On termination of log flips in dimension four

Termination of (many) 4-dimensional log flips

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.

ACC for log canonical thresholds and termination of log flips

We prove that the ascending chain condition (ACC) for log canonical (lc) thresholds in dimension $d$ and Special Termination in dimension $d$ imply the termination of any sequence of log flips starting with a $d$-dimensional lc pair of nonnegative Kodaira dimension. In particular, in characteristic zero, the latter termination in dimension 4 follows from Alexeev-Borisov's conjecture in dimension 3.

Nakayama-Zariski 分解与翻转的终止性

Nakayama-Zariski decomposition and the termination of flips

本文证明在伪有效射影对中，假设关于Nakayama-Zariski分解在极小模型程序操作下行为的自然猜想，一次翻转序列的终止性蕴含所有翻转的终止性。

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。

On termination of flips and fundamental groups

In this article, we propose a boundedness conjecture for the regional fundamental group of klt singularities. We prove that this boundedness conjecture, the Zariski closedness of the diminished base locus of $K_X$, and an upper bound for minimal log discrepancies, imply the termination of flips with scaling. We prove the boundedness conjecture in the case of toric singularities, quotient singularites, isolated $3$-fold singularities, and exceptional singularities.

关于减弱乘子理想与翻转的终止

On diminished multiplier ideal and the termination of flips

本文发展了奇异簇上的减弱乘子理想理论，并证明了在Cartier指数有界等条件下带缩放的特定翻转的终止。

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…

关于flip的终止与例外非典范奇点

On termination of flips and exceptionally non-canonical singularities

本文引入并研究了例外非典范（enc）奇点，将flip的终止问题归结为终端flip的终止及enc对mlds的ACC猜想，并在维度3和4中证明了相关结论。

On the existence of flips

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.

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…

对数典范对的特殊终止

Special termination for log canonical pairs

该研究属于代数几何领域，证明了对数典范对的特殊终止定理，并给出了该结论在广义对框架下的推广形式，完善了相关理论。

关于复解析空间上log canonical对的最简模型程序终止性

On termination of minimal model program for log canonical pairs on complex analytic spaces

本文研究复解析空间上log canonical对的最简模型程序终止性，并探讨其与项目性log canonical对理论的关系。

New outlook on the Minimal Model Program, II

We prove that the finite generation of adjoint rings proved in [Cascini and Lazić] implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

MMP for algebraically integrable foliations

We show that termination of flips for $\mathbb Q$-factorial klt pairs in dimension $r$ implies existence of minimal models for algebraically integrable foliations of rank $r$ with log canonical singularities over a $\mathbb Q$-factorial klt projective variety.

On Shokurov's Log Flips: The 3-dimensional Case

This is a survey on the recent fundamental paper by V.V. Shokurov on the existence of log flips.

正特征下三维广义对的极小模型的存在性

Existence of minimal models for threefold generalized pairs in positive characteristic

研究在特征\(p>5\)代数闭域\(\mathbb{K}\)上三维伪有效NQC lc广义对极小模型的存在性，通过\(\leq3\)维lc阈值的ACC和\(\leq2\)维广义对的全局ACC，证明其翻转终止性，提供不依赖非零定理的新证明。

Semi-stable minimal model program for varieties with trivial canonical divisor

We give a sufficient condition for the termination of flips. Then we discuss a semi-stable minimal model program for varieties with (numerically) trivial canonical divisor as an application. We also treat a slight refinement of dlt blow-ups.

Letters of a Bi-Rationalist IV: Geometry of log flips

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the existence of the flip for the contraction and another conjecture about the maximum of minimal log discripancy in given dimension; in particular, it holds in small dimensions (up to 4), and some special cases. It is illustrated by some examples.

Log minimal model program for the moduli space of stable curves: The second flip

We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for the moduli space of stable curves. In fact, our methods give a uniform, self-contained construction of the first three steps of the log minimal model program for the moduli spaces of stable pointed curves.

Positivity of the Moduli Part

We prove the Cone Theorem for algebraically integrable foliations. As a consequence, we show that termination of flips implies the b-nefness of the moduli part of a log canonical pair with respect to a contraction, generalising the case of lc trivial fibrations.

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

In this paper, I review the section 8 of V.V.Shokurov's paper '3-fold log flips'.

Log minimal models according to Shokurov

Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space.

Existence of log canonical flips and a special LMMP

Let $(X/Z,B+A)$ be a $\Q$-factorial dlt pair where $B,A\ge 0$ are $\Q$-divisors and $K_X+B+A\sim_\Q 0/Z$. We prove that any LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.

Two Two-dimensional Terminations

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates. Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. In this paper we prove that the following two sets s…

On a connectedness principle of Shokurov-Kollár type

Let $(X,Δ)$ be a log pair over $S$, such that $-(K_X+Δ)$ is nef over $S$. It is conjectured that the intersection of the non-klt (non Kawamata log terminal) locus of $(X,Δ)$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension $\leq 4$ and in arbitrary dimension assuming the termination of klt flips.

On the log minimal model program for irreducible symplectic varieties

We show the termination of any log-minimal model program for a pair $(X,Δ)$ of a symplectic manifold $X$ and an effective $\mathbb R$-divisor $Δ$.

VGIT presentation of the second flip of $\overline{M}_{2,1}$

We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus two. As a result, we give a GIT construction of the last three non-trivial log canonical models of the moduli space of pointed genus two curves, and obtain a VGIT presentation of the second flip in its Hassett-Keel program.

Log minimal model program for the moduli space of stable curves: The first flip

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…

Stable pairs and log flips

This paper has two parts. In the first part, we review stable pairs and triples on curves, leading up to Thaddeus' diagram of flips and contractions starting from the blow-up of projective space along a curve embedded by a complete linear series of the form K + ample. In the second part, we identify log canonical divisors which exhibit Thaddeus' flips and contractions as "log" flips and contractions in the sense of the log-minimal-model program.

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…

Finiteness of log abundant log canonical pairs in log minimal model program with scaling

We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs. We also discuss log MMP for slc pairs.

Supplement to the paper "On existence of log minimal models II"

We prove a stronger version of a termination theorem appeared in the paper "On existence of log minimal models II". We essentially just get rid of the redundant assumptions so the proof is almost the same as in there. However, we give a detailed proof here for future reference.

Existence of minimal models for varieties of log general type II

We prove the existence of pl-flips.

Minimal models, flips and finite generation : a tribute to V.V. SHOKUROV and Y.-T. SIU

In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs $(X/Z,B)$ with $B$ big$/Z$. This then implies existence of klt log flips, finite generation of klt log canonical rings, and most of the other results of Birkar-Cascini-Hacon-McKernan paper.

log 条件通用对的最简模型程序存在性

Existence of the minimal model program for log canonical generalized pairs

本文提出线性可分解通用对，结合改进的特殊终止和Kollár型粘合理论，证明了log canonical通用对的最简模型程序存在性，无需假设klt、NQC或$\mathbb Q$-因子性条件。