I have a question about log canonical thresholds / complex singularity exponents.
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.
I have several questions :
1) Is it true that for the singularity $x^a+y^b+z^c=0$, the complex singulary exponent is $min(1;1/a+1/b+1/c)$ ?
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) ?
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 ?
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.
Thanks in advance,