Kenta Sato, Shunsuke Takagi
In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb{Q}$-Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg and S. Ishii.
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We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
We define and calculate the weighted multiplicities of non Gorenstein terminal singularities on threefolds and some quotient singularities. We improve freeness conditions on threefolds.
We give new examples of terminal and log canonical singularities.
The first examples of exceptional terminal singularities are constructed.
In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal…
In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
We give the multiplicity of terminal singularities on threefolds by simple calculation. Then we obtain the best inequalities for the multiplicity and the index. By using this, we can improve the boundedness number of terminal weak Q-Fano 3-folds in [KMMT, Theorem 1.2]. Furthermore, we can extended [K, Theorem 3.6] for Fujita freeness conditions to nonhypersurface terminal singularities.
In this paper we give a new point of view for optimizing the definitions related to the study of singularities of normal varieties, introduced in [dFH09] and further studied in [Urb12a] and [Urb12b], in relation to the Minimal Model Program. We introduce a notion of discrepancy for normal varieties, and we define log terminal+ singularities. We use finite generation to relate these new singularities with log terminal singularities (in the sense of [dFH09]).
Divisors with minimal discrepancy over cA points are classified
Del Pezzo surfaces with four log terminal singularities
本文研究了Picard数为一且具有四个log终端奇点的Del Pezzo表面的分类问题。
In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when $X/G$ has terminal singularities. We prove that, if $G$ is cyclic of prime order and $X/G$ has terminal singularities, then $G$ has order $2,3$ or $5$.
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.
We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
Let $(Z,o)$ be a three-dimensional terminal singularity of type $cA/r$. We prove that all exceptional divisors over $o$ with discrepancies $\le 1$ are rational.
We generalize the result of Kawamata concerning the strong version of Fujita's freeness conjecture for smooth 3-folds to some singular cases, namely, Gorenstein terminal singularities and quotient singularities of type 1/r(1,1,1) and of type 1/r(1,1,-1). We generalize furthermore the result of that to projective threefolds with only canonical singularities for canonical and not terminal singularities. It turns out that the estimates in the first three cases are better than the one for the smooth case, which it is not in the fourth case. We also give explicit examples which show the estimate in…
In "Singularities on Normal Varieties", de Fernex and Hacon started the study of singularities on non-Q-Gorenstein varieties using pullbacks of Weil divisors. In "Log Terminal Singularities", the author of this paper and Urbinati introduce a new class of singularities, called log terminal+, or simply lt+, which they prove is rather well behaved. In this paper we will continue the study of lt+ singularities, and we will show that they satisfy a Bertini type result, inversion of adjunction and small deformation invariance, and that they are naturally related to rational singularities. Finally, w…
On the Nash Problem over 3-Fold Terminal Singularities of Type cAx/2
本文研究3折终端奇点类型cAx/2上的纳什估值，发现例外素数除子诱导纳什估值，并推测该结论适用于所有3折终端奇点。
This is the first of a series of papers studying real algebraic threefolds using the minimal model program. The main results are outlined in Part II. The present part I. contains the necessary preliminary work concerning terminal singularities. First I give standard forms for 3-dimensional terminal singularities over arbitrary fields. This is then used to develop a fairly complete topological classification over the reals.
Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. The first application is a Boutôt-type theorem for log-terminal singularities: given a pure morphism $Y\to X$ between affine $\mathbb Q$-Gorenstein varieties of finite type over $\mathbb C$, if $Y$ has at most a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noethe…
Rational curves on Fano threefolds with Gorenstein terminal singularities
本文研究具有Gorenstein终端奇点的Fano三维簇上的有理曲线，推广了几何Manin猜想至奇异情形，并证明了某些奇异del Pezzo三维簇的该猜想。
A 3-fold terminal quotient singularity X=C^3/G admits the economic resolution Y-> X, which is "close to being crepant". This paper proves that the economic resolution Y is isomorphic to a distinguished component of a moduli space of certain G-equivariant objects using the the King stability condition $θ$ introduced by Kędzierski.
