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In Lê-Ramanujam's paper The invariance of Milnor’s number implies the invariance of the topological type, they prove what the title says for families with isolated singularities and constant Milnor number in dimensions not equal to 3.

I'm not familiar with the literature on singularities so I'm hoping for some examples (if any are known) of such families which are nontrivial. The preprint of de Bobadilla-Pelka says that constant Milnor number also implies constant multiplicity for such families so multiplicity won't help in distinguishing two members of the family.

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    $\begingroup$ I think that if you take the family $\{xy(x-y)(x-ty) = 0\}$ (parametrised by $t \in \mathbb{C} \setminus \{0,1\}$) you win. The cross-ratio tells you that this is a non-constant family of lines in $\mathbb{C}^2$. $\endgroup$
    – Marco Golla
    Commented Dec 1, 2022 at 17:07
  • $\begingroup$ @MarcoGolla I'm a bit confused; is the cross-ratio an invariant of a singularity? I thought it only makes sense if we're embedded inside of affine or projective space (and need to remember the data of the embedding) and is an invariant for quadruples of 4 points. $\endgroup$
    – inkievoyd
    Commented Dec 1, 2022 at 20:26
  • $\begingroup$ Yup, I think I was a bit too hasty. The example only works for curve singularities in $\mathbb{C}^2$, not for abstract singularities. I guess that suspensions (branched covers) of this singularity might do the trick, but let me not say more incorrect things today :) $\endgroup$
    – Marco Golla
    Commented Dec 1, 2022 at 21:02
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    $\begingroup$ The cross ratio is an intrinsic invariant after you formulate it via the $j$-invariant. $\endgroup$
    – Jason Starr
    Commented Dec 1, 2022 at 22:41
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    $\begingroup$ In the analytic case, if one wants constant Milnor number and variation of moduli, it suffices to take a family of elliptic surface surface singularities, say $$x^3+y^3+z^3+txyz=0,$$ where $t^3 + 27 \neq 0$. Of course this is related to cross-ratio, since this affine equation in $\mathbb{C}^3$ is just the affine cone over the elliptic curve with the same equation in $\mathbb{P}^2$. $\endgroup$
    – Francesco Polizzi
    Commented Dec 2, 2022 at 13:18

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