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Homeomorphism of bases of ambient homeomorphic cones with non-smooth bases

Determine whether, for algebraic cones C(X), C(Y) ⊂ C^{n+1} of dimension n whose bases X and Y are non-smooth, ambient homeomorphism of the cones implies that the bases X and Y are homeomorphic.

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Background

For cones with smooth bases, the authors show that ambient homeomorphism of the cones implies equality of degree and, in particular, homeomorphism of the bases. The situation for cones with non-smooth bases is more complex and directly tied to the metric version of Zariski's multiplicity conjecture.

Resolving whether non-smooth bases are homeomorphic under ambient homeomorphism of the cones would imply the metric Zariski multiplicity conjecture, highlighting the significance of this open problem.

References

If bases are non-smooth, the problem whether bases are homeomorphic is a difficult open problem which implies the metric version of the Zariski multiplicity Conjecture (see ).

— Metric version of the Zariski Multiplicity Conjecture is true for multiplicity two (2509.03447 - Fernandes et al., 3 Sep 2025) in Section 1 (Introduction)