ag.algebraic geometry - complex singularity exponent, lct

Hi everybody,

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,

J-B B

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