Torical Modification of Newton non-degenerate ideals

Nonisolated forms of rational triple point singularities of surfaces and their resolutions

“…This fact was shown in [27] for isolated hypersurface singularities and generalised in [20]. In Appendix, we recall a more general definition of non-degeneracy given in [1], where it was proved that all non-degenerate singularities can be resolved by toric modifications, to show that the RTP-singularities are non-degenerate. This interesting property leads us to ask whether a singularity is non-degenerate if and only if its normalisation is non-degenerate.…”

“…Let P be the fan studied in Example 4.8 and take v 1 := v, v 2 := w, v 3 := e 1 . The regular subdivision of σ uv 1 given by {u, (uv 1 ) 1 , (uv 1 ) 2 , (uv 1 ) 3 , v 1 } = {u, (4, 3, 5), (3,2,4), (2,1,3), v 1 } (see Example 4.13). The corresponding graph Γ 1 u is then a tree consisting of 5 vertices as shown in Figure 2a.…”