Questions tagged [syzygies]
a syzygy is a relation between the generators of a module M.
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Original proof of Hilbert's syzygy theorem
Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:
the theory of projective dimension and change of rings (plus a step ...
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Degrees of syzygies of points in $\mathbb P^2$
Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when:
The syzygies of $I_X$ contains no linear forms. Since ...
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Relative version of Hilbert syzygy theorem
I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a ...
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Identities satisfied by the image of the Young symmetrizer
Consider a partition $\lambda=(r_1,\ldots,r_k)$ of an integer $n$ and the corresponding Young diagram with rows of length $r_1,\ldots,r_k$ (hence ordered in non-increasing order). Counting the column ...
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Minimal Free resolution of the ideals
What is the motivation to study the minimal free resolution of the ideals? Which geometrical information can we get from $res(I)$ for The variety of $I$?
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For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements
Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...
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Question on syzygies
Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=\operatorname{Hom}_K(A,K)$.
Do we then also have $\Omega^{-i}(A)...
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Defining ideals for rational curves in space
A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...
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Stability of syzygy bundles of smooth curves
For a very ample line bundle $L$, the kernel of the surjection $H^0(L)\otimes \mathcal{O}_X\rightarrow L \rightarrow 0$ is denoted by $M_L$ and is called syzygy bundle. In this paper authors claim in ...
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Hilbert's Syzygy Theorem in the bigraded case
I've been recently wondering how to prove the existence of a Hilbert polynomial for finitely generated bigraded modules $M$ over a polynomial ring $R=k[X_0,...,X_n,Y_0,...,Y_m]$ with the usual ...
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Normal set of points in the plane
When defining the normality of a scheme in the book The Geometry of Syzygies, Eisenbud says that there are just a few facts that are known for their bounds.
Given $X \subset \mathbb{P}^r$, we say ...
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Castelnuovo Mumford Regularity
Consider the polynomial ring $S = C[x_1,\ldots,x_n]$. Let
$$
0 \rightarrow E_{n-1} \rightarrow \cdots \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a ...
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Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$
$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained.
Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...
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Higher cohomology of projective bundles
Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let ...
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$m$-regularity of sheaves
This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...
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Geometric meaning of Koszul modules
Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\...
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A question about an irreducible polynomial in the ideal of syzygies
Let $G$ be a finite group with $n$ elements and let $\mathbb{Q}[x_1,\cdots,x_n]^G = \mathbb{Q}[g_1,\cdots,g_m]$, where $G$ acts through the regular representation. Then there exist polynomials $s_j \...
user6671
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Syzygies of determinantal varieties: Looking for English text
I would like to understand the syzygies of the determinantal ideal $I_r$, generated by the $r\times r$ minors of a matrix $(X_{ij})$ of indeterminantes in the polynomial ring over an algebraically ...
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Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module
I have asked a related question on math.SE here, but the notation is a bit different.
As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...
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The first syzygy module of a binomial ideal
It is known how you compute the first syzygy module of a monomial ideal but it seems an hard work to do the same for binomial ones. I don't know any procedure to aim that, so I would like kindly if ...
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Resolution of the ideal of a scroll
Assume that $C$ is a smooth projective curve and $p:C\to\mathbb{P}^1$ is a degree $k$ branched cover. Let $L$ be a very ample line bundle on $C$ with very large degree, defining an embedding $C\...
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Syzygies of projective varieties
I want to see some examples of syzygies of projective varieties, if possible not from Eisenbud, because I know those.
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Degree of the syzygy bundle of a curve of genus 3
Let X be an hyperelliptic curve of genus 3, 𝜔 its canonical sheaf, and M the syzygy bundle of 𝜔.
What is the degree of M?
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