Some Further Remarks on the Local Fundamental Group Scheme
Vikram Mehta
我们证明了局部基本群格式在特征p下满足Lefschetz-Bott定理，这些证明是Enriques-Severi -Zakki-Serre消灭定理和已知p-曲率事实的标准应用
We prove that the Local Fundamental Group Scheme satisfies the Lefschetz - Bott theorems in characteristic p. The proofs are standard applications of the Enriques-Severi -Zariski-Serre vanishing theorems and known facts about the p-curvature
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Some Remarks on the Local Fundamental Group Scheme
本文定义了由F-平凡向量丛定义的局部基本群方案，并给出了它基变的充要条件。
A few remarks about the Hilbert scheme of smooth projective curves
本文讨论了Francesco Severi和Joe Harris关于给定度和亏格的光滑射影曲线的Hilbert参数化格式的不可约性和维数的两个定理。
We present an elementary proof of the fact that every torsor for an affine group scheme over an algebraically closed field is trivial. This is related to the uniqueness of fibre functors on neutral tannakian categories.
Two small remarks on Nori fundamental group scheme
对于特征为$p>0$的完满域上的完备的、约化的、几何连通的$X$格式，本文分析了Nori的基本群格式的局部分解和非局部分解，并提出了群格式的分裂与几何的关系问题.我们还描述了范畴的函子对应的最大约化子群计划的列入。
A remark on isolated symplectic singularities with trivial local fundamental group
本文提出了一种简单方法构造任意维度的孤立辛奇点，其局部基本群为平凡群，回应了Beauville的问题。
In this paper, we firstly point out, by a counter example, that Proposition 6.4 of Section 6 in Bump's book (Algebraic Geometry) is error, and then give a correct statement with proof. We finally point out a gap in the proof of Theorem 3, in Chapter I Section 8, of Mumford's red book, and indicate a way to complete it.
Remark on fundamental groups and effective Diophantine methods for hyperbolic curves
我们讨论了如何非交换的基本群最终可能有助于算法找到双曲曲线上的有理点。
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
A remark on some punctual Quot schemes on smooth projective curves
本文研究光滑射影曲线上局部自由层的点状 Quot 概形，证明其除具有正规性和典范 Gorenstein 奇点外，还是 Picard 数为一的 $\mathbb{Q}$-factorial Fano 簇。
In this article, we study the log-scheme theoretic version of the Nori fundamental group scheme. Similar to the classical Nori fundamental group scheme, the log Nori fundamental group scheme classifies torsors on log flat topology. We also compute the nilpotent pro-p quotient of the log Nori fundamental group scheme with log at $\{0, 1, \infty\}$.
Some remarks on the universal cover of an open K3 surface
本文以最佳形式给出了正规K ~ 3曲面光滑轨迹的基本群有限的一个充分数值条件。我们还将证明，如果正常的K3曲面是椭圆的，上述基本群不是有限的，那么有一个有限覆盖，这是一个复杂的环面。
The Fundamental Group Scheme of a non Reduced Scheme
本文将基本群概型的定义推广到任意连通Dedekind概型上的非约化概形。然后比较了仿射格式的基本群格式和它的约化部分的基本群格式。
The tame fundamental group schemes of curves in positive characteristic
本文研究正特征代数闭域上光滑曲线的驯服基本群概形，发展余特化映射理论，证明其依赖曲线且可重构数值不变量。
Remarks on logarithmic étale sheafification
本文研究对数概形上预层的对数平展层化问题，给出预层成为全对数平展层的判定准则，明确结构层与对数结构为对数平展层的情形，并证明满足对数平展下降的对数概形上对数皮卡群是全对数平展拓扑中的栈。
Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective $0$-cycles of length $n$ on $X$. The aim of the present article is to find the $S$-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme $\mathcal{H}{\it ilb}_X^n$.
