Gap conjecture for 3-dimensional canonical thresholds

Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly

We compute global log canonical thresholds of some smooth Fano threefolds.

对数典范阈值的无穷小结构

Infinitesimal structure of log canonical thresholds

本文证明固定维数下对数典范阈值序列的聚点行为具有标准化结构，并进一步探讨了极小对数差异、典范阈值及K-半稳定阈值的类似性质。

On threefold canonical thresholds

We show that the set of threefold canonical thresholds satisfies the ascending chain condition. Moreover, we derive that threefold canonical thresholds in the interval $ (\frac{1}{2}, 1)$ consists of $ \{ \frac{1}{2}+\frac{1}{n}\}_{n \ge 3} \cup \{ \frac{4}{5}\}$.

Canonical and log canonical thresholds of Fano complete intersections

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to one, if $M\geqslant 2k+3$ and the maximum of the degrees of defining equations is at least 8. This is an essential improvements of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of K\" ahler-Einstein metrics on generic Fano complete intersections described above.

The Zariski-Lipman conjecture for log canonical spaces

In this paper we give an elementary proof of the Zariski-Lipman conjecture for log canonical spaces.

Log canonical thresholds of quasi-ordinary hypersurface 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.

Noether-Fano Inequalities and Canonical Thresholds on Fano Varieties

We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.

A note on log canonical thresholds

We prove that the largest accumulation point of the set $\mathcal{T}_3$ of all three-dimensional log canonical thresholds $c(X,F)$ is 5/6.

Log-canonical threshold for curves on a smooth surface

It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.

Log canonical threshold, Segre classes, and polygamma functions

We express the Segre class of a monomial scheme in projective space in terms of log canonical thresholds of associated ideals. Explicit instances of the relation amount to identities involving the classical polygamma functions.

Smooth 3-dimensional canonical thresholds

If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a $\Q$-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is a germ of smooth 3-dimensional variety, satisfies the ascending chain condition. We also deduce a formula for the canonical threshold of $(\C^3,S)$, where S is a Brieskorn singularity.

Accumulation points on 3-fold canonical thresholds

We obtain that the nonzero accumulation points of the set of 3-fold canonical thresholds $\ct(X,S)$ are precisely $1/k$ where $k\ge 2$ is an integer and $S$ is an effective integral divisor of a projective 3-fold $X$ with only terminal singularities. Moreover, we generalize the ascending chain condition for the set of 3-fold canonical thresholds to pair.

The Ascending Chain Condition for log canonical thresholds on l.c.i. varieties

Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties.

Log Iitaka conjecture for abundant log canonical fibrations

We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.

Xiao's Conjecture on Canonically Fibered Surfaces

A canonically fibered surface is a surface whose canonical series maps it to a curve. Using Miyaoka-Yau inequality, A. Beauville proved that a canonically fibered surface has relative genus at most 5 when its geometric genus is sufficiently large. G. Xiao further conjectured that the relative genus cannot exceed 4. We give a proof of this conjecture.

On log canonical thresholds, II

We prove that the only accumulation points of the set $T_3$ of all three-dimensional log canonical thresholds in the interval $[1/2,1]$ are $1/2+1/n$, where $n\in\ZZ$, $n\ge 3$.

Log canonical thresholds on smooth varieties: the Ascending Chain Condition

Building on results of Kollár, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.

高乘积简化平面曲线的log canonical阈值

Log canonical thresholds of high multiplicity reduced plane curves

研究高乘积简化平面曲线的log canonical阈值，确定小于2/(d-1)的可能值，并计算度数小于6的曲线阈值。

平面曲线在任意特征下的log canonical阈值

Log Canonical Thresholds for Plane Curves in Arbitrary Characteristic

本文通过估值理论将复数域上平面曲线的log canonical阈值公式推广到任意特征情况。

Green's canonical syzygy conjecture for ribbons

Green's canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz-Ramanan on Green's conjecture for general smooth curves.

Green's Conjecture for the generic canonical curve

Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's conjecture.

$K^2 = 5$ 的 Burniat 曲面的对数典范阈值

Log canonical thresholds of Burniat surfaces with $K^2 = 5$

本文计算了 $K^2 = 5$ 的次级 Burniat 曲面的全局对数典范阈值，并建立了其多重典范子线性系统成员对数典范阈值的最佳下界。

Canonical and log canonical thresholds of multiple projective spaces

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano-Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

Bounds for Behrend's conjecture on the canonical reduction

We prove Behrend's conjecture on the rationality of the canonical reduction of principal bundles and reductive group schemes for classical groups and give new bounds for the conjecture for exceptional groups. However we find a counterexample in the case of G_2-bundles in characteristic 2.

On the log canonical threshold and numerical data of a resolution in dimension 2

We show various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of Igusa on exponential sums.

Log Canonical Thresholds and Generalized Eckardt Points

Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. We prove that the log canonical threshold of $H\in|-K_X|$ is at least $\frac{n-1}{n}$. Under the assumption of the Log minimal model program, we also prove that a hyperplane section $H$ of $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hypersurface of degree $n$ in $\mathbb{P}^{n-2}$ if and only if the log canonical threshold of $H$ is $\frac{n-1}{n}$.

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.

A procedure for computing the log canonical threshold of a binomial ideal

We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials. The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function, which is defined in terms of the generators of the ideal. The minimum of this function is attained at some ray of a fan which only depends on the exponents of the monomials appearing in the generators of the ideal.

