Shihoko Ishii

The Nash problem on arc families of singularities

Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

ARCS, VALUATIONS AND THE NASH MAP

Abstract This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety. We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.

HYPERSURFACE EXCEPTIONAL SINGULARITIES

This paper studies hypersurface exceptional singularities in

Geometric Properties of Jet Schemes

\mathbb C^n

Extremal Functions and Prime Blow-ups

defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.

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

This article shows how properties of jet schemes relate to those of the singularity on the base scheme. We will see that the jet schemes properties of being ℚ-factorial, ℚ-Gorenstein, canonical, terminal, and so on are inherited by the base scheme.

Chow instability of certain projective varieties

Abstract A blow-up is called a prime blow-up, if its exceptional set is a unique irreducible divisor. This article shows that an irreducible exceptional divisor appears in a prime blow-up, if and only if there are extremal functions for this divisor.

Some projective contraction theorems

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 Reids conjecture in the

ON -K2 FOR NORMAL SURFACE SINGULARITIES

d (\geq 3)

Simultaneous Canonical Models of Deformations of Isolated Singularities

-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