# Questions tagged [hilbert-polynomials]

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### modules with the same associated variety

$S$ is a polynomial ring, $M$ is a finitely generated graded $S$-module, the associated variety of $M$ is $$\mathcal{V}\left( M\right)=V\left( Ann_S \left( M \right) \right),$$ What is the ...
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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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### 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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### The weight of a weighted filtration is given (for large $m$) by a polynomial

Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
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### Semi-stable sheaves on quadric surface

https://downloads.hindawi.com/journals/tswj/2014/346126.pdf In this paper, Stable sheaves on a smooth quadric surface with linear Hilbert bipolynomials(E. Ballico and S.Huh), I have a question. On the ...
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I have pure sheaves of dimension 1 on a ruled surface, in paticular the Hirzebruch surface F$_e$=P($O \oplus O(-e)$) with linear Hilbert bipolynomial $P(x, y)=ax+by+c$. A sheaf $E$ is pure of ...