Questions tagged [hilbert-function]
For questions on the Hilbert function and Hilbert polynomial of graded algebras over fields.
31
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
12
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2
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716
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
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When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?
For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...
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For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section?
Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is
$$p(t)=rt+r(1-g)+d$$
where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and ...
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267
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Flatness of affine cone due to semicontinuity theorem
I would like to clarify an important aspect from the discussion in this question.
The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic
Geometry Chap. III page ...
3
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0
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Growth of dimension of a good filtration on finitely generated modules over polynomial rings
Let $S=K[x_1,\cdots,x_s]$ be a polynomial ring over a characteristic zero field $K$.
For a partition $\{1, \cdots, s\} = I_1 \sqcup \cdots \sqcup I_r$ of the variable index, let $\mathbb{x}_i^{\mathbb{...
3
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On the degree of the Hilbert polynomial of a graded module over the Rees algebra
If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
4
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Computing the Hilbert series of an irreducible component of a complete intersection
There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$ in terms of the degrees of $X_1,\ldots,X_i$. Is there a way to ...
5
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The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$
We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...
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Simple proof that the arithmetic genus is non-negative
I take an irreducible and reduced closed curve $C\subseteq \mathbb{P}^n$, defined over an algebraically closed field $k$ and define the arithmetic genus $p_a(C)$ as the integer such that the Hilbert ...
0
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0
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Is the Hilbert series of an ideal related to the Hilbert series of its homogenization?
Suppose we have a field $k$ of characteristic 0, let $I$ be an ideal of $R=k[x_1,...,x_n]$, and let $H$ be the homogenization of $I$ in $S=R[z]$. Is there any relationship between the Hilbert series ...
1
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Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave
Thank you for taking the time to read this. I was hoping to get some assistance in understanding how these equations function:
$$As=\frac{\langle H(\eta)^3\rangle}{\langle\eta^2\rangle^{3/2}},\qquad ...
5
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2
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Reference book for understanding Hilbert Series/functions
For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".
https://en.wikipedia.org/wiki/...
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Relation between Hilbert function and complete intersection ideals
Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$
with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=...
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Segre embedding and Hilbert polynomial of coherent sheaves
Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, ...
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References for Hilbert schemes over non-Archimedean valuation
Can you suggest me some suitable references to learn the theory of Hilbert polynomials (or related Hilbert schemes) in the non-Archimedean setting?
Thanks.
3
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Hilbert polynomial of structure sheaf of hypersurfaces
Is there an example of a hypersurface $X$ of some projective space $\mathbb{P}^n$ such that there exists an invertible sheaf $\mathcal{L}$ on $X$, not isomorphic to the structure sheaf $\mathcal{O}_X$,...
2
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Roots of the Hilbert polynomial of an invertible sheaf
Let $X$ be a smooth, projective variety over an algebraically closed field of characteristic zero. Fix a polarisation on $X$. Let $\mathcal{L}$ be an invertible sheaf on $X$ with Hilbert polynomial, ...
4
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Examples of smooth projective varieties with "nice" Picard group
I am looking for examples of smooth projective varieties $(X,H)$ with $H$ a polarization on $X$, $\dim \mbox{Pic}^0(X)=0$, $\mbox{Pic}(X) \not= \mathbb{Z}$ satisfying the property: for any two line ...
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Hilbert functions of graded modules generated by mapped generators
I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...
33
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On the polynomial $\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$
Let $n = 2m$ be an even integer and let $F_n(X)$ be the polynomial $$F_n(X):=\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}.$$ I observed (but cannot prove) that the polynomial $F_n$ is always divisible by $...
2
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216
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Ideals with the same Hilbert series
Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their ...
28
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Which intrinsic invariants of a projective variety can you deduce from its Hilbert polynomials?
Given a projective variety $X$, each of its embeddings $i:X\hookrightarrow \mathbb P^N$ gives rise to an integer valued Hilbert polynomial $P_{X,i}(t)\in \mathbb Q[t]$.
These polynomials depend ...
0
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1
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308
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The growth of the Hilbert function of a graded ring
Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$.
In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...
3
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1
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262
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I need to refind a reference on multigraded Hilbert series
I found a theorem about multigraded Hilbert series stated as follows:
Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus_{j\in\mathbb{N}^m}{R_j}$ over $R_0=\mathbb{C}$. If $R$ is generated by $...
4
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Raising coefficients of a power series to some power
Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form
$$ \frac{1}{\prod_{i=1}^k{(1-t^{\...
4
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On the computational complexity of the Hilbert polynomial of numerical semigroup rings
Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\...
2
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1
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378
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Hilbert function of points in $\mathrm{P}^2$
Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
\mathrm{gin_{rlex}}(I)=(x_1^k,x_1^{k-1}x_2^{\lambda_{k-1}},...,x_1x_2^{\lambda_1},x_2^{\...
2
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1
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365
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Hilbert Regularity in relation to degree of generators
Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) ...
4
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1
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Hypersurfaces containing a general chain of lines
Let $X$ be a general chain of $d$ lines in $\mathbb P^n$, where $n \geq 3$. Let $I$ be the homogeneous ideal of polynomials vanishing on $X$. What is the Hilbert function
$$P(k) = \dim I_k$$
of $X$? ...
3
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1
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900
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The Hilbert function of an intersection
Assume that $X_1,\ldots,X_r\subseteq\mathbb P^n$ are irreducible, reduced hypersurfaces in complex projective space, each of the same degree $d$. In other words, $X_i=Z_\ast(f_i)$ for certain ...