Jakob Stix
An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π^{tame} of curves over a finitely generated field and its effect on the set of isomorphisms. Universal homeomorphisms are formally inverted.
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In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields o…
In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.
The Absolute Anabelian Geometry of Virtual Curves of Arbitrary Genus
本文研究虚曲线的绝对单形几何，扩展了之前关于亏格零情况的基底群理论拉回结果，引入了CAVC型包含和虚拟双螺旋范畴概念，建立虚拟曲线几何性的判据。
In this paper we prove a refined version of Uchida's theorem on isomorphisms between absolute Galois groups of global fields in positive characteristics, where one "ignores" the information provided by a "small" set of primes.
Resolution of non-singularities and the absolute anabelian conjecture
本文针对p进局部域上双曲曲线的绝对远阿贝尔猜想，将S. Mochizuki针对准Belyi型曲线的证明结果推广到包含双曲Mumford曲线在内的更广泛曲线族。
In this paper we prove a refined version of a theorem by Tamagawa and Mochizuki on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one "ignores" the information provided by a "small" set of primes.
Anabelian properties of tame fundamental groups of singular curves
本文利用 Das 的定理和已知 anabelian 结果，在特定条件下确定了多种域上奇异曲线正规化的同构类，从而推进了奇异曲线 anabelian 性质的研究。
Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves over number fields can be reduced to the case of elliptic curves. In this case we prove that, as a consequence of a result of Stoll, a section of the exact sequence of the absolute Galois group of an elliptic curve over a number field arises from a rational point if and only if the induced section of the corresponding (geometrically abelianised) arithmetic fun…
In this survey, we introduce the three theorems about the m-step solvable Grothendieck conjecture in anabelian geometry of hyperbolic curves by H. Nakamura, S. Mochizuki, and the author. We also give sketches of the proofs of these theorems.
We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arise from embeddings between the function fields.
In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $π_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary p…
Etale descent obstruction and anabelian geometry of curves over finite fields
本文通过建立平展基本群之间的态射与全局函数域上曲线的局部常值阿代尔点之间的双射，将Grothendieck的阿纳贝尔猜想与Sutherland和第二个作者的最新猜想联系起来，为阿纳贝尔猜想提供了进一步证据。
Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whethe…
Topological rigidity of maps in positive characteristic and anabelian geometry
研究正特征域上有限型整概形间的非常值映射对，通过添加纯拓扑判据推广玉川关于映射相差Frobenius幂次的等价性结果。
In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by André, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language o…
We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suff…
Anabelian properties of infinite algebraic extensions of finite fields
本文研究有限域无限代数扩张上的远阿贝尔几何，将 Tamagawa 与 Sarashina 关于双曲曲线的结果推广至此类域，并用绝对 Galois 群刻画其（环向）Kummer-faithful 的充要条件。
The geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields
本文在非交换几何中针对素域上有限生成域上的仿射双曲曲线，通过先证有限域情形、再证Oda-Tamagawa准则的几何m步可解版本，最终证明了（弱与强双非交换的）几何m步可解Grothendieck猜想。
Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves
本文证明了定义在sub-$p$-adic域上的曲线模栈的有限连通étale覆盖满足Grothendieck猜想意义下的“几乎”anabelian性质，建立了其自同构群与几何基本群外部自同构子群之间的自然同构，将双曲曲线的anabelian性质部分扩展至曲线模空间。
A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic
本文针对1维情形，证明了代数闭域上曲线的函数域可由其绝对Galois群与仅固定基域的更大域自同构群构成的群对恢复，推进了Bogomolov纲领。
We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.
We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.
On the infinite Frobenius action on de Rham fundamental groups of affine curves
本文研究仿射曲线de Rham基本群上的无穷Frobenius作用，计算相关实混合Hodge结构的扩张类，并在模曲线情形下将其与Rankin-Selberg L-函数特殊值联系，证明扩张非分裂。
Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.
Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve
本文证明了在仿射曲线与投影曲线乘积上，关于有理数闭包上内射映射的动力学Mordell-Lang猜想。
I will give a new way to understand the Geometric Langlands Program.
In this series of papers, we investigate a new anabelian phenomenon of curves over algebraically closed fields of positive characteristic. Let $\overline M_{g, n}$ be the moduli space of curves of type $(g, n)$ over $\overline {\mathbb{F}}_p$. We introduce a topological space $\overline Π_{g, n}$ which can be determined group-theoretically from admissible fundamental groups of pointed stable curves of type $(g, n)$. By introducing a certain equivalence relation $\sim_{fe}$ on the underlying topological space $|\overline M_{g, n}|$ of $\overline M_{g, n}$, we obtain a topological space $\overli…
It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then for some $n\gg 1$ there exists a counterexample $F\in \mathbb{Z}[X]^n$ of the form $F_i(X)=X_i+ (a_{i1}X_1+\dots+a_{in}X_n)^{d_i}$, $a_{ij}\in \Z$, $d_i=2;3 $, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\dots=F_n(X)$ has no non-zero integer points.
Using the approach of Barkatou and El Kaoui, we classify certain affine curves over discrete valuation rings having a free additive group action. Our classification generalizes results of Miyanishi in equi-characteristic 0.
We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.
We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.
Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$ passing through all the $\mathbb{F}_q$-points of the plane admita representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their degenerations by means of the matrix representation, and also discuss the similar problem for the affine-plane filling curves of degree $q+1$.
It is shown that the commutator subgroup of the fundamental group of a smooth affine curve over an uncountable algebraically closed field $k$ of positive characteristic is a profinite free group of rank equal to the cardinality of $k$.
