# Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

49 questions
76 views

101 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, ...
60 views

190 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 ...
84 views

74 views

149 views

89 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 ...
34 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 $r_J(I) = n(I) + d$?
225 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. ...
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 ...
77 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 $\sum_{n>0}H_S(n)t^n$...
119 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 $1+t+t^2+\dots = 1/(1-t)$.
732 views

### 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? ...
155 views

54 views

### 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 ...
66 views

### 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 ...
154 views

### 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 ...
229 views

### 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 ...
466 views

### 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 ...
183 views

### 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}-\...
364 views

131 views

### 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 ...
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 ...