaves on a smooth variety
论文作者
论文摘要
我们研究了局部免费的捆绑纸条$ e $ of $ \ nathcal {o} _x $ -Modules的函数$ \ operatorAname {def} _e^k $的无限型捆绑$ e $ e $ e $ e $ e $ e $ e $ e $ e $ e $ e $ e $ e $ - _x $ -Modules $ x $，因此至少$ k $独立的部分将$ k $独立地提升至$ h^0（e）$ h^0（e）。我们推断出一些有关$ e $的$ k $ -th brill-noether基因座的信息，例如在某些平稳的点上对切线锥的描述，在某些光滑的空间中对切线空间的描述，以及函数$ \ operator $ \ operatorname {def} _e _e^k $的平滑度之间的某些链接，以及某些知识渊博的变形者的平稳性和与其相关的MODULI SPES的平稳性。作为调查$ \ operatorname {def} _e^k $的工具，我们研究了Pairs $（E，U）$的无穷小变形，其中$ u $是$ e $的部分的线性子空间。我们概括了$ e $具有任何等级和$ x $任何维度的情况，许多经典结果都涉及相干系统的模量空间，例如对其切线空间的描述以及其平滑度和Petri Map的注入性之间的链接。
We study the functor $\operatorname{Def}_E^k$ of infinitesimal deformations of a locally free sheaf $E$ of $\mathcal{O}_X$-modules on a smooth variety $X$, such that at least $k$ independent sections lift to the deformed sheaf, where $h^0(E) \geq k$. We deduce some information on the $k$-th Brill-Noether locus of $E$, such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor $\operatorname{Def}_E^k$ and the smoothness of some well know deformations functors and their associated moduli spaces. As a tool for the investigation of $\operatorname{Def}_E^k$, we study infinitesimal deformations of the pairs $(E,U)$, where $U$ is a linear subspace of sections of $E$. We generalise to the case where $E$ has any rank and $X$ any dimension many classical results concerning the moduli space of coherent systems, like the description of its tangent space and the link between its smoothness and the injectivity of the Petri map.
