The geometry of Hilbert functions

Smooth Hilbert schemes: their classification and geometry

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.

Hilbert functions and geometry

This note is devoted to the study of the links between the Hilbert function of a subscheme X of the projective space, and its geometric properties. We will assume that X is arithmetically Cohen-Macaulay, which allows us to characterize its Hilbert function by a d integers, where d is the degree of X. We study in this context the geometric description of special linear systems of dimension maximal with respect to their degree on projective Gorenstein curves.

关于奇异曲线上点希尔伯特方案几何及其动机ζ函数的几何

On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions

本文研究奇异曲线点希尔伯特方案的几何结构及其动机ζ函数，通过定义树结构和计算ζ函数揭示奇点的几何特性。

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. The above module is usually called the Hilbert module over $\mathbb{C}[z_1, \ldots, z_n]$. Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ (or natural function algebras) …