Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

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Question about Hilbert Series and SU(n) characters

The following question stems from page 10 of this dissertation. The Hilbert series is defined for $\mathbb{C}^n$ by $$HS(t_1, t_2, \ldots, t_n; \mathbb{C}^n) = \sum_{i_1, \ldots, i_n = 0}^{\infty} ...
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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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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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Non-zero divisor and Hilbert function

Consider $T=k[y_1,\ldots,y_n]$ the polynomial ring in $n$ variable over $k$. I want to prove that, if $I\subset T$ is an ideal and $y_1$ is not a zero-divisor in $T/I$, then $$HF(T/I,t)=\sum_{i=0}^t ...
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Question on flat morphisms and Hilbert polynomials

I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in lecture 16 about which I feel confused. Оказывается, что для пучков на проективном пространстве, ...
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7 points in $\mathbb {P}^3$ in general linear position

Let 7 points in $\mathbb {P}^3$ in general linear position, i.e. no 3 are on a line, no 4 are one a plane, etc. If I denote with $R $ the ring associated to this variety show that $H_R (0)=1$ $...
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Hilbert function of a primary ideal

I'm reading Geramita's lectures on fat points, but I found a passage a bit confusing. (I'll post the reference as soon as I'll find the link). I read that a fat point (of order $t$, defined by the ${p}...
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Hilbert function for points in projective space

In Lecture 13 of Harris's Algebraic Geometry, we have the Hilbert functions, $h_X$, for the following two cases: If $X=\{p_1,p_2,p_3\}\subseteq \mathbb{P}^2$ then there are two possible Hilbert ...
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Why is : $ \mathrm{dim} \ ( S_X )_d = \begin{pmatrix} n+d \\ n \end{pmatrix} - \begin{pmatrix} n+d-s \\ n \end{pmatrix} $? [duplicate]

In page : $206$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf , the author says that the Hilbert function of $ S_X $ is : $$ \mathrm{dim} \...
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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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How do I compute the reduced Groebner Basis?

I am trying to do a question on Hilbert's Weak Nullenstratz theorem for the 3 colouring of vertices and i know i need to compute the reduced Groebner Basis (GB) for the following Ideal: $$I:= <x_{...
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Hilbert polynomial for $X=Z(x^2+y^2+z^2+w^2)\subset \mathbb{P}^3$

I'm beginning to learn about Hilbert polynomials and I'm trying to find it for the variety $X=Z(x^2+y^2+z^2+w^2)\subset \mathbb{P}^3$. I know that the leader term must be of the form $\frac{2}{2!}t^...
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Ideal associated to a set of points of $\mathbb{P}^n$ in general position

In The homogeneous ideal of $2n$ points in general position in $\mathbb{P}^n$, we let $\Gamma$ be a set of $d=2n$ points in general position in $\mathbb{P}^n$, and we want to show that the associated ...
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Nakajima's Hilbert Polynomial

Nakajima gives a definition of the Hilbert polynomial on p. 5 here that I can't explain or find explanation for anywhere else. It goes, $X$ is a projective scheme over algebraically closed field with ...
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Degree of Projections???

Let $\ f_1,\cdots, f_s \in K[x_0, x_1, \cdots, x_n]$ and $F_1 = f_1, \cdots, F_s = f_s \in K[x_0, x_1, \cdots, x_n, x_{n+1}]$, also let $X = V(f_1,\cdots, f_s) \subset \mathbb{P}^n$ and $X' = V(F_1,\...
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leading coefficient of Hilbert polynomial

Let $R=\mathbb{C}[x_0,x_1,...,x_n]$ and $I$ be a homogeneous $J$-primary ideal, where $J=\sqrt{I}$ and $J$ is a homogeneous prime ideal. Assume $V(J)$ is a $d$-dim projective subvariety inside $\...
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Hilbert polynomial and dimension of $M \otimes K(x_1,\dots,x_n)$

Let $K$ a field and $M$ a finetely generated graded module over $K[x_1,\dots,x_n]$. My goal is to study the dimension of $M \otimes_{K[x_1,\dots,x_n]} K(x_1,\dots,x_n)$ as vector space over $L=K(x_1,\...
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Hilbert polynomial of the pullback of a line bundle, a particular case

I am working on a particular case of the following problem. Let $X$ be a projective algebraic surface, $L$ a base point free invertible sheaf on $X$ and $\varphi:X\rightarrow \mathbb{P}^n$ the ...
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Question about Hilbert polynomials and Reduction of ideals

