# Questions tagged [syzygies]

a syzygy is a relation between the generators of a module M.

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### Degree of the syzygy bundle of a curve of genus 3

Let X be an hyperelliptic curve of genus 3, đťś” its canonical sheaf, and M the syzygy bundle of đťś”.
What is the degree of M?

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $m$-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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### 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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### 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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### 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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### 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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### 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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### 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 \...

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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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### Castelnuovo Mumford Regularity

Consider the polynomial ring $S = C[x_1,\ldots,x_n]$. Let
$$
â€Žâ€Ź0 â€Žâ€Ž\rightarrowâ€Žâ€Žâ€Ź â€ŽE_{n-1} â€Žâ€Ž\rightarrow â€Ž\cdots â€Žâ€Žâ€Ž\rightarrow â€ŽE_1 â€Žâ€Žâ€Ž\rightarrow â€ŽE_0 \rightarrow I \rightarrow 0â€Ž â€Ž,â€Ž
$$
be a ...

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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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### 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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### 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 $I$?