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.



multiplicities of rational singularities in higher dimension
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 This is an unpublished result of Huneke and Watanabe (currently under review).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.


