## Nonisolated forms of rational triple point singularities of surfaces and their resolutions.(English)Zbl 1368.32019

Let $$S$$ be a germ of a normal surface embedded in $${\mathbb C}^N$$ with a singularity at the origin, and let $$\pi:\widetilde{S}\to S$$ be a resolution of $$S$$. The singularity of $$S$$ is called rational if $$H^1(\widetilde{S},{\mathcal O}_{\widetilde{S}})=0$$. This condition is known to imply a number of combinatorial results on the invariants obtained from the resolution graphs of such surfaces. In the present paper, the authors give a list of non-isolated hypersurface singularities in $${\mathbb C}^3$$ such that their normalizations are rational surface singularities of multiplicity $$3$$ (those can be defined by three equations in $${\mathbb C}^4$$). Using a method introduced by Oka for isolated complete intersections, they construct the corresponding minimal resolution graphs. They also show that both normal surfaces in $${\mathbb C}^4$$ and their non-isolated forms in $${\mathbb C}^3$$ are Newton non-degenerate, which means, roughly speaking, that they can be resolved by toric modifications well-behaved with respect to the Newton polygons.

### MSC:

 32S25 Complex surface and hypersurface singularities 32S45 Modifications; resolution of singularities (complex-analytic aspects) 58K20 Algebraic and analytic properties of mappings on manifolds

Gfan; SINGULAR
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### References:

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