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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 general $d$-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 This is an unpublished result of Huneke and Watanabe (currently under review).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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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 general $d$-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 This is an unpublished result of Huneke and Watanabe (currently under review).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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That particular bound doesn't hold if I recall correctly, but it's possible the following bound does:

Conjecture

Theorem : (C. Huneke and K.-i. Watanabe) The multiplicity of a general $d$-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. The evidence I know for this conjecture This is as follows.

For any minimal log canonical center of dimension $d$ in a smooth variety an unpublished result of dimension $n$, Helmke proved that the above bound holds (see Corollary 4.6 in On Fujita's conjecture). Such centers always have rational singularities (in fact, they are always log terminal, which is stronger, but I suspect this bound holds only for rational singularities).

You could also try asking Craig Huneke or Kei-ichi and Watanabe (or email me, if you'd like me to introduce you -- I've also heard that Craig reads math overflow, so perhaps he'll read this). I know they wrote a paper in characteristic $p > 0$ on some related bounds for $F$-singularities (singularities defined by Frobenius, as in tight closure theory or Frobenius splitting theory)currently under review). In particular, if they proved the same bound for $F$-rational singularities, it probably should imply your bound You could certainly ask them for rational singularities (although I'd need to think through this)a preprint.

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