关于Hilbert格式上Borel-不动点的Galligo定理的推广
On an extension of Galligo's theorem concerning the Borel-fixed points on the Hilbert scheme
摘要：
给定一个理想$I$和一个权向量$w$，它对单项式进行部分排序，我们可以考虑具有相同希尔伯特函数的初始理想$\init_w (I)$。一个公知的构造通过一个$\GL_{n+1}$的单参数子群来实现这一点，然后可以将其视为相应希尔伯特方案上的曲线。Galligo \cite{galligo}证明，如果$I$是在一般坐标，如果$w$诱导单项式阶到足够大的程度，那么$\init_w(I)$是固定的行动的博雷尔子群的上三角形矩阵。我们证明了路径的方向接近这个Borel不动点的希尔伯特计划也是Borel固定的。
英文摘要：
Given an ideal $I$ and a weight vector $w$ which partially orders monomials we can consider the initial ideal $\init_w (I)$ which has the same Hilbert function. A well known construction carries this out via a one-parameter subgroup of a $\GL_{n+1}$ which can then be viewed as a curve on the corresponding Hilbert scheme. Galligo \cite{galligo} proved that if $I$ is in generic coordinates, and if $w$ induces a monomial order up to a large enough degree, then $\init_w(I)$ is fixed by the action of the Borel subgroup of upper-triangular matrices. We prove that the direction the path approaches this Borel-fixed point on the Hilbert scheme is also Borel-fixed.
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