multiplicities of rational singularities in higher dimension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:47:13Z http://mathoverflow.net/feeds/question/112464 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112464/multiplicities-of-rational-singularities-in-higher-dimension multiplicities of rational singularities in higher dimension Fei YE 2012-11-15T08:20:15Z 2012-11-18T21:10:06Z <p>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").</p> <p>I am wondering if this inequality also holds in higher dimension. If not, what can we say about the multiplicities.</p> http://mathoverflow.net/questions/112464/multiplicities-of-rational-singularities-in-higher-dimension/112477#112477 Answer by Karl Schwede for multiplicities of rational singularities in higher dimension Karl Schwede 2012-11-15T12:53:42Z 2012-11-18T21:10:06Z <p>That particular bound doesn't hold if I recall correctly, but the following bound does:</p> <p><strong>Theorem :</strong> (C. Huneke and K.-i. Watanabe) <em>The multiplicity of a \$d\$-dimensional variety with rational singularities and embedding dimension \$n\$ is at most</em> \$\${n - 1 \choose d - 1}.\$\$</p> <p>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 This is an unpublished result of Huneke and Watanabe (currently under review).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.</p> <p><strong>EDIT:</strong> 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.</p>