The Kobayashi pseudometric in the presence of log-terminal singularities
本文证明Kobayashi伪度量在对数终端奇点解消下保持良好性质，从而回答了Kamenova和Lehn提出的问题。
General n-canonical divisors on two-dimensional smoothable semi-log-terminal singularities
在本文中，我们计算一般的n-典型因子的光滑半对数终端奇异在2维，换句话说，全层相关的双重对偶的第n张量的权力，这些奇异的对偶层。作为其应用，我们给出了以这些奇点的n-典型因子的局部自交数来约束Gorenstein指数的不等式。
Log terminal singularities, platonic tuples and iteration of Cox rings
本文将曲面奇点作为有限可解群商并对应 Cox 环迭代的图景推广到任意维数、复杂度为一的环面作用情形，并对复合 du Val 三维奇点给出完整分类及所有迭代 Cox 环链。
This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.
On the Picard Number of Fano 3-Folds with Terminal Singularities
本文证明了如下主要结果：设X是一个具有终端Q-阶乘奇点的Fano 3-fold，且X的Mori多面体不存在小的极值射线和Kodaira维数为1或2的面。则Picard数ρ（X）< 8。
We show that there exists a natural number $p_0$ such that any three-dimensional Kawamata log terminal singularity defined over an algebraically closed field of characteristic $p>p_0$ is rational and in particular Cohen-Macaulay.
Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.
We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an…
A Gorenstein criterion for strongly $F$-regular and log terminal singularities
本文在反典范覆盖 Noetherian 的条件下证明 Hirose–Watanabe–Yoshida 猜想，并将阈值判据推广到正规 $F$-纯与对数典范奇点的拟 Gorenstein 判定。
We prove that a Kawamata log terminal pair has the canonical model.
Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities form an unfamiliar class and are rarely even log-canonical, we prove the Theorem. A rational homology disk smoothing of a weighted homogeneous surface singularity can always be chosen so that the total space is log-terminal. In particular, this smoothing is $\Q$-Gorenstein. The key idea is to define a finite "graded discrepancy" of a normal graded domain with…
Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.
Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method
本文利用圆法研究低阶超曲面上曲线模空间的奇点，证明了当映射阶数 $e$ 相对亏格 $g$ 足够大时，该模空间至多具有终端奇点。
Let X be a projective variety with terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray $ R \subset \bar {NE(X)}$ such that R.(K_X+(n-2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R.(K_X+rL)<0, where r is a non-negative rational number, and the fibres of f have dimension less or equal to r+1.
We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $π_*ω_{X'}(G) \simeq ω_X$ for a log resolution $π: X' \to X$, where $G$ is the reduced exceptional divisor of $π$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for …
Crepant resolutions of log-terminal singularities via Artin stacks
本文证明任意 log-terminal 奇点簇均可由光滑 Artin 叠实现 crepant 消解，并建立相应 McKay 对应，用光滑叠弧空间上的动机积分表达 Q-Gorenstein 簇的弦不变量。
We prove the boundedness theorem for Fano threefolds with log-terminal singularities of any fixed index. This is an improvement of our earlier result, where we required additionally that the variety is Q-factorial, with Picard number 1. The new ideas of the paper include the following. 1. Using Alexeev Minimal Model program with suitable boundary to find horizontal extremal contractions. 2. Using Kollár's effective Base Point Freeness theorem. 3. Using Kawamata's result on the length of extremal curves with suitable boundary to avoid gluing curves in some cases.
In this paper, we prove that Mather-Jacobian (MJ) singularities are preserved under the process of generic linkage. More precisely, let $X$ be a variety with MJ-canonical (resp. MJ-log canonical) singularities. Then a generic link of $X$ is also MJ-canonical (resp. MJ-log canonical). This further leads us to a result on minimal log discrepancies under generic linkage.
We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over $\mathbb C$ by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.
Blow-ups of three-dimensional toric singularities
该研究刻画了三维终端环面奇点的纯对数终端爆破，以及满足特定条件的三维除子收缩的结构，为代数几何中环面奇点的研究提供了相关理论结果。
We initiate a systematic investigation of F-theory on elliptic fibrations with singularities which cannot be resolved without breaking the Calabi-Yau condition, corresponding to $\mathbb Q$-factorial terminal singularities. It is the purpose of this paper to elucidate the physical origin of such non-crepant singularities in codimension two and to systematically analyse F-theory compactifications containing such singularities. The singularities reflect the presence of localised matter states from wrapped M2-branes which are not charged under any massless gauge potential. We identify a class of …
Terminal 3-folds that are not Cohen-Macaulay
本文证明在特征 $p$（$p=2,3,5$）或混合特征 $(0,p)$ 下，3 维终端奇点可非 Cohen-Macaulay，通过正则概形关于 $p$ 阶循环群的商构造，并给出仅具环面奇点的充分条件。
This is a survey of some recent developments in the study of singularities related to the classification theory of algebraic varieties. In particular, the definition and basic properties of Du Bois singularities and their connections to the more commonly known singularities of the minimal model program are reviewed and discussed.
