Manish Kumar
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 $k$.
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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 $q+1$.
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…
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.
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.
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.
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。
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。
We prove that every projective variety of dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and a chosen smooth point not on the divisor to some point not in H.
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…
Our aim in this paper is to compute the Poincaré series of the derivation module of the projective closure of certain affine monomial curves.
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 …
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.
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…
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
Motivated by amplitude calculations in string theory we establish basic properties of homotopy invariant iterated integrals on affine curves.
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。
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.
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.
I will give a new way to understand the Geometric Langlands Program.
Positivity of polynomials on the nonnegative part of certain affine hypersurfaces
研究多项式在特定仿射超曲面非负部分上的非负性，证明严格正多项式可由正系数多项式表示，推广Pólya定理。
We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H. This improves our earlier result in math.AG/0207150, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.
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.
We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwartz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwartz-Halphen's Theorem for Pham-Brieskorn surfaces.
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 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.
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.
Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of birational endomorphisms f defined by requiring that the missing curves or contracting curves of f are lines. The last part of the article is concerned with the monoid structure of the set of birational endomorphisms of A^2.
In this article we continue from \cite{sub2-1} the study of normal analytic compactifications of $\cc^2$ from the point of view of their associated pencils of jets of curve germs centered at infinity. If $\bar X$ is a normal analytic compactification of $\cc^2$ which is {\em primitive}, i.e.\ $\bar X \setminus \cc^2$ is irreducible curve, then we show that $\bar X$ is projective iff $\bar X$ is algebraic iff at least one of the jets in the associated pencil of jets of curve-germs can be represented by a planar curve with one place at infinity. As a result we show that there are primitive norma…
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.
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.
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 $K$.
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…
Let $π_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $π_1(C)$. Under the hypothesis that the quotient $π_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.
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…
We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study of affine schemes.
Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces
本文旨在发展正特征下高维变分上的射影和仿射结构理论，通过Berthelot的高阶微分算子描述此类结构，并推导出Chern类的正特征版本公式，进而利用射影结构对射影空间进行刻画。
Affine Deligne-Lusztig Varieties of Positive Coxeter Type
本文提出正 Coxeter 类型的仿射 Deligne-Lusztig 变种，揭示其简单几何结构，并推广了 He-Nie-Yu 的工作，应用于 Shimura 变种。
Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m_0 modulo n. We first show that the defining ideal of the monomial curve can be written as a sum of two determinantal ideals. Using this fact, we describe the minimal free resolution of the coordinate ring in the following three cases: when m_0 is 1 modulo n (determinantal), when m_0 is n modulo n (almo…
We provide criteria for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. These criteria reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem can be addressed using Cartan's moving frame method. This leads to a novel algorithmic solution of the projection problem for curves. The computational advantage of the algorithms presented here, in comparison to algorithms based on a straightforwar…
Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex for curves with at least two punctures, as well as a new proof of Matsumoto's theorem that the congruence kernel depe…
In this paper we present three related results on the subject of fields of parametrization. Let C be a rational curve over a field of characteristic zero. Let K be a field finitely generated over Q, such that it is a field of definition of C but not a field of parametrization. It is known that there are quadratic extensions of K that parametrize C. First, we prove that there are infinitely many quadratic extensions of K that are fields of parametrization of C. As a consequence, we prove that the witness variety, that appear in the context of the parametric Weil's descente method, is always a s…
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…
On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, general…
Normality of Schubert varieties in affine Grassmannians
本文分类了任意域上半单群仿射格拉斯曼流形中的正规Schubert簇，特别关注小正特征，通过切空间计算、Levi引理和极小退化分类给出初等证明。
Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A^{(1)}_1$ and the two birational maps on the …
Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of these configurations. The canonical functions on triples of flags that were defined by Fock and Goncharov in \cite{FG1} have a tropicalization that gives functions on triples of points in the affine Grassmannian. One expects that these functions, though of algebro-geometric origi…
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…
This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.
On the topology of fiber-type curves: a Zariski pair of affine nodal curves
本文研究光滑射影曲面中曲线补空间基本群为循环群自由积的条件，构造无穷多纤维型曲线Zariski对，含复仿射平面内仅含结点的曲线对，证明结点位置影响射影曲线嵌入拓扑，并用有限SU(2)表示的扭曲Alexander多项式区分同阿贝尔不变量的基本群。
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