Paul Hacking
The semistable minimal model program is a special case of the minimal model program concerning 3-folds fibred over a curve and birational morphisms preserving this structure. We classify semistable divisorial contractions which contract the exceptional divisor to a normal point of a fibre. Our results can be applied to describe compact moduli spaces of surfaces.
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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.
Every three-fold divisorial contraction to a non-Gorenstein point is a weighted blow-up.
We study Calabi--Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi--Yau 3-folds with the infinite automorphism group.
Divisorial contractions to singularities, defined by equations $xy+z^n+u^n=0$ $n\ge3$ and $xy+z^3+u^4=0$ are classified.
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 point.
We treat threefolds with divisorial contractions whose exceptional divisors contract to compound Du Val points. We prove that general elements in their anticanonical systems around the exceptional divisors have at worst Du Val singularities. As applications to classification, we describe divisorial contractions to compound A_n points, and moreover we conclude that discrepancies of divisorial contractions to compound D_n or E_n points are at most four.
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 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.
Counting divisorial contractions with centre a $cA_n$-singularity
本文简化中心为 $cA_n$ 奇点的三维除子收缩分类，刻画局部解析等价类对应的整体代数收缩，并证明差异至少为2时此类收缩若存在则不可数多。
On the divisorial contractions to curves of threefolds
本文证明终端三维簇到曲线的除子收缩在适当嵌入下均为加权爆破，并在曲线光滑条件下给出其完整分类。
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 singularity.
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 general hypersurface section S of X through C. We completely classify such contractions in the case that S has A_i, i < 4 and D_2n for any n, type singularities.
In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth projective toric varieties which can be equivariantly blown-up to Fano along curves.
This paper has been withdrawn by the author. It is not true that plt blow-up of toric singularity is toric (in dimension 3 too) and the first main theorem is incorrect also. The error is in deformation argument. I am grateful to Ivan Cheltsov for discussions.
Extended Abstract of Counting Divisorial Contractions with Centre a $cA_n$-Singularity
本文为《代数几何ZAG手册》第149章，综述了发表于《京都大学数学研究所出版物》的、关于以$cA_n$-奇点为中心的除子收缩计数的论文的主要结果。
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。
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 of type c$A/n$ then any such contraction is a suitable weighted blow-up; and that if otherwise then the discrepancy is $1/n$ with a few exceptions. Every such exception has an example. Some exceptions allow the discrepancy to be arbitrarily large, but any contraction in this case is described as a weighted blow-up of a singularity of type c$D/2$ embedded int…
Divisorial Mori contractions of submaximal length
本文改变极大长度假设，证明次极大长度的除子型Mori收缩的例外轨迹双有理等价于射影丛或二次曲面丛，拓展了Mori收缩相关刻画研究。
Let X/T be a one parameter family of canonical 3-folds and let D be a Weil divisor on it flat over T. We study the problem of when the D_t-minimal models of X_t form a family and we obtain conditions for this to happen. As an application of this we classify terminal divisorial contractions Y-->X that contract an irreducible divisor E onto a smooth curve C in the case when the general hyperplane section S of X through C is a D5 DuVal singularity.
Toric Landau-Ginzburg models in threefold divisorial contractions
本文研究了终端Fano三维流形的除子收缩下的量子周期和托里纤维Landau-Ginzburg模型，证明了正则化周期恒等式，为计算三维中心模型的Sarkisov链接和高阶syzygies提供了镜像方法。
In this paper, we will list up all the cases for the ray contractions of divisorial and fiber types for smooth projective varieties of dimension five. These are obtained as a corollary from the lists of n-dimensional k-th adjoint contractions f: X -> Y of the same types for k=1,2,3 and 4 (n> or =5). The lists for k=1,2 and 3 have previously been obtained in [Na], Proposition 1.2 and Theorem 1.3. The main task will be to have such a list for k=4, where one case in the list fails to show that a positive-dimensional general fiber F of f is irreducible when n>5. This assertion will, however, be pr…
In this paper, we initiate our investigation of log canonical models for the moduli space of curves with the boundary divisor $\a \d$ as we decrease $\a$ from 1 to 0. We prove that for the first critical value $\a = 9/11$, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that $\a = 7/10$ is the next critical value, i.e., the log canonical model stays the same in the interval $(7/10, 9/11]$. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in …
We study Mori fiber spaces over a two-dimensional base which satisfy the semistability assumption. As an application of our technique we give a new proof of the existence of semistable 3-fold flips.
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.
Relative stability theory and properness of K-moduli spaces
本文定义Fano簇族的相对稳定性阈值，通过除子赋值计算并迭代替换特殊纤维，仅用双有理几何论证给出了K-模空间固有性的新证明。
We prove that every log crepant birational morphism between log terminal surfaces is decomposed into log-flopping type divisorial contraction morphisms and log blow-downs. Repeating these two kinds of contractions we reach a minimal log minimal surface from any log minimal surface.
On the factorization of three-dimensional terminal flops
本文研究三维终端翻折的结构，将其分解为到点的除子收缩和光滑曲线胀开的复合，揭示了终端翻折的几何分解方式。
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.
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。
Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then the Picard number rho(X) of X is at most 12. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case rho(X)=12 an open subset of X has a smooth fibration with fiber the projective line. Together with previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of fam…
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementar…
On the Minimal Model Program for Kähler 3-folds
本文证明了紧 Kähler 簇在维度 n 下的 pl-flipping、除子收缩及 pl flip 的存在性，并给出了维度 3 下锥定理、flip 及极小模型存在的自包含证明。
We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(α)$ of the pairs $(\bar M_g, αδ)$ for $α\in (7/10 - ε, 7/10]$ for small $ε\in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety bicanonical curves; $\bar M_g(7/10-ε)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic t…
In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.
In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n at most 9, we explicitly …
The log minimal model program for Kähler $3$-folds
本文针对紧 Kähler 3维流形上的 Q-factorial dlt 配对，证明经有限次除子收缩和翻转可得到对数极小模型或 Mori 纤维空间，并建立无基点定理。
Derived deformation theory of crepant curves
本文确定了Calabi-Yau三维流形中光滑有理曲线的完整导出形变理论，给出非交换形变的超势代数描述，验证弦理论预测，并提出推广Artin曲面结果的三维流形有理曲线可收缩性猜想。
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