This paper shows the affirmative answer to the local Nash problem for a toric singularity and analytically pretoric singularity. As a corollary we obtain the affirmative answer to the local Nash problem for a quasi-ordinary singularity.
Let be a Gorenstein, -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.
We describe the set of minimal log discrepancies of toric log varieties, and study its accumulation points.
We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips.
These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gelfand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.
We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485- 503. In the first procedure the singularity is...