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For a normal surface rational singularity, we know that the multiplicity of is bounded by $e-1$ where $e$ is the embedding dimension (See for example Miles Reid's book "Chapters on algebraic surfaces").

I am wondering if this inequality also holds in higher dimension. If not, what can we say about the multiplicities.

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up vote 4 down vote accepted

That particular bound doesn't hold if I recall correctly, but the following bound does:

Theorem : (C. Huneke and K.-i. Watanabe) The multiplicity of a $d$-dimensional variety with rational singularities and embedding dimension $n$ is at most $${n - 1 \choose d - 1}.$$

In the case of a surface, this reduces to the bound you mentioned above. This is an unpublished result of Huneke and Watanabe (currently under review). You could certainly ask them for a preprint.

EDIT: My previous answer said that this was a conjecture, and that Huneke and Watanabe proved something related to this, but I wasn't sure if they actually proved this. It turns out that they did indeed prove this, and I got their permission to post that this was indeed a theorem of theirs.

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Graham, sorry, the word general has no meaning. Let me remove it. –  Karl Schwede Nov 18 '12 at 21:09

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