3-fold log flips according to V.V.Shokurov

Termination of 4-fold canonical flips

We prove the termination of 4-fold canonical flips.

On termination of 4-fold semi-stable log flips

In this paper, we prove the termination of 4-fold semi-stable log flips under the assumption that there always exist 4-fold (semi-stable) log flips.

Factoring 3-fold flips and divisorial contractions to curves

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.

伪有效四维翻转移除的终止性

Termination of pseudo-effective 4-fold flips

本文证明特征零代数闭域上对数典范四维簇的任意伪有效翻转移除序列必然终止。

Normality of general elephants on 3-fold terminal flips

We prove that the general elephant $E_{X^+}\in |-K_{X^+}|$ is normal where $X--> X^+$ is a 3-fold terminal flip. Hence $E_{X^+}$ has at worst Du Val singularities. As a corollary, there exists no non-Gorenstein singularity of type $cE/2$, $cD/3$, nor $cAx/4$ on the flipped curve $C^+$.

Fano 4-folds, flips, and blow-ups of points

In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest.

Existence of flips and minimal models for 3-folds in char p

We will prove the following results for $3$-fold pairs $(X,B)$ over an algebraically closed field $k$ of characteristic $p>5$: log flips exist for $\Q$-factorial dlt pairs $(X,B)$; log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$; the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a big $\Q$-divisor; semi-ampleness holds for a nef and big $\Q$-divisor $D$ if $D-(K_X+B)$ is nef and big and $(X,B)$ is projective klt; $\Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pair…

Flips from semi-stable 4-folds whose degenerate fibers are unions of Cartier divisors which are terminal factorial 3-folds

We shall investigate flipping contractions from a semi-stable 4-fold $X$ whose degenerate fiber is a union of Cartier divisors which are terminal factorial 3-folds. Especially we shall prove that $X$ is smooth along the flipping locus, and that the flip exists for such contractions.

On termination of log flips in dimension four

We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $κ\ge 2$.

Semistable 3-fold flips

We piece together ingredients, which are well known and documented in the literature, into a new proof of the existence of semistable 3-fold flips

Mld's vs thresholds and flips

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the acc conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension $\le d$ imply the following: the acc conjecture for $a$-lc thresholds, in particular, for canonic…

Birational maps of 3-folds

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.

Flips from 4-folds with isolated complete intersection singularities whose downstairs have rational bi-elephants

We shall investigate a flipping contraction g : X -> Y from a 4-fold X with at most isolated complete intersection singularities. If Y has an anti-bi-canonical divisor (=bi-elephant) with only rational singularities, then g carries an inductive structure chained up by blow-ups (La Torre Pendente), and in particular the flip exists. This naturally contains Miles Reid's `Pagoda' as an anti-canonical divisor (=elephant) and its proper transforms.

The LMMP for log canonical 3-folds in char $p$

We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

关于 Kähler 3-fold 的极小模型纲领

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。

On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension $3$ over algebraically closed fields, they have applications to tight closure in dimension $4$.

Letters of a Bi-Rationalist: VII. Ordered termination

To construct a resulting model in LMMP is sufficient to prove existence of log flips and their termination for certain sequences. We prove that LMMP in dimension $d-1$ and termination of terminal log flips in dimension $d$ imply, for any log pair of dimension $d$, the existence of a {\em resulting} log model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply existence of log flips in dimension $d+1$. As consequence, we prove existence of a resulting model of 4-fold log pairs, existence of log flips in dimension 5, and some fragments of Geography of log model…

Kähler 3维流形的对数极小模型纲领

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。

Kähler 3折上广义对的MMP

MMP for Generalized Pairs on Kähler 3-folds

本文研究Kähler 3折的广义对MMP，证明了极值定理、翻转存在性、log终端模型等核心结果。