complex singularity exponent, lct - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:42:27Z http://mathoverflow.net/feeds/question/37269 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37269/complex-singularity-exponent-lct complex singularity exponent, lct Jean-Baptiste Bouly 2010-08-31T15:23:53Z 2011-03-04T18:22:15Z <p>Hi everybody,</p> <p>I have a question about log canonical thresholds / complex singularity exponents. </p> <p>If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in dimension 1 (\$x^3+y^2=0\$) has \$c_0 = 5/6\$ and the ordinary quadratic singularity has c_0=1.</p> <p>I have several questions :</p> <p>1) Is it true that for the singularity \$x^a+y^b+z^c=0\$, the complex singulary exponent is \$min(1;1/a+1/b+1/c)\$ ?</p> <p>2) If it's true, how to distinguish \$x^3+y^2+z^2=0\$ and the ordinary quadratic surface singularity ? Is there another invariant (besides the Milnor number) ?</p> <p>3) If I have a variety with isolated singularities, does it make sense to try to measure its 'singularity' by adding the exponents at all the singular points ? Or taking the inf ?</p> <p>4) compared to the Milnor number, what is the advantage of working with this invariant ? I thought it would be a way to say that a curve with a cusp is 'more singular' than a curve with two ordinary quadratic points (with the Milnor number they are 'equaly singular'), but I don't know if that makes sense.</p> <p>Thanks in advance, </p> <p>J-B B</p> http://mathoverflow.net/questions/37269/complex-singularity-exponent-lct/54340#54340 Answer by Karl Schwede for complex singularity exponent, lct Karl Schwede 2011-02-04T17:54:52Z 2011-02-04T18:14:01Z <p>With regards to your questions.</p> <p>1) Yup, see Example 9.3.31 and Theorem 9.3.37 in Lazarsfeld's book, "Positivity in Algebraic Geometry II".</p> <p>2) You might not be aware of things like (minimal) log discrepancies. Of course there are various properties of the minimal resolutions too.</p> <p>3) Do you mean exponents of different variables or the same variables, what did you have in mind? I don't know of any good geometric interpretation of doing this in general.</p> <p>4) This log canonical threshold also appears in many other unexpected contexts? See for example:</p> <p>Budur's initial question in "Singularity invariants related to Milnor fibers: survey". <a href="http://arxiv.org/pdf/1012.3150" rel="nofollow">http://arxiv.org/pdf/1012.3150</a></p>