Ziquan Zhuang
We show that the set of Fano varieties (with arbitrary singularities) whose anticanonical divisors have large Seshadri constants satisfies certain weak and birational boundedness. We also classify singular Fano varieties of dimension $n$ whose anticanonical divisors have Seshadri constants at least $n$, generalizing an earlier result of Liu and the author.
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We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$. In characteristic $p>5$ and dimension $3$, we also show that Fano varieties $X$ with Seshadri constants $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ (for some fixed $ε>0$) have bounded anti-canonical degrees.
We give a lower bound of the $δ$-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.
Given $ε>0$, we show that over an algebraically closed field of characteristic $p>5$, the anticanonical volume of a Fano threefold $X$ (with arbitrary singularities) whose anticanonical divisor has Seshadri constant $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ is bounded from above.
Seshadri constants, introduced by Demailly, measure the local positivity of a nef divisor at a point. In this paper, we compute the Seshadri constants of the anticanonical divisors of Fano manifolds with coindex at most $3$ at a very general point. As a consequence, if $X$ is a nonsingular Fano threefold which is very general in its deformation family, then $\varepsilon(X,-K_X;x)\leq 1$ for all points $x\in X$ if and only if $\vert-K_X\vert$ is not base point free.
We prove that the Seshadri constant of a polarized abelian variety is equal to the Seshadri constant of its abelian subvariety if the Seshadri constant is relatively small with respect to its degree, or it contains an abelian divisor which has sufficiently small degree. As an application of these results, we show that the Seshadri constant of a polarized abelian surface is computed by the unique Seshadri curve if the Seshadri constant is small enough. As another application, we compute the Seshadri constants of some non-simple polarized abelian threefolds which contain principally polarized ab…
Seshadri constants and degrees of defining polynomials
本文研究了Seshadri常数与定义多项式次数之间的关系。特别是，我们计算的Seshadri常数的Fano品种作为完整的交点在理性齐性空间的皮卡德数1。
Seshadri constants and periods of polarized abelian varieties
考虑复数域上的极化阿贝尔簇$(A,L)$。根据Demailly，可以将一个实数$ε(A,L)$与$(A,L)$相关联，它的{ \em Seshadri常数}实际上测量了$L$的阳性有多少可以集中在$A$的任何给定点。最近有相当大的兴趣在寻找界的Seshadri常数的阿贝尔簇和光滑的射影品种一般。本文首先推广Buser和Sarnak的方法，根据极化类型给出了$(A,L)$的最小周期长度的下界，并由Lazarsfeld的结果得到了$(A,L)$的Seshadri常数的下界.其次，我们认为Prym品种，并表明，他们有小Seshadri常数，因此不寻常的小周期。最后，我们得到了精致的结果的情况下，阿贝尔表面，这意味着特别是令人惊讶的事实，Seshadri常数总是合理的，在这种情况下。
Seshadri constants of indecomposable polarized abelian varieties
本文研究不可分解极化阿贝尔簇的Seshadri常数下界猜想，受Debarre思想启发，并借助Newton-Okounkov体的凸几何方法，推广作者先前关于阿贝尔三维簇合冲的工作。
We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.
Seshadri constants via toric degenerations
我们给出了一种方法来估计Seshadri常数的复曲面品种在任何点。利用这些估计和环面退化，我们可以得到非环面簇上Seshadri常数的一些新的计算或估计。特别地，我们详细地研究了射影空间和Picard数为1的Fano 3-folds中超曲面上的Seshadri常数.
We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.
We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose ta…
Seshadri constants and the generation of jets
在本文中，我们探讨Seshadri常数和产生的喷流之间的联系。众所周知，观察塞沙德里常数的一种方法是把它们看作是测量多个线丛产生的喷流数量的增长率。这里我们反过来问，一旦知道了塞沙德里常数，我们能对喷流的数量说些什么。作为结果的应用，我们证明了在所有Fano簇中射影空间的Seshadri常数刻画.
Jets-separation thresholds, Seshadri constants and higher Gauss-Wahl maps on abelian varieties
本文在极化阿贝尔簇上定义闭子概形的消失阈值与射流分离阈值，并利用Fourier-Mukai方法建立其与高阶Gauss-Wahl映射满射性及Seshadri常数之间的联系，给出满射性判别准则。
Seshadri constants and hyperelliptic curves on abelian varieties
研究极化阿贝尔簇上光滑曲线的Seshadri常数与曲线度数之比的上界，通过曲线的高阶Gauß-Wahl映射的非满射性定义内在不变量，并证明超椭圆曲线的Castelnuovo型不等式及其等号条件。
Seshadri constants and Gromov widths of toric varieties associated with generalized permutohedra
本文研究极化的环面簇的Seshadri常数和Gromov宽度，其矩多面体为广义置换体，证明两个不变量等于矩多面体的格宽度，并给出基于定义子模函数的显式公式。
Seshadri constants in finite subgroups of abelian surfaces
给定光滑射影簇的一个标商q：X->Y，我们将Y上的线丛M的单Seshadri常数与q*M在纤维上的点的重Seshadri常数联系起来.我们应用这种方法计算了有限子群点上的极化阿贝尔曲面的Seshadri常数。
We prove that an $n$-dimensional complex projective variety is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$. We also classify complex projective varieties with Seshadri constants equal to $n$.
