论文标题
边界与标记基地之间的密切关系
The close relation between border and Pommaret marked bases
论文作者
论文摘要
在多项式环$ k [x_1，\ dots，x_n] $中，给定有限订单，在$ k $上，让$ \ partial \ mathcal o $是$ \ mathcal o $和$ \ \ \ \ \ \ \ \ \ m nathcal p _ {\ mathcal o} $ pommaret $ $ o $ $ o pommare $ of fequial o $ $ o pommare $ of的$ o $ o y $ o $的边界。在2019年Ceria，Mora，Roggero引入的还原结构的框架中，我们研究了$ \ partial \ Mathcal O $ $ $ $ $ $ $ $ $ $ $ $ $ \ MATHCAL P _ {\ MATHCAL P _ {\ MATHCAL O} $ - 标记的集合（base）之间的关系。 我们证明，仅当$ \ narccal p _ {\ Mathcal p _ {\ Mathcal O} $ - 标记的$ b $中包含的$ p $的标记$ b $是标记的$ b $时，$ \ partial \ mathcal o $ $ $ $ $ b $是标记的基础。使用对这些标记碱的功能描述，作为副产品，我们获得了仿射方案分别参数化$ \ partial \ natercal o $ $ $标记的基础和$ \ Mathcal p _ {\ Mathcal O} $ - 标记的基础是异形的。我们能够将这种同构描述为可以在不使用Gröbner消除技术的情况下明确构建的投影。特别是，我们在较小的仿射空间中直接嵌入边界方案。此外，我们观察到，标有pommaret方案的守时，守恒的希尔伯特计划的封面。在所有论文中给出了几个例子。
Given a finite order ideal $\mathcal O$ in the polynomial ring $K[x_1,\dots, x_n]$ over a field $K$, let $\partial \mathcal O$ be the border of $\mathcal O$ and $\mathcal P_{\mathcal O}$ the Pommaret basis of the ideal generated by the terms outside $\mathcal O$. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $\partial\mathcal O$-marked sets (resp. bases) and $\mathcal P_{\mathcal O}$-marked sets (resp. bases). We prove that a $\partial\mathcal O$-marked set $B$ is a marked basis if and only if the $\mathcal P_{\mathcal O}$-marked set $P$ contained in $B$ is a marked basis and generates the same ideal as $B$. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $\partial\mathcal O$-marked bases and $\mathcal P_{\mathcal O}$-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in smaller affine spaces. Furthermore, we observe that Pommaret marked schemes give an open covering of punctual Hilbert schemes. Several examples are given along all the paper.
