On submaximal plane curves
Joaquim Roé
我们证明了次极大平面曲线（即，Nagata猜想的一个不可约反例）具有r个奇点，具有重数序列（m，n，.，n），其中m<sn，对于每个整数，（（s-1）（s+2））^2 > 6.76（r-1）。
We prove that a submaximal plane curve (i.e., an irreducible counterexample to Nagata's conjecture) with r singular points has sequence of multiplicities (m, n, ..., n) with m<sn for every integer with ((s-1)(s+2))^2 > 6.76(r-1).
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The main purpose of the note is to exclude the existence of certain submaximal curves in fake projective planes. This will lead to lower bounds on multipoint Seshadri constants of `fake' $\mathcal{O}(1)$ on fake projective planes.
On the 7th order ODE with submaximal symmetry
我们找到了唯一的七阶常微分方程的一般解，该方程允许十维接触对称群。该常微分方程的积分曲线是$\PP^3$中的有理接触曲线，它产生六次有理平面曲线。这些曲线的模空间是齐性空间$Sp(4)/SL(2)$的实形式。
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 $q+1$.
Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics
简单六次曲线是一个简化的复杂的射影平面曲线的次数为6，只有简单的奇点。本文对沿单六次线分支的射影平面的双重覆盖引入Z-分裂曲线的概念，并利用Z-分裂曲线研究了单六次线的格Zaprikik-ples。我们定义了格型的特化，并将度小于或等于3的Z-分裂曲线的所有格型分类为特化。
Enumerative geometry of plane curves of low genus
我们收集了各种已知的结果（关于平面曲线和稳定映射的模空间），导出了新的递归公式，列举了具有各种行为的任何次数的低亏格平面曲线。给出了有理平面曲线、椭圆平面曲线和具有固定复结构的椭圆平面曲线示性数的递推公式。还给出了具有各种“余维1”行为（尖点、三角点、三尖点等）的椭圆（和有理）平面曲线的数目的递归。以及塞维里变种的几何和算术部分属）。对于亏格2和亏格3的平面曲线，我们也计算后者的数。我们依靠Caporaso，Diaz，Getzler，Harris，Ran的结果，特别是Pandharipande。
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
The splitting number of a plane irreducible curve for a Galois cover is effective to distinguish the embedded topologies of plane curves. In this paper, we define a connected number of any plane curve for a Galois cover whose branch divisor has no common components with the plane curve, which is similar to the splitting number. We classify the embedded topology of Artal arrangements of degree $b\geq 4$ by the connected number, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of dgree $b$ and three total inflectional tangents.
Simultaneous Galois points for a reducible plane curve consisting of nonsingular components
本文将伽罗瓦点概念推广至可约平面曲线，定义同时伽罗瓦点，并研究非奇异分量约化曲线的此类点的数量。
We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. Then we classify parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.
We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.
Hyperelliptic plane curves of type (d,d-2)
在前一篇文章中，我们对（d，d-2）型有理平面曲线进行了分类和构造。本文将这些结果推广到亏格为正的（d，d-2）型不可约平面曲线。
Microlocal Versal Deformations of Plane Curves
本文引入了平面曲线微局部非线性变形的概念。我们构造了Legendrian曲线的等奇异的变形，这些变形是半拟齐次分支的余法线。
It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2+q+1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2+q+1$ and the Ballico-Hefez curve.
We show that the resolution graph of a plane curve singularity admits a canonical decomposition into elementary graphs.
We prove that if two plane curve singularities are equisingular, then they are topologically equivalent. The method we will use is P.~Fortuny~Ayuso's who proved this result for irreducible plane curve singularities.
Automorphism groups of smooth plane curves
确定了至少四次光滑平面曲线的自同构群的结构。此外，还给出了光滑平面曲线自同构群阶的上界，并对自同构群较大的情况进行了分类。本文还简单地证明了具有极大序模全自同构群的光滑平面曲线的唯一性。
Recovering plane curves of low degree from their inflection lines and inflection points
本文研究了由曲线的拐点恢复至少三次光滑的平面曲线的问题。我们肯定地回答所提出的问题，一般光滑的平面四次曲线，使额外的假设，也有一个拐点，并为任何光滑的平面三次曲线。
Enumeration of rational plane curves tangent to a smooth cubic
本文利用扭曲稳定映射计算了与光滑平面三次曲面有预定接触的有理d次平面曲线的个数。
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
In the projective plane PG(2,q) over a finite field of order q, a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree q+2 containing all points of PG(2,q). Such curves were investigated by G. Tallini in 1961, and by Homma and Kim in 2013. Our results concern the automorphism groups, the Weierstrass semigroups, the Hasse-Witt invariants, and quotient curves of the Tallini curves.
In the study of hyperelliptic curve cryptography, presentations of semi-reduced divisors on a hyperelliptic curve play important roles. In this note, we give an interpretation for such presentations from view points of Gröbner bases. As applications, (i) we give a method to construct weak $n$-contact curves and (ii) we study plane curves satisfying quasi-toric relations of type $(2, n, 2)$
We study the relation between the type of a double point of a plane curve and the curvilinear 0-dimensional subschemes of the curve at the point. An Algorithm related to a classical procedure for the study of double points via osculating curves is described and proved. Eventually we look for a way to create examples of rational plane curves with given singularities $A_s$.
It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\geq 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\leq 7+1/9$.
