Questions tagged [syzygies]

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

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1answer
173 views

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 ...
8
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2answers
1k views

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 ...
8
votes
1answer
236 views

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 ...
2
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0answers
61 views

$m$-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 $...
1
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0answers
72 views

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\...
2
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0answers
126 views

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\...
4
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0answers
125 views

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)...
3
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0answers
72 views

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 ...
0
votes
1answer
166 views

Syzygies of projective varieties

I want to see some examples of syzygies of projective varieties, if possible not from Eisenbud, because I know those.
2
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0answers
140 views

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 \...
5
votes
0answers
425 views

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 ...
2
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1answer
431 views

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 ...
1
vote
3answers
572 views

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 ...
3
votes
1answer
349 views

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 ...
4
votes
1answer
456 views

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 $I$?