A Remark on Higher Homotopy Sheaves of Derived Arc Spaces
本文探讨导出弧空间在光滑方案与局部完整交方案中的同调sheaf特性，指出其与经典几何空间无异。
A remark on the full support property
该研究证明域上射影概型数值预稳定条件的质量-Hom界蕴含全数值Grothendieck群下的支撑性质，结合稳定条件构造得到域上所有射影概型的全支撑稳定条件，还建立相对统一版本并确定稳定流形的典范分支。
In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic cur…
Notes On Fundamental Groupoid Schemes
本文研究了对应于特定向量丛Tannakian范畴的基本群胚概形，计算了各向异性圆锥、Klein瓶和阿贝尔簇的基本群胚概形，并通过表示探讨了它们之间的关系。
The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme
定义了任意（非空）基概型上约化群概型的代数基本群函子，并证明了该函子是正合的。
Let $k$ be an algebraically closed field. Let $C$ be an irreducible smooth projective curve over $k$. Let $E$ be a locally free sheaf on $C$ of rank $\geq 2$. Fix an integer $d \geq 2$. Let $\mathcal{Q}$ denote the Quot scheme parameterizing torsion quotients of $E$ of degree $d$. In this article we compute the $S$-fundamental group scheme of $\mathcal{Q}$.
Birational invariance of the $S$-fundamental group scheme
设$X$和$Y$是正特征代数闭域$k$上的非奇异投射簇.如果$X$和$Y$是双有理的，我们证明了它们的$S$ -基本群格式是同构的.
In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental group-scheme, and of finite group-schemes in general to prove that the kernel of such a fibration is finite, and that the homotopy exact sequence holds in this case. As an application, we describe the fundamental group-schemes of certain proper varieties that are connected by curves or are related to them.
On the fundamental group scheme of rationally chain connected varieties
设$k$是一个代数闭域. Chambert-Loir证明了$k$上正规有理链连通簇的顶点基本群是有限的.本文证明了$k$上正规有理链连通簇的基本群概型是有限的和无尾的.特别地，Fano簇的基本群概型是有限的且不稳定的。
Weak Density of the Fundamental Group Scheme
关于代数闭域上的$X$投射光滑，证明了若Nori的基本群概型是平凡的，则不存在非平凡的0次Nori半稳定丛，即Langer特别研究的群概型$π^S(X)$是平凡的.
On the S-fundamental group scheme II
S-基本群格式是对应于数值平坦向量丛的Tannaka范畴的群格式。本文利用行列式线丛证明了两个完全簇的乘积的S-基本群是它们的S-基本群的乘积。我们还计算了交换簇或具有平凡根基本群的簇的S-基本群概型和S-基本群概型的交换部分。
W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport-Zink space. In the minuscule case, Rapoport-Terstiege-Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.
Finite group schemes as fundamental group schemes of smooth projective varieties
本文研究将有限群概形实现为光滑射影簇基本群概形的问题，通过贝尔蒂尼型结果、戈多-塞雷构造等方法，证明在无限完美域k上任意有限群概形G/k均可作为对应类型的基本群概形实现于特定维数的连通光滑射影簇。
Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $A$ be an abelian surface over $k$. Fix an integer $n \geq 1$ such that $p \nmid n$ and let $K^{[n]}$ be the $n$-th Generalized Kummer Variety associated to $A$. In this article we aim to find the $S$-fundamental group scheme and Nori's fundamental group scheme of $K^{[n]}$.
A Simple Application of Enriques-Severi-Zariski-Serre to the Local Fundamental Group Scheme
证明了局部基本群格式满足Lefschetz-Bott-Grothendieck条件。
A smooth counterexample to Nori's conjecture on the fundamental group scheme
我们证明了Nori的基本群方案π（X，x）在特征为2的域上定义的亏格为2的光滑射影普通曲线X的基域扩张下不正确地基变换.
Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.
In this expository paper we present a proof of the equivalence of the standard definition of descent data on schemes with another one mentioned in the literature that involves certain cartesian diagrams. Using this equivalence, we discuss the Galois descent of both schemes and morphisms of schemes.
We follow the pattern in a recent paper of Otabe [Ota15] to define an action of the étale fundamental group scheme $π^\text{et}(X)$ on the local component of the essentially finite fundamental group scheme $π^{\mathrm{EF}}(X)$ of Nori. We show that the associated representation is faithful when $X$ is a curve of genus $\geq 2$.
Let $(X ,x_0)$ be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for $(X ,x_0)$ produces a homomorphism from the abelianization of the $F$-divided fundamental group scheme of $X$ to the $F$-divided fundamental group of the Albanese variety of $X$. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.
