Florin Ambro
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log discrepancies for smooth varieties in positive characteristic.
We prove the ideal-adic semi-continuity of minimal log discrepancies on surfaces.
Minimal log discrepancies on a fixed threefold
本文证明任意固定三维簇上最小对数差异满足ACC，为三维簇奇点不变量的有界性提供结论。
We discuss the ideal-adic semi-continuity problem for minimal log discrepancies by Mustata. We study the purely log terminal case, and prove the semi-continuity of minimal log discrepancies when a Kawamata log terminal triple deforms in the ideal-adic topology.
We describe the set of minimal log discrepancies of toric log varities, and study its accumulation points.
We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.
Minimal log discrepancies on smooth threefolds
本文证明了光滑三维簇上极小对数差异的ACC，并推导出a-lc阈值ACC、一致m-adic半连续性及计算除子有界性等推论。
On the ACC for Minimal Log Discrepancies for Bounded Generalized Sub-Pairs
本文证明了最小对数差异的ACC猜想在任意维有界广义子对及有界三维簇上成立。
In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for $n$-dimensional minimal log discrepancies of regularity one holds around zero. Furthermore, we show that there exists an upper bound for the minimal log discrepancy of any $n$-dimensional klt singularity of regularity one.
We formulate a comparison of minimal log discrepancies of a variety and its ambient space with appropriate boundaries in terms of motivic integration. It was obtained also by Ein and Mustaţǎ independently.
On semi-continuity problems for minimal log discrepancies
本文证明了具有Crepan解析的簇的极小对数差异的半连续性，并据此解决了环面对的理想-adic半连续性问题。
This paper shows that Mustata-Nakamura's conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.
On minimal log discrepancies on varieties with fixed Gorenstein index
本文推广了对数典范阈值聚点的有理性定理，并将其应用于极小对数差异的 ACC 问题，证明了固定 Gorenstein 指数簇上具有固定系数的三维典范对的极小对数差异满足 ACC。
On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.
In this article we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities, with standard coefficients, which admit an $ε$-plt blow-up have minimal log discrepancies belonging to a finite set which only depends on $d,a$ and $ε$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities relate…
In this paper, we show that Shokurov's conjectures on the ACC for $a$-lc thresholds and the ACC for minimal log discrepancies are equivalent in the interval $[0,1)$. That is, the conjecture on ACC for $a$-lc thresholds holds for every $0\leq a<1$ if and only if the set of minimal log discrepancies for pairs with DCC coefficients do not have an accumulation point from below which belongs to $[0,1)$.
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.
A counterexample to the PIA conjecture for minimal log discrepancies
本文通过构造反例，推翻了关于最小 log 抛物线差异的 PIA 猜想，并对家庭中的 LSC 猜想也提供了反例。
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.
We introduce an approach of Riemann--Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that its minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded. Secondly we recover the characterisation of a Gorenstein terminal three-fold singularity by Reid, and the precise boundary of its minimal log discrepancy by Markushevich, without explicit classification. Finally we provide the precise boundary for a special four-fold singularity, whose …
We prove that for each positive integer $n$ there exists a positive number $ε_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,ε_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.
For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.
We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-δ,+\infty)$, where $δ>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's conjecture on the existence of $(ε,n)$-complements for threefolds for any $ε\geq 1$. As a key important step, we prove the uniform boundedness of divisors computing minimal log discrepancies for terminal threefolds. We show the ACC for threefold canonical thresholds, and that the set of accumulation points of threefold canonical thresholds is equal to $\{0\}\cup\{\frac{1}{n}\…
Bounding minimal log discrepancies of general arrangement varieties
本文证明了在一般排列簇上，最小对数歧变可由簇的维数上界限制，扩展了关于该猜想的先前结果。
Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$. If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$.
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-…
This is the major revision. The main purpose of this paper is to prove that minimal discrepancies of $n$-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less than $n$. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.
The second largest accumulation point of the set of minimal log discrepancies of threefolds is $\frac{5}{6}$. In particular, the minimal log discrepancies of $\frac{5}{6}$-lc threefolds satisfy the ACC.
A boundedness conjecture for minimal log discrepancies on a fixed germ
本文提出klt芽上极小对数差异的有界性猜想，证明其蕴含Shokurov的ACC猜想，并给出部分结果。
We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov's conjecture is true for log-terminal threefolds.
