Three dimensional divisorial extremal neighborhoods

3-fold divisorial extremal neighborhoods over cE7 and cE6 Compound DuVal singularities

Let X be the germ of a Gorenstein 3-fold singularity and C a smooth curve through it such that the general hyperplane section S of X containing D is DuVal of type E6 or E7. In this paper we obtain criteria for the existence of a terminal divisorial extremal neighborhood f: Y-->X contracting an irreducible divisor E onto C and classify all such neighborhoods.

到余维三轨道的除子收缩

Divisorial contractions to codimension three orbits

本文研究连通代数群作用下中心为余维三单连通轨道的等变极值收缩，证明此类收缩均为加权爆破，推广了 Kawakita 的经典结果。

某些 Hyper-Kähler 四维流形的两条极射线

The two extremal rays of some Hyper-Kähler fourfolds

研究形变等价于 K3 曲面 Hilbert 平方的四维射影 Hyper-Kähler 流形，在 Picard 秩为二且具两个除子收缩时，确定了例外二次曲线丛的七种类型，并给出三次四维流形 Fano 簇的四种实例。

On the behavior of formal neighborhoods in the Nash sets associated with toric valuations: a comparison theorem

We show that there exists a strong connection between the generic formal neighborhood at a rational arc lying in the Nash set associated with a toric divisorial valuation on a toric variety and the formal neighborhood at the generic point of the same Nash set. This may be interpreted as the fact that, analytically along such a Nash set, the arc scheme of a toric variety is a product of a finite dimensional singularity and an infinite dimensional affine space.

On Classification of the extremal contraction from a smooth fourfold

We classify extremal divisorial contraction which contracts a divisor to a curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb P^2$bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve.

Flipping surfaces

We study semistable extremal threefold neighborhoods following earlier work of Mori, Kollár, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in …