Desingulization of Quasi-smooth Derived Schemes and Virtual Fundamental Classes
证明拟光滑导出概形的去奇异化定理，并提出其虚拟基本类K理论积分与光滑概形上完美复形楔幂积分等价的猜想。
On an extension of Nori and local fundamental group schemes
本文探讨了Abelian簇的扩展Nori和局部基本群方案，分析了其birational不变性及在Albanese和étale映射下的行为。
On the S-fundamental group scheme
本文对定义在正特征的代数闭域上的簇引入了一个新的基本群方案，并利用它研究了C. Simpson的结果为阳性特征。我们还研究了这个群的性质，并证明了Lefschetz型定理。
Comparison of fundamental group schemes of a projective variety and an ample hypersurface
设$X$是定义在代数闭域上的光滑射影簇，$L$是$X$上的充裕线丛.证明了对完备线性系统$| L^{\otimes d}|$中$X$上的任意光滑超曲面$D$，只要$d$足够大且$\dim X \geq 3$，包含映射$D\hookrightarrow X$诱导基本群方案同构.如果$\dim X = 2$和$d$足够大，则基本群格式的诱导同态保持满射。我们给出一个例子，说明由光滑射影曲面中约化的充分曲线的包含映射导出的基本群格式的同态一般不是满射的。
Some remarks on biequidimensionality of topological spaces and Noetherian schemes
有很多例子可以证明维数和余维数的行为有些违反直觉。在EGA中指出，拓扑空间是等维、等余维和悬链线的当且仅当每个不可约闭子集的极大链具有相同的长度。我们构造的例子表明，这是不正确的，甚至频谱的诺特环。这就产生了两个概念的biequidimensionality，我们展示了这些涉及到的尺寸公式和存在的余维函数。
Loops on schemes and the algebraic fundamental group
本文通过引入区间概形重新解释真概形的代数基本群，定义代数环路与同伦关系，并利用麦考利化和勒夫谢茨型结果建立主要结论。
On the bumpy fundamental group scheme
在这篇短文中，我们首先回顾了在已知的两种情形下：定义在域上和定义在Dedekind概型上的概型$X$的基本群概型的定义和构造。它对$X$上的所有有限（或准有限）fpqc torsor进行分类。当$X$被定义在任意维数的诺特正则模式$S$上时，我们不知道这样的对象是否可以被构造。这就是为什么我们引入一个新的类别，包含fpqc torsors，其对象是新拓扑的torsors。我们证明了这个新的类别是cofiltered从而产生一个基本群计划在$S$，说\textit{bumpy}，因为它可能不是平坦的一般。我们证明了当$S$是Dedekind格式时，它是平坦的，从而与\textit{classical}格式一致.
On Nori's Fundamental Group Scheme
我们确定了Nori的基本群方案到其顶点和局部的同态的核的表示范畴--商范畴。Pierre Deligne指出了本文第一版中的一个错误。我们深深地感谢他，特别是他给我们的启示的例子复制在备注2.4 2）。
Higgs bundles and fundamental group schemes
本文利用数值有效性概念证明数值平坦希格斯向量丛构成Tannaka范畴，引入希格斯基本群概形，并将其性质与陈类消失猜想联系起来。
In this note we present a notion of fundamental scheme for Cohen- Macaulay, order 1, irreducible congruences of lines. We show that such a congruence is formed by the k-secant lines to its fundamental scheme for a number k that we call the secant index of the congruence. If the fundamental scheme X is a smooth connected variety in PN, then k = (N-1)/(c-1) (where c is the codimension of X) and X comes equipped with a special tangency divisor cut out by a virtual hypersurface of degree k-2 (to be precise, linearly equivalent to a section by an hypersurface of degree (k-2) without being cut by on…
We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697). We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas fo…
On base change of the fundamental group scheme
我们提供了所有素数$p$的例子，光滑的射影曲线在一个领域的特征$p$的基础变化的基本群计划失败。这与$F$ -平凡向量丛（即被弗罗贝尼乌斯态射的幂平凡化的丛）在（平凡）族中的行为密切相关。最后，我们研究的行为$F$ -平凡（不一定平凡）的家庭。
On the fundamental group schemes of certain quotient varieties
本文在$F$-divided和本质有限基本群概形的设定下，证明了Armstrong关于商空间基本群定理的类似结果。
Fundamental group schemes of globally F-regular varieties
该研究证明正特征代数闭域上连通正规整体F-正则射影簇的连通拟挠子覆盖塔稳定，其正则轨迹的Nori基本群概型有限且线性约化，有限性是Xu和Braun相关定理的正特征类比。
Remarks on cycle classes of sections of the arithmetic fundamental group
给定域k上的光滑可分离K（pi，1）簇X，我们将一个具有紧支集的环上同调的“圈类”与从X的算术基本群到k的绝对伽罗瓦群的自然映射的任何连续截面相关联.我们讨论了这类曲线在p-adic域上的代数性，并特别推出了Stix定理的一个新证明，根据该定理，只要从X的算术基本群到k的绝对伽罗瓦群的自然映射允许一个截面，则p-adic域k上的曲线X的指数必须是p的幂。最后，贝林森的拓扑基本群的幂零完备化的几何化的一个Etale适应允许我们在合适的上同调群中提升这个循环类。
Relations for virtual fundamental classes of Hilbert schemes of curves on surfaces
在[DKO]中，我们构造了曲面V上拓扑型m因子的希尔伯特方案的虚拟基本类$[[ Hilb^m_V ]]$，并使用这些类定义V的庞加莱不变量：（P^+_V，P^-_V）：H^2（V，Z）--> Λ^* H^1（V，Z）x Λ^* H^1（V，Z）Z）我们推测这个不变量与关于正则方向数据计算的完整的Seiberg-Witten不变量一致。本文证明了积分曲线$C \subset V$的存在性导致了某些虚基本类$[[Hilb^m_V ]]$之间的关系。Poincare不变量的对应关系可以看作是[OS]中基本关系的代数类比。
In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null locus. We also show a generalized Fujita approximation for the arithmetic restricted volumes.
