Related papers: On the factorization of three-dimensional terminal…
We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a…
We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.
We show that terminal 3-fold divisorial contraction to a point of index $>1$ with non-minimal discrepancy may be factored into a sequence of flips, flops and divisorial contractions to a point with minimal discrepancies.
We prove that each divisorial contraction to a curve between terminal threefolds is a weighted blow-up under a suitable embedding. Moreover, we give a classification of the weighted blow-ups assuming that the curve is smooth.
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a smooth point. We prove that any such contraction can be obtained by a suitable weighted blow-up.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions $f\colon (Y,E)\to (X\ni P)$ are described provided that $\Exc f=E$ is an irreducible divisor,…
We prove that any three dimensional terminal singularity $P \in X$ can be resolved by a sequence of divisorial contractions with minimal discrepancies which are weighted blowups over points.
In this paper the three-dimensional divisorial contractions $f\colon Y\to (X\ni P)$ are classified provided that $\Exc f=E$ is an irreducible divisor, $f(E)=P$, the variety $Y$ has canonical singularities and $(X\ni P)$ is a toric terminal…
We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips.
The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point a matrix factorization of the equation of the rational double…
Let C be a smooth curve on an index 1 terminal 3-fold. We investigate the existence of extremal terminal divisorial contractions Y-->X that contract an irreducible surface E to C. We consider cases in respect to the singularities of the…
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a cA_1 point. We prove that any such contraction is obtained by a suitable weighted blow-up.
We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.
We give a simplified algorithm of the functorial weak factorization of birational morphisms of nonsingular varieties over a field of characteristic zero into a composite of blow-ups and blow-downs with smooth centers.
We complete the explicit study of a three-fold divisorial contraction whose exceptional divisor contracts to a point, by treating the case where the point downstairs is a singularity of index $n \ge 2$. We prove that if this singularity is…
This is an overview article, based on my 2018 Kinosaki lecture, that surveys and announces work on 3-fold flopping contractions, their affine combinatorics, stability conditions, tilting bundles and autoequivalences. Some first applications…
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
We describe an explicit algorithm to factorize an even antisymmetric N^2 matrix into triangular and trivial factors. This allows for a straight forward computation of Pfaffians (including their signs) at the cost of N^3/3 flops.
Every three-fold divisorial contraction to a non-Gorenstein point is a weighted blow-up.
This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It…