No abstract

In this paper we study the quotient of an isolated strictly logcanonical singularity by a finite group. As a result, we obtain the boundedness of indices of these singularities of dimension 3 and determine all possible indices. We also determine the ramification indices of the quotient map of a 2-dimensional strictly log-canonical singularities by a finite group.

A goal of this paper is a characterization of singularities according to a new invariant, Mather discrepancy. We also show some evidences convincing us that Mather discrepancy is a reasonable invariant in a view point of birational geometry.

This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the codimension for this subset and give a formula of the codimension in terms of "relative Mather canonical class". By using this subset, we prove that a divisorial valuation is determined by assigning the values of finite functions. We also have a criterion for a divisorial valuation to be a monomial valuation by assigning the values of finite functions.

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.