Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

49 questions
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How to compute this Hilbert polynomial?

I am trying to solve Hartshorne's exercise V $1.2$. Let $H$ be a very ample divisor on the surface $X$, corresponding to a projective embedding $X\subseteq P^N$. Then the Hilbert polynomial of $X$ ...
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On the definition of degree of closed subschemes

$\underline {Background}$:We know that,for a projective variety $X \subset\mathbb{P}^{n}=(\mathbb{K}^{n+1}-{0})/\sim$ we define , degree($X$)=$(r!)$.(leading coefficient of the hilbert polynomial of ...
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Calculating the Hilbert Series for symmetric polynomials

Let $S = \mathbb{C}[x_1,...,x_n]$ be the polynomial ring in $n$ variables, $S_d \subset S$ the subspace of homogeneous polynomials of degree $d$, i.e., the polynomials with the property \begin{align} ...
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Hilbert polynomials in two variables with Macaulay2

In J. Symb. Comput. (1999) 28, 681-710, Levin worked with bifiltered, finitely generated $R$-modules ($R$ being a polynomial ring in two sets of variables) and he found an analogue of the Hilbert ...
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Blow up of a planar curve at a singularity

Assume I have a planar projective curve $C\subseteq \mathbb{P}^2$. Furthermore, assume $C$ has a nodal singularity at some point $Q\in\mathbb{P}^2$. I am looking to resolve $C$'s nodal singularity by ...
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A question about a definition of the Hilbert scheme $\mathcal{H}_P ( \mathbb{P}^n )$.

First of all, i'm sorry for my bad english. I'm from a foreign country. :-) I have some questions about a paragraph appearing in page : $205$ of the following electronic textbook : https://scholar....
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Prove an ideal $I$ defines a finite set of points

Let me start with some background: I am studying defining ideal of finite sets of points and have recently noticed the importance of the Hilbert function and polynomial in investigating this problem. ...
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Binomial coefficients undefined in in the Hilbert polynomial for projective space

Let $k$ be a field and let $X= \mathbb{P}_{k}^{r}$ be the projective space (as a scheme) of dimension $r$ over $k$. Let $\mathcal{O}(d)$ denote the degree $d$ twisted structure sheaf. Then we define ...
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Hilbert Series of $k[x_1, … x_n]$

For the algebra $A = k[x_1, \dots, x_n]$ graded by degree. How does one find the Hilbert series. For a single variable, the hilbert series is simply $1+t+t^2+\dots = 1/(1-t)$.
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Hilbert Polynomial of zero dimensional module

Let $M$ be a finitely generated graded $A$-module with $\dim M=0.$ Let $M$ be $\mathbb Z$-graded and $A$ be $\mathbb N$-graded Noetherian ring. Then how can we say that the Hilbert function of $M$ is ...
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Turing Reduction, Hilbert's 10th Problem

I have the following problem : Using the fact the following language is undecidable H: The set of all multivariate polynomial with integer coefficients [p] such ...
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Hilbert polynomial for a dimension zero projective variety by taking an affine chart

I am looking at exercise 12.21 from Gathmann's notes on algebraic geometry. I am given a homogeneous ideal $$I \unlhd k[x, y, z]$$ with a dimension $0$ projective locus. WLOG, we assume that this ...
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Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
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Hilbert polynomial of iterated Veronese embedding

Let $X=\mathbb{V}(x^2-yz)\subset\mathbb{P}^2$ and consider the Veronese embedding $Y=\mathcal{v}_2(X)\subset\mathbb{P}^5$. Find the Hilbert polynomial, and thus the degree, of $Y$. I know how we can ...
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Hilbert function and series.

If $f$ is a homogeneous polynomial of degree $d$ in a polynomial ring in $t$ variables over a field, and it generates an ideal $I$. Then the Hilbert function of $R/I$ is H(R/I,n) = \binom{n+t-1}{t}-\...
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Is the Hilbert Scheme $\operatorname{Hilb}_P(X)$ independent of the choice of ample line bundle?

The construction of the Hilbert scheme of a projective scheme $X$ requires us to fix an ample line bundle $L$ on $X$ in order to define the Hilbert polynomial $P$. Suppose that $S \subset X$ is a ...
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Is the Hilbert series of graded module interpreted as a rational function meaningful everywhere?

Say we have a graded module $M$ over a field $k$. In my mind I have $M=k[x,y,z]/(y^2-xz)$ graded by rank. The Hilbert series of this is $\sum_{n=0}^{\infty} (2n+1)t^n$. You can represent this as a ...
Why does $\dim \Gamma(X)_n=\chi(\mathcal{O}_X(n))$ (probably, for sufficiently large $n$), where $\Gamma(X)_n$ is $n$-dimensional component of homogeneous coordinate ring of $X$ and $\mathcal{O}_X(n)$ ...
We have the following condition: For each $i=2,...,m$ multiplication by $f_{i}$ is injective on $S/(f_{1},...,f_{i-1})$, where $S=k[T_{0},...,T_{n}]$, $m \leq n$, and the $f_{i} \in S_{d_{i}}$ are ...