On smooth affine surfaces with the cohomology of a smooth projective curve
本文针对具有正亏格光滑射影曲线有理混合霍奇结构的光滑复仿射曲面，证明其存在到该曲线的光滑态射，纤维为仿射直线，是Jouanolou技巧在二维的逆命题。
Detecting affine equivalences between certain types of parametric curves, in any dimension
本文提出一种计算任意维度特定类型参数曲线间仿射等价性的算法，适用于有理曲线及含亚纯分量且有理逆的曲线，通过二元分解避免多项式系统求解，并在Maple中实现。
We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many non-equivalent embeddings of the affine line in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfa…
Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by Lê for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case…
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
We consider log deformations of affine surfaces with fibrations by the affine lines. Such a fibration is of affine type (resp. of complete type) if the base curve of the fibration is an affine curve (resp. a complete curve). The case of affine type is mainly treated in the article.
Motivated by amplitude calculations in string theory we establish basic properties of homotopy invariant iterated integrals on affine curves.
Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's étale fundamental group of $U$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme, and give a partial answer to it.
For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.
This paper has been withdrawn by the auther due to an error
We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine monomial curve. We also describe the irregularity complex of such a system with respect to its singular support. We use restriction and some results in D-module theory to reduce our study to the two dimensional case.
It is proved in this paper that a locally complete intersection curve in a smooth affine C-algebra with trival conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck Group is torsion.
Purity of generalized affine Springer fibers from generic planar curve singularities
本文通过构造显式仿射铺砌，证明了广义不可约平面曲线奇点上点状Hilbert概型的上同调纯性，由此确立了广义$GL_N$-仿射Springer纤维纯性猜想的一个新情形，并推广了关于紧化Jacobian的组合学结果。
For an affine double plane defined by an equation of the form z^2 = f, we study the divisor class group and the Brauer group. Two cases are considered. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface P^1 x C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f is the equation of an affine hyperelliptic curve. On the open set where the cover is unramified, we compute the groups of divisor classes, the Brauer groups, the relative Brauer group, as well as all of the terms in he exact sequence of Chase, Harrison…
Equi-affine minimal-degree moving frames for polynomial curves
本研究提出仿射空间多项式曲线的极小次数等变活动标架构建理论与算法，给出将任意矩阵补全算法转化为等变活动标架算法的等变化流程并证明次数极小性，同时提出新型极小次数矩阵补全算法，补充相关领域研究。
We provide the existence, for every complex rational smooth affine curve $Γ$, of a linear action of $\mathrm{Aut}(Γ)$ on the affine 3-dimensional space $\mathbb{A}^3$, together with a $\mathrm{Aut}(Γ)$-equivariant closed embedding of $Γ$ into $\mathbb{A}^3$. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.
This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^2$, over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automor…
The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of cha…
Let k be an algebraically closed field of positive characteristic. The goal of this paper is to characterize the proper smooth curves X/k of positive genus g equipped with a k-rational point P such that X¶can be realized as an etale cover of the affine line. A first elementary analysis shows that such a cover exists if and only if there exists an exact regular differential form on X having a unique zero at P (of order 2g-2). The main inconvenient of this approach is that it doesn't give any control on the degree of the cover. A finer investigation, essentially based on the duality between the …
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $π_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $π_1(C)$. In "Normal subgroups of fundamental groups of affine curves in positive characteristic" we proved the same result under the additional hypothesis that $k$ had countable cardinality.
A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$ covers of affine curves gives a moduli space for $\mathsf{p}'$-by-$\mathsf{p}$ covers of an affine curve. A local moduli space is also constructed, for cyclic-by-$\mathsf{p}$ covers of $Spec(k((x)))$, which is the same as the global moduli space for cyclic-by-$\mathsf{p}$ covers of $\mathbb{P}^1-\{0\}$ tamely ramified over $\infty$ with the same Galois grou…
We define the affine stratification number asn X of a scheme X. For X equidimensional, it is the minimal number k such that there is a stratification of X by locally closed affine subschemes of codimension at most k. We show that the affine stratification number is well-behaved, and bounds many aspects of the topological complexity of the scheme, such as vanishing of cohomology groups of quasicoherent, constructible, and l-adic sheaves. We explain how to bound asn X in practice. We give a series of conjectures (the first by E. Looijenga) bounding the affine stratification number of various mod…
We prove that the bracket width of the simple Lie algebra of vector fields $\rm{Vec}(C)$ of a smooth irreducible affine curve $C$ with a trivial tangent sheaf is at most three. In addition, if $C$ is a plane curve, the bracket width of $\rm{Vec}(C)$ is at most two and if moreover $C$ has a unique place at infinity, the bracket width of $\rm{Vec}(C)$ is exactly two. We also show that in case $C$ is rational, the width of $\rm{Vec}(C)$ equals one.
We give criteria for the Jacobian of a singular curve $X$ with at most ordinary $n$-point singularities to be anti-affine. In particular, for the case of curves with single ordinary double point we exhibit a relation with torsion divisors. If the geometric genus of the singular curve is atleast 3 and the normalization is non-hyperelliptic and non-bielliptic, then except for finitely many cases the Jacobian of $X$ is anti-affine. Furthermore, if the normalization is a general curve of genus atleast 3 then the Jacobian of $X$ is always anti-affine.
Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration γ: R \to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of γinto Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define γin the framework of the Grassmannian description of R due to Cannings and Holland.
Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmülle…
Let $C$ be a complex affine reduced curve, and denote by $H^1(C)$ its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant $μ'(C,x)$ that measures the complexity of the singularity of $C$ at the point $x$. Then, if $H_1(C)$ denotes the first singular homology group of $C$ with complex coefficients, we establish the following formula: $$ dim H^1(C)=dim H_1(C) + \sum_{x\in C} μ'(C,x) $$ Second we consider a family of curves given by the fibres of a dominant morphism $f:X\to \mathbb{C}$, where $X$ is a…
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