# Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

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### 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 $0$,...
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 $K:=V(X_{k+1},\cdots,X_r)\... 1answer 69 views ### Hilbert polynomials of closed subschemes$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$... 0answers 99 views ### Hilbert scheme of hypersurfaces in Nitsure’s Construction of Hilbert and Quot Schemes I have a few questions on the construction of Hilbert scheme of hypersurfaces I found recently in Nitin Nitsure’s Construction of Hilbert and Quot Schemes (page 6 part 4) In his paper Nitsure added a ... 2answers 128 views ### Resolving indeterminacy of map on projective spaces induced by a linear map Let$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$naturally induces a rational ... 0answers 50 views ### Hilbert functions for affine vs projective varieties and filtered vs graded algebras. I'm somewhat confused about how to define Hilbert functions for affine varieties and for filtered rings in a compatible way. I'm familiar with how they are defined for projective varieties: Let$X$be ... 0answers 37 views ### What is the Hilbert polynomial of this simple scheme Denote by$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)... 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^... 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 ... 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] ... 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 ... 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_{... 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[... 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 ... 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,... 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 \... 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/... 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 ... 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) . ... 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 ... 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 ... 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 ... 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 ... 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 ... 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} ... 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 ... 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 ... 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}... 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 ... 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, ... 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} \... 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.... 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. ... 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 ... 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_{... 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 ... 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,\... 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 \... 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,\... 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 ... 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^... 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? 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. ... 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 ... 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 ... 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... 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} ... 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$1+t+t^2+\dots = 1/(1-t)$. 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$...
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 ...