Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

Filter by
Sorted by
Tagged with
1
vote
1answer
54 views

Hilbert series of $k[x,y]/(x^2,y^3)$

This is Example 1.7 in Graded Syzygies by Peeva, and I am wondering how to work this out. The author writes Let $A=k[x,y]$ and let $J=(x^2,y^3)$. Then $A/J$ is graded with basis $\{1\}$ in degree $0$,...
0
votes
0answers
75 views

Hilbert polynomial of the blow-up of a projection

Let $p:\mathbb{P}^r\dashrightarrow\mathbb{P}^{r-k-1}$ with $0\le k<r$ be the projection $(a_0:\cdots:a_r)\longmapsto (a_{k+1}:\cdots:a_r)$. Such a rational map has base $K:=V(X_{k+1},\cdots,X_r)\...
2
votes
1answer
69 views

Hilbert polynomials of closed subschemes $Y \subset \mathbb{P}^n_k$

Let $k$ any field and $Y \subset \mathbb{P}^n_k$ closed subscheme. We are going to study the Hilbert polynomials in reversal sense: Let recall a Hilbert polynomial associated to a coherent sheaf $F$ ...
2
votes
0answers
99 views

Hilbert scheme of hypersurfaces in Nitsure’s Construction of Hilbert and Quot Schemes

I have a few questions on the construction of Hilbert scheme of hypersurfaces I found recently in Nitin Nitsure’s Construction of Hilbert and Quot Schemes (page 6 part 4) In his paper Nitsure added a ...
1
vote
2answers
128 views

Resolving indeterminacy of map on projective spaces induced by a linear map

Let $V$ and $W$ be two $k$-vector spaces, where $k$ can be assumed as $\mathbb{C}$, and $f:V\longrightarrow W$ a non null $k$-linear map with kernel $K$. The map $f$ naturally induces a rational ...
1
vote
0answers
50 views

Hilbert functions for affine vs projective varieties and filtered vs graded algebras.

I'm somewhat confused about how to define Hilbert functions for affine varieties and for filtered rings in a compatible way. I'm familiar with how they are defined for projective varieties: Let $X$ be ...
0
votes
0answers
37 views

What is the Hilbert polynomial of this simple scheme

Denote by $S'$ the follow subset of $(\mathbb{P})^{k+1}$ $$\{((a_0:\cdots:a_r),(0:\cdots:0:a_{r_1}:a_{r_1+1}:\cdots:a_r),\dots,(0:\cdots:0:a_{r_1+r_2+\cdots+r_k}:a_{r_1+r_2+\cdots+r_k+1}:\cdots:a_r)...
1
vote
1answer
31 views

What is the Hilbert Polynomial of the diagonal inside the product os projective space

Let $\Delta$ be the diagonal in $\mathbb{P}^r\times\cdots\times\mathbb{P}^r$ ($k$ times), that is, $\Delta=\mbox{Proj}S/I$, where $S$ is the multi-graded k-algebra $S=k[X_0^1,\cdots,X_r^1,\cdots, X_0^...
1
vote
1answer
85 views

Hilbert Polynomial at a point?

One of the review problems in my final review is the following: Let $X\subset\mathbb P_{\mathbb C}^n$ be a hypersurface, and $P\in X$ a singular point. Let $L$ be a line not contained in $X$ that ...
1
vote
1answer
37 views

Hilbert series of agebraically independents polynomial - problem with a proof

I'm studying Sturmfels's "Algorithms in Invariant theory", and in particolar this result (page 30) Lemma. Let $p_1,\ldots,p_m$ be algebraically independent elements of $\mathbb{C}[x_1,\ldots,x_n]$ ...
1
vote
0answers
43 views

Grassmannian as Hilbert scheme.

Given a Hilbert polynomial $f(x)=\binom{x+r}{r}\in \mathbb{Q}(x)$ where $r>0$. It’s said that Grassmannian scheme $Grass(n+1,r+1)$ over $\mathbb{Z}$ parameterises a tautological family of subscheme ...
0
votes
1answer
37 views

Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed

Let $R=\oplus_{i\ge 0} R_i $ be a graded domain such that $R_0=k$ is an algebraically closed field, $R$ is finitely generated $k$-algebra and $R=k[R_1]$. Let $d=\dim R>0$. Let $\mathfrak m=\oplus_{...
1
vote
1answer
56 views

Dimension in Hilbert function definition. And “for large s” statement confusion in Eisenbud commutative algebra

On page 42 of Eisenbud's commutative algebra textbook I ran into some problems I would like some expert guidance with. We have the definition: Let $M$ be a finitely generated graded module over $K[...
1
vote
0answers
110 views