We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. We also study the case of arbitrary codimension.
Lengths of Periods and Seshadri Constants of Abelian Varieties
本说明的目的是指出一个基本的，但有点令人惊讶的工作之间的联系Buser和Sarnak的长度期间的阿贝尔品种和Seshadri常数测量当地积极的θ因子。联系是通过辛爆破建立的，在McDuff和Polterovich的精神。作为主要不等式的一个应用，我们得到了Buser-Sarnak型的一个命题的一个简单的新证明，其大意是曲线的雅可比矩阵具有异常小长度的周期。我们还推导出一个下界的Seshadri常数的一个非常一般的p. p. a. v.不同的最大可能的一个因素小于4。
Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties
本文将不变量$α_{x}(L)$与代数簇$X$上的代数点$x$联系起来，该代数簇具有一个充满线丛$L$ .不变量$α$测量相对于与$L$相关联的高度函数，$x$可以被$X$上的有理点近似的程度。我们发现，这个不变量是密切相关的Seshadri常数$ε_{x}(L)$测量局部积极性的$L$在$x$，特别是罗斯定理$\mathbf{P}^1$概括为这两个不变量之间的不等式有效的任意投影品种。
Multiprojective Seshadri stratifications for Schubert varieties and standard monomial theory
本文借助 Seshadri 分层，为 Lakshmibai-Seshadri 表及标准单项式基建立几何解释，并通过 Schubert 簇多齐次坐标环的滤过得到由标准表索引的一维子商。
Seshadri stratifications and Schubert varieties: a geometric construction of a standard monomial theory
该研究结合 Schubert 簇的 Seshadri 分层几何与 Demazure 模的 LS-路径特征公式，构造 Schubert 簇的标准单项式理论，并通过任意线性化和弱化平衡条件改进一般分层理论。
The theory of Seshadri stratifications has been developed by the authors with the intention to build up a new geometric approach towards a standard monomial theory for embedded projective varieties with certain nice properties. In this article, we investigate the Seshadri stratification on a Schubert variety arising from its Schubert subvarieties. We show that the standard monomial theory developed in [32] is compatible with this new strategy.
Let $X$ be a smooth projective variety defined over a field $k$ of characteristic $0$ and let $\mathcal{L}$ be a nef line bundle defined over $k$. We prove that if $x\in X$ is a $k$-rational point then the Seshadri constant $ε(X, \mathcal{L}, x)$ over $\overline{k}$ is the same as that over $k$. We show, by constructing families of examples, that there are varieties whose global Seshadri constant $ε(X)$ is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.
Combinatorial Seshadri stratifications on normal toric varieties
本文提出一种纯组合方法，将 Seshadri 分层理论应用于与正规格多胞形相关的嵌入环面簇，揭示了多胞形三角剖分、Seshadri 分层与退化半环面簇间的联系，并证明该方法与现有文献产生相同的拟赋值和退化。
Moving Seshadri constants and effective Fujita-type conjectures
本文提出并探讨了新的 Fujita 猜想变体，涉及线性系统形式 |K_X + aK_X + bL| 的无基点性、非常量性和喷射放大性，并扩展了移动塞沙迪常数理论至任意域上的完备变种。
The Seshadri Constants of Tangent Sheaves on Toric Varieties
本文研究环面簇上切丛的 Seshadri 常数，给出其正性的判别条件，并证明光滑射影环面簇若在某点处该常数为正则同构于射影空间，验证了 Fulger-Murayama 猜想的一个特例。
Seshadri constants of Higgs Vector bundles
本文在特征零代数闭域上的光滑射影簇上定义了Higgs丛的Seshadri常数，证明了其与普通Seshadri常数类似的性质，包括Higgs丰沛性的Seshadri判别法以及可通过限制到曲线来计算。
Some results on Seshadri constants of vector bundles
本文研究射影簇上丰富向量丛的Seshadri常数，给出了使其在所有点处至少为1的若干条件，并证明其可任意小。
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine 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.
Varieties of Fano type are very well behaved with respect to MMP, and they are known to be rationally connected. We study the relation between classes of rationally connected varieties and varieties of Fano type. We give examples of rationally connected varieties of dimension $\geqslant 3$ which are not birationally equivalent to Fano type varieties, thereby answering the question of Cascini and Gongyo.
