log-canonical threshold

Question

hello. i'm reading the book of Lazarfeld "positivity in algebraic geometry" and in particular i'm studying the multiplier ideals $\mathcal{I}(a^{c})$ of an ideal sheaf $a \subset\mathcal{O}_{X}$ and its invariant the log-canonical threshold (lct). i recall that the lct of an multiplier ideal is defined as:

$sup\{c\geq 0:\mathcal{I}(a^{c})\neq\mathcal{O}_{X}\}$.

So my question is the following: is it true that if $a\subset b$ is strictly contained in $b$ then $lct(a)<lct(b)$ ? can anyone can suggest me some references or a contre-examples?

Answer 1

I don't think that it's true: take $a=(x^2-y^3)$ and $b=(x^2,y^3)$ in $k[x,y]$. Then $a$ is strictly contained in $b$ but $lct(a)=lct(b)=5/6$. This example can be found e.g. in "An informal introduction to multiplier ideals" by Blickle and Lazarsfeld, see here.