Hypersurface non-rational singularities which look canonical from their Newton boundaries

Abstract

We study the Newton boundaries of hypersurface singularities from the viewpoint of the theory of filtered blowing-ups. We can construct counter examples to Reid's conjecture in the d(≥ 3)-dimensional cases which is related to the problem about the existence of good embedding for the criterion about the rationality (or the non-rationality). For 3-dimensional case, our examples contains a simple K3 singularity which does not belong to the famous list consisting of 95 types.

Gorenstein Fano threefolds with base points in the anticanonical system

“…The cases (a) and (b) are as in Iskovskikh's list. In a different context case i) appears in [Me99] and [IT01], and apparently also in [M88].…”

Blowing Ups of 3-dimensional Terminal Singularities, II

“…These are the main theorems in this paper. There is a related work by [IT99] which also uses embeddings into C 5 in order to study hypersurfaces in C 4 . This paper, combined with the results in [Hay99], covers all 3-dimensional terminal singularities of indices m > 2.…”