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.
On fillings of contact links of quotient singularities
该研究利用Floer理论探讨商奇点接触链环的各类填充问题，证明多类孤立终端商奇点的接触链环不存在恰当填充并解决相关猜想，同时得到部分情形下恰当轨形填充的轨形微分同胚类型唯一性。
A singularity in characteristic zero is said to be of dense F-pure type if its modulo p reduction is locally F-split for infinitely many p. We prove that if $x \in X$ is an isolated log canonical singularity with $μ(x \in X) \le 2$ (see Definition 1.4 for the definition of the invariant $μ$), then it is of dense F-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense F-pure type in the case of three-dimensional isolated singularities.
For a characteristic $p > 0$ variety $X$ with controlled $F$-singularities, we state conditions which imply that a divisorial sheaf is Cohen-Macaulay or at least has depth $\geq 3$ at certain points. This mirrors results of Kollár for varieties in characteristic zero. As an application, we show that that relative canonical sheaves are compatible with arbitrary base change for certain families with sharply $F$-pure fibers.
In this paper we apply Shokurov's inductive method to study terminal and canonical singularities. As an easy consequence of the Minimal Model Program we show that for any three-dimensional log terminal singularity there exists some special, so called, plt blow-up. We discuss properties of them and construct some examples.
In characteristic zero, quotient singularities are log terminal. Moreover, we can check whether a quotient variety is canonical or not by using only the age of each element of the relevant finite group if the group does not have pseudo-reflections. In positive characteristic, a quotient variety is not log terminal, in general. In this paper, we give an example of the quotient variety which is not log terminal such that the quotient varieties associated to any proper subgroups are canonical. In particular, we cannot determine whether a given quotient singularity is canonical by looking at prope…
Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $π\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $π(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.
We study smoothings of Fano threefolds. We prove that the Picard number remains constant in the case of terminal Gorenstein singularities.
If $X$ is Frobenius split, then so is its normalization and we explore conditions which imply the converse. To do this, we recall that given an $\mathcal{O}_X$-linear map $ϕ: F_* \mathcal{O}_X \to \mathcal{O}_X$, it always extends to a map $\barϕ$ on the normalization of $X$. In this paper, we study when the surjectivity of $\barϕ$ implies the surjectivity of $ϕ$. While this doesn't occur generally, we show it always happens if certain tameness conditions are satisfied for the normalization map. Our result has geometric consequences including a connection between $F$-pure singularities and sem…
Kawamata-Miyaoka-type inequality for $\mathbb Q$-Fano varieties with canonical singularities II: Terminal $\mathbb Q$-Fano threefolds
本文证明了 Fano 指数至少为 $3$ 的终端 $\mathbb Q$-Fano 三维簇的最优 Kawamata-Miyaoka 型不等式，并应用得到所有终端 $\mathbb Q$-Fano 三维簇满足 $c_1(X)^3 < 3c_2(X)c_1(X)$。
We prove a conjecture of Batryev which states that the family of all Fano varieties with kawamata log terminal singularities and fixed index, forms a bounded family.
Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram Method)
在我们的预印本中：“Algebras surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces”，Preprint Max-Planck-Institut für Mathematik，Bonn，（1989）MPI/89-28（Russian），我们已经将图解方法推广到具有nef anticanonical class的代数曲面（之前，它被应用于Del Pezzo曲面）。这里的主要问题是，森多面体可能不是有限多面体在这种情况下。在本预印本的§4中，我们已经证明了这种方法对具有nef反正则类和对数终端奇点的代数曲面有效。最近，Valery Alexeev（Boundedness and K^2 for log surfaces，Preprint（1994），alg-geom/942007）特别利用我们的这些结果，对具有半对数正则奇点的一般类型曲面获得了强有力的重要结果。原因是德尔佩佐曲面（图方法是特别有效的）和一般类型的表面“连接”通过表面的数字零规范类。这是我们决定将上面预印本的第一部分（§1-§3）翻译成英语的主要原因，其中有一些扩展（我们省略了证明）和小的更正。这一部分包含了罗巴切夫斯基空间中的广义反射群的图解方…
We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.
We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.
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