Dimensions of jet schemes of log singularities
Takehiko Yasuda
本文利用 jet 概型的维数刻画 Kawamata 对数终端奇点和对数典范奇点，推广了 Mustata 的已有结果。
本文旨在通过 jet 概型的维数刻画 Kawamata 对数终端奇点与对数典范奇点，这是 Mustata 结果的推广。
The aim of the paper is to characterize Kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes. It is a generalization of Mustata's result.
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We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical singularities if and only if the log jet schemes of X are irreducible.
Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.
We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the condition that X has irreducible jet schemes in terms of data coming from an embedded resolution of X in Y. We show that this condition is equivalent with having canonical singularities. In the appendix, this result is used to prove a generalization of Kostant's freeness theorem to the setting of jet schemes.
Kempf proved that the theta divisor of a smooth projective curve C has rational singularities. In this paper we estimate the dimensions of the jet schemes of the theta divisor and show that all these schemes are irreducible. In particular, we recover Kempf's theorem in this way. For general projective smooth curves, our method also gives a formula for the log canonical threshold of the pair (\pic^d(C), W^r_d(C)).
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary characteristic. Based on this fact we will reduce many geometric properties of the singularities into the problem on jet schemes and try to avoid discussions which are distinctive for characteristic 0.
We give a self-contained presentation of the basic results on jet schemes of singular varieties. Applications are given to invariants of singularities, such as minimal log discrepancies. We simplify our older approach to Inversion of adjunction for locally complete intersection varieties, and we prove a more general result for arbitrary Q-Gorenstein varieties embeded in nonsingular varieties (this statement has been proved independently also by Kawakita) .
Let X be a smooth variety and Y a closed subscheme of X. By comparing motivic integrals on X and on a log resolution of (X,Y), we prove the following formula for the log canonical threshold of (X,Y): c(X,Y)=dim X-sup_m{(dim Y_m}/(m+1)}, where Y_m is the mth jet scheme of Y. We show how this formula can be used to study the log canonical threshold. In particular, we give a proof of the Semicontinuity theorem of Demailly and Kollár.
On jet closures of singularities
本文引入与射流闭包、射流支撑闭包相关的局部代数并证明其为奇点不变量，研究单项式理想等情形的不变量，提出新滤子与射流指标并分析其性质。
In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity. We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on $\mathbb{A}^3$), that computes the log canonical threshold of the singularity embedded in $\mathbb{A}^3$.…
This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense $F$-regular type. This provides us with examples of singular varieties whose $m$-jet schemes have rational singularities for every $m$.
We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the first author.
We give new examples of terminal and log canonical singularities.
This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035).
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]).
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.
In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.
The LCS locus is an essential ingredient in the proof of fundamental results of Log Minimal Model Program, such as nonvanishing and base point freeness theorems. We prove in this paper that the LCS locus of a log canonical variety has seminormal singularities.
The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed, using an explicit list of pole candidates for the motivic zeta function found by the last two authors.
In this paper the detailed classification of three-dimensional exceptional canonical hypersurface singularities which don't satisfy the condition of well-formedness is given. This result completes the classification of three-dimensional exceptional log canonical hypersurface singularities started in [4] (math.AG 0201025).
We point out an interesting relation between hypersurface elliptic singularities and log Enriques surfaces: with a few exceptions, every hypersurface elliptic singularity define some klt log Enriques surface $(S,Diff)$. In many cases, the log canonical cover of $(S,Diff)$ is a Du Val K3 surface with ``large'' nonsymplectic automorphism group.
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…
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。
We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…
This paper characterizes singularities with Mather minimal log discrepancies in the highest unit interval, i.e., the interval between $d-1$ and $d$, where $d$ is the dimension of the scheme. The class of these singularities coincides with one of the classes of (1) compound Du Val singularities, (2) normal crossing double singularities, (3) pinch points, and (4) pairs of non-singular varieties and boundaries with multiplicities less than or equal to 1 at the point. As a corollary, we also obtain one implication of an equivalence conjectured by Shokurov for the usual minimal log discrepancies.
In this paper we study $(ε,δ)$-lc singularites, i.e. $ε$-lc singularities admitting a $δ$-plt blow-up. We prove that $n$-dimensional $(ε,δ)$-lc singularities are bounded up to a deformation, and $2$-dimensional $(ε,δ)$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(ε,δ)$-lc singularities are not bounded in higher dimensions, even in the analytic sense.
Let $(P\in X,Δ)$ be a three dimensional log canonical pair such that $Δ$ has only standard coefficients and $P$ is a center of log canonical singularities for $(X,Δ)$. Then we get an effective bound of the indices of these pairs and actually determine all the possible indices. Furthermore, under certain assumptions including the log Minimal Model Program, an effective bound is also obtained in dimension $n\geq 4$.
We show that if $(X,B)$ is a log canonical pair with $\dim X\geq d+2$, whose non-klt centers have dimension $\geq d$, then $X$ is has depth $\ge d+2$ at every closed point.