Vector bundles trivialized by proper morphisms and the fundamental group scheme
设X是定义在代数闭域k上的光滑射影簇。Nori在X上构造了一个向量丛范畴，称为本质有限向量丛，这让人想起基本群（特征为零）的表示范畴。事实上，这个范畴等价于控制所有有限扭子的亲有限群概型的表示范畴。我们证明了本质有限的向量丛与那些在被某些真满射拉回到X之后变得平凡的向量丛是一致的。
Some Remarks on Beilinson Adeles
本文给出了两个关于在概型的De Rham复形中取值的Beilinson三角形的注记。第一个是一个解释，在adeles方面，分解的德拉姆复杂的计划定义模$p^{2}$（结果Deligne-Illusie）。第二个注记是关于Adele分解与Hodge分解之间的可能关系。我们要找一个反例。
Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_η$. Let $f: Y \rightarrow X_η$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_η$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $ψ^{(p')}: π(X_η,x_η)^{(…
Vector bundles trivialized by proper morphisms and the fundamental group scheme, II
设$X$是代数闭域$k$上的投射光滑簇.设$f:Y\rightarrow X$是$k$ -簇的真满射.假设$f$是可分的，我们证明了与$X$上的向量丛$E$相关联的Tannakian范畴，使得$f^*E$是平凡的，等价于一个有限的代数群概型的表示范畴.在不存在可分性的情况下，我们给出了这个结论的一个反例。
Let $S$ be a connected Dedekind scheme and $X$ be a proper smooth connected scheme over $S$ . Let $D$ a divisor with no multiplicity of $X$ such that the irreducible components of $D$ and as well their intersections are smooth over $S$. Now if we endow $X$ with the log structure associated with $D$ then the structure morphism from $X$ to $S$ is log-smooth. Let $x: S \to X$ be a $S$-point such that it doesn't intersect $D$. Then we prove that the maximal abelian quotient of the log Nori fundamental group scheme of $X$ fits in to an exact sequence of the form $0 \rightarrow (\mathbf{NS}^τ_{X/S,D…
Let $X$ be any scheme defined over a Dedekind scheme $S$ with a given section $x\in X(S)$. We prove the existence of a pro-finite $S$-group scheme $\aleph(X,x)$ and a universal $\aleph(X,x)$-torsor dominating all the pro-finite pointed torsors over $X$. Though $\aleph(X,x)$ may not be unique in general it still can provide useful information in order to better understand $X$. In a similar way we prove the existence of a pro-algebraic $S$-group scheme $\aleph^{\rm alg}(X,x)$ and a $\aleph^{\rm alg}(X,x)$-torsor dominating all the pro-algebraic and affine pointed torsors over $X$. The case where…
Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the latter, we will give an explicit way of calculating the fundamental group using methods from 2-category theory. Indeed, we will calculate the more general fundamental groupoid. As a specialisation, we will also look at the Stanley Reisner Rings. Our method simplifies in this case, allowing us to describe the fundamental groupoid in terms of the s…
The Künneth Formula Of Fundamental Group Schemes
针对proper态射，给出Tannakian基本群概形同伦序列正合的统一判据，进而得到两个proper连通概形乘积的基本群概形Künneth公式的等价条件，并应用于多种基本群概形。
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