Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ideal of $\mathcal{O}_K$. I want to get a upper bound of the local Hilbert-Samuel function of $X$ at the point $\xi$, the upper bound is related to the degree of $X$. Attention: $\xi$ may not be regular. I know such a result for $X$ is a general projective variety which published is JAG of 1997, but it is too coarse, and I want a better approximation. By the way, if I know a approximation better enough, can I use the Koszul-complexes to estimate the case of complete intersection?
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$\begingroup$ PS. The result is also related to its multiplicity. $\endgroup$– varJul 14, 2013 at 19:40
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$\begingroup$ I think you should edit this question and give precise references. Why is the bound from the JAG paper too coarse? $\endgroup$– Karl SchwedeJul 15, 2013 at 6:48
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$\begingroup$ The reference is: V. Srinivas, V. Trivedi, On the Hilbert function of a Cohen-Macauley local ring, Journal of Algebraic Geometry 6(1997), No. 4, 733-751. $\endgroup$– varJul 15, 2013 at 10:31
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$H_\xi(k)={n+k\choose k}-{n-\mu_\xi+k\choose k}$, where $\mu_\xi$ is the multiplicity of $\xi$, and $H_\xi(k)$ is its Hilbert-Samuel function.