Let $(R, m)$ be a C-M local ring with infinite residue field and $I$ an $m$-primary ideal. Does there always exist a minimal reduction $J$ of $I$ such that $r_J(I) = n(I) + d$?
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Proof of Theorem 9.9 in Part III of Hartshorne's Algebraic Geometry

I am studying section III.9 on flat morphisms of Hartshorne's Algebraic Geometry and stuck in the proof of the following Theorem 9.9 (Hartshorne, page 261). Let $T$ be an integral noetherian scheme. ...
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$\sum_n \dim(R_n)z^n $

I solved the following problem using Molien's theorem, but I would like to know another solution. Let $R = \oplus R_n$ be the subring of the graded ring $\mathbb{C}[x,y]$ of polynomials invariant ...
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Computation of Hilbert series involving floor

Let $S$ be the graded polynomial ring $k[x_1,x_2]$ such that $x_i$ has degree $i$. Then it's pretty easy to show that $H_S(n)=\text{floor}(n/2)+1$. Now I'm trying to show that $\sum_{n>0}H_S(n)t^n$...
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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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Usefulness of the notion of Hilbert scheme in algebraic geometry.

Could someone tell me why and how Hilbert schemes and relative Hilbert schemes are important and useful in algebraic geometry? Could anyone give me some applications of this notion in concrete terms? ...
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Computation of basic Hilbert functions

I want to compute the Hilbert function for the ring $$M:=\frac{k[x,y,z,w]}{(x,y)\cap(z,w)}$$ and compare it to the Hilbert function for the ring $$N:=k[x,y].$$ I tried computing the bases for each $...
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Push-forward of coherent sheaves and Hilbert polynomials

Let $k$ be an algebraically closed field of characteristic zero, $X, Y$ be projective $k$-schemes. Fix closed immersions $i:X \hookrightarrow \mathbb{P}^n$ and $j:Y \hookrightarrow \mathbb{P}^m$ for ...
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Multiplicity is always positive

Let $R$ be a homogeneous noetherian ring with $R_0$ artinian (e.g. $R = k[x_1,\dots, x_n]$), and $M$ a finitely generated graded $R$-module. I want to show that $e(M)$, the multiplicity of $M$, is ...
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In zero dimensional graded module the degree $ 1$ component is non zero.

I am reading the commutative algebra book by W. Bruns and H. Herzog. I am stuck at the Corollary 4.1.8 which comes from lemma 4.1.7. Actually in the corollary $d=0$ case does not come from the ...
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Lemma 4.1.7. in Bruns and Herzog, Cohen-Macaulay Rings

I am reading Hilbert Polynomial chapter from Commutative Algebra book by Bruns and Herzog. I stuck on Lemma 4.1.7. I failed to understand how the proof goes in the second line i.e., $(1-t)^d H(t)=\...
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Zero divisor on $S/I$ when $\mathrm{depth}(S/I) >0.$

Consider the polynomial ring $S=K[X_1,\ldots,X_n]$, where $K$ is a field and let $I \subset S$ be a homogeneous ideal under standard grading. If $\mathrm{depth}(S/I) >0$, then how can I show that ...
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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 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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if the Hilbert polynomial of coherent sheaf is constant number,then locally free?

I'm reading "The Geometry of Moduli Spaces of Sheaves" (Huybrechts Lehn). I want to prove [$Quot_{X/S}(H,l)=Grass_S(H,l)$]where X=S and l is a number. Let T be a S-scheme and [ρ:$H_T \rightarrow F$]$...
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Two definitions of Hilbert series/Hilbert function in algebraic geometry

In classical algebraic geometry, suppose $I$ is a reduced homogeneous ideal in $k[x_0,\cdots,x_n]$, where $k$ is algebraically closed field, then $I$ cuts out a projective variety $X$, whose Hilbert ...
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Poincaré series pole at $1$

Let $A$ be a graded ring and $M$ a graded $A$-module. By $P(M,t)$ we denote the Poincaré series for $M$. In Atiyah and Macdonald, theorem 11.1 claims $P(M,t)=\dfrac{f(t)}{\prod _{i=1}^n (1-t^{k_i})}$...
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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 ...
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Hartshorne problem III.5.2(a)

Consider problem III.5.2(a) in Hartshorne's Algebraic Geometry: Let $X$ be a projective scheme over a field $k$, let $\mathcal O_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let $\...
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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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Hilbert polynomial: two definitions

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 $\mathcal{O}_X(n)$ ...
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Computing Hilbert polynomial

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 ...