We prove that the cohomology sheaves of the relative dualizing complex of a flat family of varieties with semi-log-canonical or Du Bois singularities are flat and commute with base change. This is a local version of our earlier similar result where the family was assumed to be projective. We also derive several consequences for deformations of semi-log-canonical and of Du Bois 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 recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the modu…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic desingularization of logarithmic schemes and morphisms, weighted methods: logarithmic and non-logarithmic ones, desingularization in wider contexts: quasi-excellent schemes and formal schemes, complex and non-archimedean spaces.
We introduce new notions of log jet spaces. Mildly singular spaces are ``smooth'' in log geometry, so their log jet spaces behave like the jet spaces of smooth varieties. Myriad examples contrast log jet spaces with the usual jet spaces of schemes. We then compute the log Kähler differentials of the log jet and arc spaces after de Fernex and Docampo. We obtain some of their applications to the structure of the log jet space. We conclude with comparison remarks with other log jet spaces.
We give a method to investigate isolated log canonical singularities with index one which are not log terminal. Our method depends on the minimal model program. One of the main purposes is to prove that our invariant coincides with Ishii's Hodge theoretic invariant.
For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of $V\backslash \{0\}$. For this purpose we introduce sheaves of $m$-canonical and $L^{2,m}$-canonical forms on normal complex spaces. For the case of affine varieties with good ${\bf C}^*$-action we give an explicit formula for these sheaves in terms of the grading of t…
We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Γ_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.
We show that the minimal log discrepancy of any $\mathbb Q$-Gorenstein non-canonical threefold is $\leq\frac{12}{13}$, which is an optimal bound.
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。
We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.
We prove semi-rationalification and semi-log-canonicalization for Gorenstein demi-normal surfaces. That is, given a Gorenstein demi-normal surface X with semi-rational (respectively, semi-log canonical) singularities in an open set U with complement a finite set of points, there is a proper birational morphism f : Y --> X such that f is an isomorphism over U and Y has only semi-rational (respectively, semi-log canonical) singularities. We proceed by passing to the normalization and then gluing along the conductor in an appropriate rationalification or log-canonicalization of the normalization …
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…
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…
Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Rieme…
We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.
Classification of toric surface singularities
本文以最小对数偏差为不变量，完成了二维环面对数芽的分类。
We study the singularities of the isospectral Hilbert scheme $B^n$ of $n$ points over a smooth algebraic surface and we prove that they are canonical if $n \leq 5$, log-canonical if $n \leq 7$ and not log-canonical if $n \geq 9$. We describe as well two explicit log-resolutions of $B^3$, one crepant and the other $\mathfrak{S}_3$-equivariant.
Given a scheme X over a field k, a generalized jet scheme parametrizes maps from Spec(A) to X, where A is a finite-dimensional, local algebra over k. We give an overview of known results concerning the dimensions of these schemes when A has embedding dimension one, when they are related to invariants of singularities in birational geometry. We end with a discussion of more general jet schemes.
Log canonical threshold of du Val singularities
本文计算了du Val型奇点曲面的对数典范阈值，涵盖$A_n, D_n, E_6, E_7, E_8$，并指出复数域结果与实数域奇异学习理论中的结果不同。
Motivated by Shokurov's ACC Conjecture for log canonical thresholds, we propose an inductive point of view on singularities of pairs, in the case when the ambient variety is smooth. Our main result characterizes the log canonicity of a pair in dimension N+1 by the log canonicity of another pair (not effective, in general) in dimension N.
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.
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If …
We show that there exists a positive real number $δ>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-δ$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.
We give a precise formulation of the modularity principle for the log canonical models of $\bar{M}_g$. Assuming the modularity principle holds, we develop and compare two methods for determining the critical alpha-values at which a singularity or complete curve with G_m-action arises in the modular interpretations of log canonical models of $\bar{M}_g$. The first method involves a new invariant of curve singularities with G_m-action, constructed via the characters of the induced G_m-action on spaces of pluricanonical forms. The second method involves intersection theory on the variety of stabl…
Let $E$ be the essential part of the exceptional locus of a good resolution of an isolated, log canonical singularity of index one. We describe the dimension of the dual complex of $E$ in terms of the Hodge type of $H^{n-1}(E, O_E)$, which is one of the main results of the paper [1] of Fujino. Our proof uses only an elementary classical method, while Fujino's argument depends on the recent development in minimal model theory.
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。
In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-…
In 2002, Alexeev provided a modular interpretation for the toroidal compactification of $A_g$ for the 2nd Voronoi fan. In this paper we show that pairs $(X,Θ)$ in the boundary of this moduli space are semi-log canonical, the analog of log canonical in the non-normal setting. This extends the result of Koll${á}$r stating that principally polarized abelian pairs $(X,Θ)$ are log canonical.
For a toric log variety with standard coefficients, we show that the minimal log discrepancy at a closed invariant point bounds the Cartier index of a neighbourhood.
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