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On higher dimensional Hirzebruch-Jung singularities. (English) Zbl 1076.32022

A germ of irreducible normal complex analytic space is called Hirzebruch-Jung singularity if it is the normalization of an \(n\)-dimensional irreducible quasi-ordinary germ (i.e., there exists a finite morphism to a smooth space of the same dimension). Let \(W\) be a lattice and \(\sigma\) a strictly convex finite rational polyhedral cone in \(W_\mathbb R=W\otimes \mathbb R\), \(M\) the dual lattice and \(\check{\sigma}\) the dual cone, \[ Z(W, \sigma)=\operatorname{Spec} \mathbb C\;[\check{\sigma}\cap M]. \] \((W, \sigma)\) is called a maximal simplicial pair if \(\sigma\) and \(W_\mathbb R\) have the same dimension and \(\sigma\) is simplicial. Hirzebruch-Jung singularities are analytically isomorphic with the germ \((Z(W,\sigma), 0)\) at the \(0\)-dimensional orbit of an affine toric variety defined by a maximal simplicial pair \((W, \sigma)\). It is proven that the analytical type of a Hirzebruch-Jung singularity \((Z(W,\sigma), 0)\) determines the pair \((W, \sigma)\) up to isomorphism. A normalization algorithm for quasi-ordinary hypersurface singularities is given.

MSC:

32S10 Invariants of analytic local rings
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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