Bounds for log canonical thresholds with applications to birational rigidity

We use intersection theory, degeneration techniques and jet schemes to study log canonical thresholds. Our first result gives a lower bound for the log canonical threshold of a pair in terms of the log canonical threshold of the image by a suitable smooth morphism. This in turn is based on an inequality relating the log canonical threshold and the Samuel multiplicity, generalizing our previous result from math.AG/0205171. We then give a lower bound for the log canonical threshold of an affine scheme defined by homogeneous equations of the same degree in terms of the dimension of the non log te…

Theta-regularity and log-canonical threshold

We show that an inequality, proven by Küronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over $\mathbb{P}^n$ and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with $Θ$-regularity of Pareschi-Popa.

The log-canonical threshold of a plane curve

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in a paper by Kuwata in Amer. J. Math. 121.

叶状对数典范阈值的ACC

ACC for foliated log canonical thresholds

针对叶状对数典范阈值是否满足升链条件的问题，受叶状极小模型纲领启发，研究其叶状版本性质，在曲面和三维簇情形下证明了该性质成立。

On real log canonical thresholds

We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the $b$-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distribution defined by the complex power of the absolute value of the function. However, this number may be greater than 1 if the codimension of the real zero locus of the function is greater than 1. So it does not necessarily coincide with the maximal r…

GIT of complete intersections and log canonical thresholds

We establish a link between the GIT stability of complete intersections of same degree hypersurfaces in the same ambient projective space, which can be parametrised as tuples in a Grassmanian scheme, and the log canonical thresholds of hypersurfaces contained in these tuples. We also establish a similar link in the case of Variations of GIT quotients, where we consider tuples of complete intersections and distinct hyperplanes.

Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities.

On log canonical rings

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.

典范丛公式与Schnell关于某类代数纤维空间的猜想

Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell

本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。

The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold

Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to I, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of I. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.

G-stable rank of symmetric tensors and log canonical threshold

Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the G-stable rank introduced by Derksen is true: the symmetric G-stable rank and G-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the G-stable rank of the defining ideal.

Burniat 曲面的全局对数典范阈值及作为 $\mathbb{Z}_2^n$-覆盖的一般型曲面的某些无限族的全局对数典范阈值

Global log canonical thresholds for Burniat surfaces and some infinite families of surfaces of general type as $\mathbb{Z}_2^n$-covers

本文完成 Burniat 曲面全局对数典范阈值计算，并确定作为 $\mathbb{P}^2$ 或 $\mathbb{P}^1 \times \mathbb{P}^1$ 的 $\mathbb{Z}_2^n$-覆盖的一般型曲面无限族的相应阈值。

三重典范阈值的分类

Classification of threefold canonical thresholds

本文分类了三重典范阈值，证明光滑情形与二维超曲面对数典范阈值一致，并给出完整集合为{0}、{4/5}与光滑情形的并集。

Elliptic Gorenstein singularities, log canonical thresholds and log Enriques surfaces

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.

Remarks on the abundance conjecture

We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.

Limits of log canonical thresholds

Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n lies in T_{n-1}, proving in this setting a conjecture of Kollár. We also show that T_n is a closed subset in the set of real numbers; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov's ACC Conjecture for all T_n, it is eno…

Log pluricanonical representations and abundance conjecture

We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if so is the log canonical divisor of its normalization. We also treat many other applications.

On a conjecture of Teissier: the case of log canonical thresholds

For a smooth germ of algebraic variety $(X,0)$ and a hypersurface $(f=0)$ in $X$, with an isolated singularity at $0$, Teissier conjectured a lower bound for the Arnold exponent of $f$ in terms of the Arnold exponent of a hyperplane section $f\vert_H$ and the invariant $θ_0(f)$ of the hypersurface. By building on an approach due to Loeser, we prove the conjecture in the case of log canonical thresholds.

du Val奇点的对数典范阈值

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。

平面曲线束的稳定性、对数典范阈值与重数

Stability of pencils of plane curves, log canonical thresholds and multiplicities

该研究借助几何不变量理论，通过关联曲线束稳定性与生成元稳定性、对数典范阈值、基点重数，得到平面d次曲线束的显式稳定性判据，解决相关分类问题。

The relative canonical resolution: Macaulay2-package, experiments and conjectures

This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2-package which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on a dedicated webpage. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all…

Log canonical thresholds of del Pezzo surfaces

We study global log canonical thresholds of del Pezzo surfaces.

Log canonical thresholds of Del Pezzo Surfaces in characteristic p

The global log canonical threshold of each non-singular complex del Pezzo surface was computed by Cheltsov. The proof used Kollár-Shokurov's connectedness principle and other results relying on vanishing theorems of Kodaira type, not known to be true in finite characteristic. We compute the global log canonical threshold of non-singular del Pezzo surfaces over an algebraically closed field. We give algebraic proofs of results previously known only in characteristic $0$. Instead of using of the connectedness principle we introduce a new technique based on a classification of curves of low degre…

The log canonical threshold of homogeneous affine hypersurfaces

We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in P^{n-1}. In the case when Y is a hyperplane section of a smooth hypersurface in P^{n+1}, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that…

ACC for log canonical thresholds for complex analytic spaces

We show that log canonical thresholds for complex analytic spaces satisfy the ACC.

Multiplicities and log canonical threshold

If R is a local ring of dimension n, of a smooth complex variety, and if I is a zero dimensional ideal in R, then we prove that e(I)\geq n^n/lc(I)^n. Here e(I) is the Samuel multiplicity along I, and lc(I) is the log canonical threshold of (R,I). We show that equality is achieved if and only if the integral closure of I is a power of the maximal ideal. When I is an arbitrary ideal, but n=2, we give a similar bound involving the Segre numbers of I.