Upper bound of discrepancies of divisors computing minimal log discrepancies on surfaces
本文研究了曲面上计算极小对数差异性的除子差异上界问题，给出了一个显式上界$\ell(\gamma)\in O(1/\gamma^2)$，并通过实例证明该界是最优的。
Minimal log discrepancies of hypersurface mirrors
本文研究加权射影空间中拟光滑 Calabi-Yau 超曲面商的最小对数差异，证明其与 BHK 镜像的权和次数相关，并构造出已知最小对数差异的 klt Calabi-Yau 簇。
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.
We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum α_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal subvarieties. Our result implies the semicontinuity conjecture for minimal log discrepancies of such pairs. For these computations, we use the description of minimal log discrepancies via codimensions of cylinders in the space of jets; this necessitates the computations of an explicit generator for the canonical differential forms and the Na…
Exceptional Fano varieties with small minimal log discrepancy
本文构造了加权射影空间中的良构超曲面作为例外Fano簇，其最小对数差异随维度双指数衰减，并在二维达到最优值。
It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.
Let $Γ$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:=\sum_{i} b_iB_i)$ such that $b_i\in Γ$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X\ni x$, such that $a(E,X,B)={\rm{mld}}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura's conjecture to the setting that $X\ni x$ is not necessarily fixed and $Γ$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X…
A Sharp inequality between local volumes and minimal log discrepancies
该论文解答了Li–Liu–Xu的问题，证明了n维klt芽满足局部体积与极小对数偏差的严格不等式，还证明了相关比值的离散性，并将其应用于得到对数Fano对极小对数偏差的下界。
We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.
On boundedness of singularities and minimal log discrepancies of Kollár components
本文研究klt奇点的有界性问题，证明在三维或Kollár分量极小对数差异有上界时，局部体积远离零的klt奇点有界至多相差特殊退化，并回答了关于对数典范阈值与局部体积关系的一个问题。
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.
Complements, index theorem, and minimal log discrepancies of foliated surface singularities
本文将配对与簇的若干结果推广到叶状曲面对，证明局部补集有界性、局部指标定理及mlds一致有界性，否定Cascini-Spice的两个问题，并恢复叶状曲面mlds升链条件猜想的证明。
The stringy Euler number and E-function of Batyrev for log terminal singularities can in dimension 2 also be considered for a normal surface singularity with all log discrepancies nonzero in its minimal log resolution. Here we obtain a structure theorem for resolution graphs with respect to log discrepancies, implying that these stringy invariants can be defined in a natural way, even when some log discrepancies are zero, and more precisely for all normal surface singularities which are not log canonical. We also show that the stringy E-functions of log terminal surface singularities are polyn…
On boundedness of singularities and minimal log discrepancies of Kollár components, II
本文证明K-半稳定log Fano锥奇点的有界性由局部体积下界和Kollár分量极小对数差异上界决定，并据此确认了三维情形的有界性猜想及klt奇点局部体积的聚集性质。
We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist, then our definition agrees with previous approaches to log discrepancies of semivaluations that these resolutions. We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong F-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity …
As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.
This paper contains a number of observations on the {$F$-signature} of triples $(R,Δ,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,Δ,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we …
On structures and discrepancies of klt Calabi--Yau pairs
本文研究 klt Calabi–Yau 对的结构，证明固定维数下其对数中心差异取值有限，并据此得出四维非典范 Calabi–Yau 簇指数有界。
Discrepancies of $p$-cyclic quotient varieties
本文研究特征 p>0 下 p 阶循环群线性表示所对应商簇的奇异性，通过估计例外除子的最小差异，给出终端、典范和对数典范的判别准则，并构造了新的非 Cohen-Macaulay 终端奇异性例子。
On generalized minimal log discrepancy
本文研究广义配对的最小对数差异，将通常配对下 ACC 与 LSC 猜想的已知结果推广到广义配对，并探讨其补理论。
We explicitly compute the Hodge ideals of Q-divisors in terms of the V-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein-Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.
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.
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţǎ-Nakamura.
We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by a sequence of flops, and the two log minimal models share some properties. We also give examples of two log minimal models of an lc pair which have different properties.
We discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.
A non-klt counterexample to Shokurov's index conjecture
该研究针对非klt情形的Shokurov指标猜想，构造了反例，为相关代数几何问题提供了否定性依据。
We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.
In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications …
We prove the existence of good log minimal models for dlt pairs of numerical log Kodaira dimension 0.
We prove $\mathbb{R}$-boundary divisor versions of results proved by Birkar or Hacon-Xu on special kinds of the relative log minimal model program.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。