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Questions tagged [hilbert-function]

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For questions on the Hilbert function and Hilbert polynomial of graded algebras over fields.

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Score of 4
0 answers
111 views

This old MO question deserves renewed attention in my opinion. The original question defined $$F_n(X) := \sum_{k=0}^n (-1)^k \binom{n}{k} X^{k(n-k)}$$ where $n=2m$ is an even integer, and asked for a ...
Score of 3
0 answers
115 views

Let $G$ be a finitely generated abelian group and $R$ a finitely generated $G$-graded ring. Then we can define the Hilbert function $H_R : G \to \mathbb{N}$ given by $g \mapsto \dim R_g$. I believe it ...
Score of 0
0 answers
134 views

Are there algorithmic procedures for constructing finitely-generated $\mathbb{Z}$-graded $\mathbb{C}$-algebras $R$, given knowledge of the Hilbert function? That is, say I have a sequence $n_1, n_2, \...
Score of 2
0 answers
207 views

Let $I$ be a graded polynomial ideal of $\mathcal{S} = \mathbb{C}[x_1, \ldots, x_n]$, generated by homogeneous polynomials of degree $\leq d$. As I understand, the Hilbert function of $I$ is the ...
Score of 3
0 answers
169 views

Recently I have come across the paper Multigraded Hilbert Schemes by Mark Haiman and Bernd Sturmfels, where they construct a very general object parametrizing homogeneous ideals with fixed Hilbert ...
Score of 1
1 answer
193 views

Let $k$ be a field (assume $k=\mathbb{C}$ if necessary) and let $A=\bigoplus_{d=0}^\infty A_d$ be a finitely generated Gorenstein graded commutative $k$-algebra of Krull dimension $n$ with $A_0=k$. ...
Score of 4
1 answer
266 views

Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
Score of 2
1 answer
303 views

Let $\newcommand{\frakm}{\mathfrak{m}}(R,\frakm)$ be a reduced Noetherian local ring of dimension $d$ and $f\in\frakm^\alpha\setminus\frakm^{\alpha+1}$ a parameter of $R$, i.e. $\dim R/(f)=d-1$. Let $...
Score of 1
0 answers
87 views

I've came across this problem while thinking about some properties of fat schemes. Let me give you an explicit (motivating) example: We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
Score of 12
2 answers
966 views

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
Score of 2
1 answer
578 views

For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...
Score of 3
1 answer
585 views

Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is $$p(t)=rt+r(1-g)+d$$ where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and ...
Score of 3
0 answers
115 views

Let $S=K[x_1,\cdots,x_s]$ be a polynomial ring over a characteristic zero field $K$. For a partition $\{1, \cdots, s\} = I_1 \sqcup \cdots \sqcup I_r$ of the variable index, let $\mathbb{x}_i^{\mathbb{...
Score of 3
1 answer
243 views

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
Score of 4
0 answers
281 views

There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$ in terms of the degrees of $X_1,\ldots,X_i$. Is there a way to ...

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