Computing Hilbert-Samuel multiplicity of $k[[f_1(t),…,f_n(t)]]$

Let $k$ be a field and $k[[t]]$ be the formal power series ring over $k$. Consider the subring $R=k[[f_1(t),...,f_n(t)]]$ for some $f_1(t),...,f_n(t)\in k[[t]]$ such that $k[[t]]\subseteq Q(R)$ and ...
2
votes
0answers
67 views

Hilbert polynomial of algebraic variety with 4 generators

I wanted to compute the Hilbert polynomial of the algebraic variety $Z(f,f_1, f_2, f_3) \subset \mathbb{P}^5$. Here $f = a_0^3 + 4a_0^2a_5 + 2a_0a_3a_2 + 4a_0a_1a_2 + 4a_3a_5a_2 + a_1a_2^2 \in K[a_0,...
0
votes
0answers
56 views

Algebraic characterization (or sufficient condition) when a (graded) local hypersurface has rational singularity

Let $(S, \mathfrak n)$ be a regular local ring of dimension $d\ge 4$ and let $R=S/(f)$ , where $0\ne f \in \mathfrak n^2$. Then $\dim R=d-1\ge 3$. If $\mathfrak m$ is the maximal ideal of $R$ then $\...
0
votes
1answer
95 views

On the intersection of integral closures of all powers of maximal ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring of positive dimension. For an ideal $J$ of $R$, let $\bar J$ denote the integral closure of $J$ (https://en.m.wikipedia.org/wiki/...
2
votes
1answer
121 views

On the $k$-vector space dimension of graded pieces of an Artinian $k$-algebra $k[x,y]/J$

Let $R=k[x,y]$ be a polynomial ring in two variables over an infinite field $k$. Let $\mathfrak m=(x,y)$. Let $J$ be a homogeneous ideal whose radical is $\mathfrak m$. Consider the standard grading ...
2
votes
1answer
120 views

Hilbert-Samuel multiplicity of a local ring of positive dimension is positive?

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Then $e(R)=(d-1)!\lim_{n\to \infty} \mu (\mathfrak m^n)/n^{d-1}$. (Here $e(R)$ denotes Hilbert-Samuel multiplicity of $R$) . ...
1
vote
0answers
45 views

Writing down the graded pieces of $k[x_1,…,x_n]/(x_1^2,…,x_n^2) $ in terms of the irrelevant maximal ideal

Let $k$ be a field of characteristic $0$. Consider the $\mathbb N$-graded (polynomial) ring $R=k[x_1,...,x_n]$ graded in the standard way that is the $d$-th graded piece $R_d$ is the $k$-vector space ...
5
votes
0answers
102 views

On a special kind of local Gorenstein ring of dimension $2$

Let $(R, \mathfrak m,k)$ be a local Gorenstein ring of dimension $2$ such that $\mu (\mathfrak m^2)(=\dim_k \mathfrak m^2/\mathfrak m^3) =3$ . Then is it true that $R$ is regular ? Or at least is it ...
5
votes
0answers
67 views

Demystify the Hilbert Function

Let $I \subseteq k[x_1,\dots,x_n]$ be an ideal for a field $k$ and let $A = k[x_1,\dots,x_n]/I$. For $d \geq 0$ let $$ A_{\le d} := \{f + I : f \in k[x_1,\dots,x_n], \deg{f} \leq d \}.$$ The Hilbert ...
0
votes
1answer
138 views

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 $X$ ...
2
votes
1answer
123 views

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 ...
1
vote
0answers
82 views

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} ...
1
vote
0answers
29 views

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 ...
1
vote
1answer
82 views

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 $H_R (0)=1$ $...
3
votes
0answers
91 views

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}...
0
votes
0answers
347 views

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 ...
2
votes
1answer
133 views

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, ...
1
vote
1answer
122 views

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} \...
0
votes
0answers
54 views

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....
1
vote
0answers
110 views

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. ...
2
votes
1answer
99 views

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 ...
1
vote
1answer
210 views

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_{...
1
vote
0answers
85 views

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 ...
0
votes
0answers
151 views

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,\...
3
votes
0answers
202 views

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 $\...
3
votes
1answer
121 views

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,\...
2
votes
0answers
119 views

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 ...
1
vote
1answer
99 views

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^...
0
votes
0answers
38 views

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 $r_J(I) = n(I) + d$?
5
votes
2answers
278 views

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. ...
1
vote
0answers
109 views

$\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 ...
6
votes
1answer
224 views

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 ...
0
votes
1answer
108 views

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 $\sum_{n>0}H_S(n)t^n$...
2
votes
1answer
152 views

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} ...
1
vote
2answers
160 views

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 $1+t+t^2+\dots = 1/(1-t)$.
2
votes
1answer
222 views

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 $...
0
votes
0answers
148 views

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 ...