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Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.

These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.

The typical situations where these notions are used are the following:

  • The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree.
  • The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
  • The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.

The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.

Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.

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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 $\mathcal F$ be a coherent sheaf on $X$. Show that…
Potato
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Computation of a Hilbert Samuel function

I am trying to solve the following exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry: Exercise 12.1: Let $f\in R = k[x,y,z]_{(x,y,z)}$ be a homogeneous form of degree $d$, monic in $x$. Show that $(y,z)$ is an ideal…
user940160
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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 function is defined…
Wenzhe
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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 characteristic positive? Thank you
Ella Smith
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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? Thanks a lot for your answers.
Lina45
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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 homogeneous ideal $I(X)$ is generated by…
GSF
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Hilbert scheme of points of complex surface

Let $X$ be a smooth algebraic surface over $\mathbb{C}$ i.e a smooth and connected scheme of finite type over $\operatorname{spec}(\mathbb{C})$ of dimension $=2$. I denote with $X^{[n]}$ the Hilbert scheme of $0$-dimensional subscheme of length…
Tommaso Scognamiglio
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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 true that $R$ has minimal multiplicity i.e.…
user
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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 function of $I$ is $$ h_I: \mathbb{N}_0 \to…
Qi Zhu
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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. Let $X \subseteq \mathbb{P}^n_T$ be a closed…
Jupp
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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 has non-vanishing $z$ coordinate, and hence we can…
Luke
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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 closed subscheme of $X$. Suppose $L_1$ and $L_2$ are…
Schemer
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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 $\mathcal{O}_X(n)$ is $n$-twisting sheaf on $X$? On the one hand,…
evgeny
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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 non-zero homogeneous elements of degree…
user 3462
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Complete local Cohen-Macaulay ring of dimension $1$ whose type equals $1$ less than embedding dimension

Let $(R,\mathfrak m,k)$ be a complete local Cohen-Macaulay ring of dimension $1$. The type of $R$ is then given by $\dim_k \text{Ext}^1_R(k,R)=\mu(\omega)$, where $\omega$ is the canonical module of $R$. My question is: If $R$ has minimal…
Snake Eyes
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