This paper is a survey about cylinders in Fano varieties and related problems.
Fano varieties with conjecturally largest Fano index
本文构造了具有典范和末端奇点的 Fano 簇中大 Fano 指标的示例，并基于低维证据猜想这些示例在所有维度上达到最大 Fano 指标。
Rigidity properties of Fano varieties
我们概述了最近的一些结果Fano品种的证据，他们的刚性性质下的小变形。
Semisimple quantum cohomology of some Fano varieties
给出了Picard秩为1的Fano簇的量子上同调半单性的充分条件。我们应用这些技术来证明新的半简单性结果的一些Fano品种的皮卡德秩1和大指数。我们也给出了Fano簇具有非半单小量子上同调但具有半单大量子上同调的例子。
Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian étale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian étale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.
The pseudo-index of horospherical Fano varieties
证明了L.Bonavero，C.卡萨格兰德岛Debarre和S. Druel，关于光滑Fano品种的伪指数，在horospheric品种的特殊情况下。
Fano symmetric varieties with low rank
根据Akhiezer的一个经典结果，秩为1的对称射影簇都是光滑的和Fano的。我们分类的局部阶乘（分别光滑）的射影对称$G$ -品种的秩2是Fano。当$G$是半单的，我们也分类的局部阶乘（分别顺利）射影对称$G$ -品种秩2，这只是准法诺。此外，我们分类的Fano对称$G$ -品种的秩3可从一个奇妙的品种的爆破沿$G$ -稳定品种的序列。最后，我们对任意秩的Fano对称簇进行了分类，这些簇可以通过沿闭轨道的一系列爆破从一个奇妙簇得到。
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
A characterization of Fano type varieties
本文通过证明Fano型变种的特征化，揭示了其几何结构的核心属性。
Multiprojective Seshadri stratifications and Young-tableaux
本文通过将 Seshadri 分层推广至多重射影簇，为经典半标准 Young 表提供了代数几何解释，揭示了其与 Schubert 簇上有理函数消失重数的对应关系。
Fano Varieties with Large Degree Endomorphisms
我们给出了Picard数为1的Fano簇$X$的例子，这些簇具有终端奇点，并且允许度大于1的自同态。
Boundedness of Q-Fano varieties of Picard number one
证明了对于每个$n$，Picard数为1且维数为$n$的Q-Fano簇的双有理有界性.我们不使用森理论。
Birational geometry of singular Fano varieties
证明了具有初等奇点的一般位置的Fano超曲面和对偶空间的除子标准性。
Toric Fano varieties with divisorial contractions to curves
在本文中，我们得到了一个完整的分类光滑复曲面Fano品种配备极值收缩因子曲线的任何维。作为应用，我们得到了沿曲线等变爆破到Fano的光滑投射环面簇的一个完全分类。
We give a simple necessary and sufficient condition for uniform K-stability of $\mathbb{Q}$-Fano varieties.
In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields.
Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds
我们对光滑三次四重Fano簇的极化辛自同构进行了分类（配备Plücker极化），并研究了其不动轨迹.
Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties
推广Batyrev和Borisov的自反多面体和nef划分的概念，我们提出了一个镜像对称的Fano复曲面簇中的Calabi-Yau完全相交。
Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called Q-Fano threefolds. The classification of Q-Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.
On Fano varieties whose effective divisors are numerically eventually free
本文对具有最大Picard数且其有效因子在数值上最终自由的轻度奇异Fano簇进行了分类。
On Landau--Ginzburg models for Fano varieties
我们观察到一种方法找到弱朗道-金斯堡模型的法诺品种，并找到他们光滑法诺三倍的属9，10和12。
Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{φ_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.
This is a survey paper about a selection of recent results on the geometry of a special class of Fano varieties, which are called of K3 type. The focus is mostly Hodge-theoretical, with an eye towards the multiple connections between Fano varieties of K3 type and hyperkähler geometry.
We show that the Picard rank is constant in families of Fano varieties (in arbitrary characteristic) and we moreover investigate the constancy of the index.
We prove results about 1-cycles on certain Fano varieties using techniques that rely on rational curves. Firstly, we show that Fano weighted complete intersections with index bigger than half their dimension have trivial first Griffiths group. Secondly, we prove that the first Chow group of most $2$-Fano weighted complete intersections, and of $2$-Fano conic-connected varieties in $\mathbb{P}^n$ of high enough index (with $3$ obvious exceptions), are generated by lines. Furthermore, if the Fano variety of lines is irreducible, the first Chow group is isomorphic to $\mathbb{Z}$.
We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
Seshadri stratifications and standard monomial theory
本文提出嵌入射影簇上的 Seshadri 分层，用以构造 Newton-Okounkov 单纯复形与平坦环面退化，并为标准单项式理论及 Schubert 簇中 LS 路径的几何解释提供统一框架。
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