On the genus of a cyclic plane curve over a finite field
循环曲线，即由循环直射变换群固定的曲线，在研究狄氏射影平面中的循环弧中起着中心作用。本文计算了由Singer型循环k-弧生成的循环曲线的亏格。
Noncommutative plane curves
本文研究非交换平面曲线，即一维单模构成一条平面曲线的非交换k-代数。我们研究简单模的扩张，并试图启发完备化问题，即理解不同维数的简单模之间的联系。
Braid monodromy and Alexander polynomials of real plane curves
本文研究实平面曲线（包括无实点的曲线）的辫子单值性分解的对称性，并证明其Alexander不变量的新整除关系。
Let $K$ be a number field and $O_K$ the ring of integers of $K$. In the spirit of Siegel's theorem on integral points on affine algebraic curves, the plane Jacobian conjecture over $K$ is equivalent to the following statement: if $P,Q\in O_K[x,y]$ and $P_xQ_y-P_yQ_x\equiv 1$, then the curve $P=0$ has more than one integral point.
In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.
A Compactness Criterion for Real Plane Algebraic Curves
利用定义多项式的牛顿多边形，给出实平面代数曲线紧的两组条件，一个充分条件，一个必要条件。
The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail.
On the Saito basis for plane curves
本文研究解析平面曲线的Saito模与挠子模，提出计算方法，并用于计算重数不超过三的曲线的解析不变量。
Drawing real plane algebraic curves in OSCAR
本文展示如何利用OSCAR计算机代数系统生成拓扑正确或视觉美观的实平面代数曲线图示。
We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.
Integration over Tropical Plane Curves and Ultradiscretization
本文从超离散化的角度研究热带平面曲线上的全纯积分。证明了热带曲线上的格积分可以作为黎曼曲面上的复积分的一个极限而得到。
Enumerative geometry of hyperelliptic plane curves
我们递归计算的Gromov-Witten不变量的希尔伯特计划的两个点在平面上。通过对稳定映射空间的研究和虚贡献的计算，我们利用这些不变量来枚举通过3d+1个一般点的亏格为g的d次超椭圆平面曲线.
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically equivalent.
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A'Campo and Gusein-Zade.
On the addition technique for Betti and Poincaré polynomials of plane curves
本文利用加法技术推导出平面曲线的Betti和Poincaré多项式多项式恒等式。
We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.
Topology of plane algebraic curves: the algebraic approach
本文讨论了直纹曲面中平面六次曲线和三角曲线的分类和拓扑问题。
This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.
The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve
一条d次射影非奇异平面代数曲线称为极大对称曲线，如果它达到d次复非奇异平面代数曲线的自同构群的最大阶.对于d<=7，所有这些曲线都是已知的。直到射影，它们是d= 5，7的费马曲线，见\cite{kmp1,kmp2}，d=4的克莱因四次曲线，见\cite{har}，d=6的威曼六次曲线，见\cite{doi}。本文研究定义在特征为零的代数闭域上的射影平面曲线，并将这一结果推广到任意d>=8，证明了Fermat曲线是唯一的d次极大对称非奇异曲线，且d>=8，直到射影性。对于d= 11，13，17，19，费马曲线的这种特征已经得到，见\cite{kmp2}。
Defect of irreducible plane curves with simple singularities
本研究围绕Dimca提出的平面曲线亏量展开，推导了含结点、普通尖点与普通三重点的不可约平面曲线的亏量下界，证明高Arnold指数的约化简单奇异平面曲线不可能是自由曲线。
Limits of PGL(3)-translates of plane curves, II
每个复平面曲线C确定3 × 3矩阵的$P^8$的一个子模式S，其射影法锥（PNC）捕获C的微妙不变量。在“PGL（3）-translates of平面曲线，I的极限”中，我们得到了PNC的集合论描述，从而确定了一般元素同构于C的平面曲线族的所有可能极限。本文的主要结果是确定PNC作为一个周期，这是一个重要的组成部分，在我们的计算“线性轨道的任意平面曲线“的程度的PGL（3）-轨道封闭的任意平面曲线，不变量的自然枚举意义。
Let $C$ be a generic complex plane plane curve with a given Newton polygon $P$. We compute the number of its inflection points and bitangents (equivalently, the number of singularities of the projectively dual curve $C^\vee$). We also prove that $C^\vee$ has no singularities other than nodes and cusps for large enough polygons $P$.
The projective characterization of elliptic plane curves which have one place at infinity
本文考虑在无穷远处有一个位置的光滑仿射椭圆平面曲线。我们将它们与只有一个尖点作为奇点且只在尖点处与无穷远无穷远直线相交的椭圆射影平面曲线等同起来。我们通过严格变换的自相交数来刻画这类曲线，并通过其尖点的最小嵌入分辨来刻画这类曲线。并证明了它们的自交数是所有只有一个尖点的椭圆平面曲线的自交数中的最大值。
Fibrations by plane projective rational quartic curves in characteristic two
本文在特征二下对平面射影有理四次曲线纤维化进行了完整的双有理等价分类。
This note is devoted, after the result of Harui, arXiv:1306.5842, to solve some natural questions for non-singular plane curves of degree $d$ over an algebraically closed field $K$ of zero characteristic.
We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, \mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, \mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.
We construct a well-behaved compactification of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in $\overline{M}_6$ of the locus of smooth plane quintic curves.
Linear systems associated to unicuspidal rational plane curves
射影平面上的曲线C称为非负的，如果C的奇点的最小奇点的分解后的自相交数非负。给定一条具有奇点P的单尖有理平面曲线C，研究了在满足C在Lambda_C内且P是其唯一基点的射影平面上的唯一束Lambda_C.我们证明了Lambda_C的一般成员是一条有理曲线当且仅当曲线C是非负的。我们还表明，在这种情况下，则Lambda_C有一个diccritical度1。注意，所有当前已知的射影平面中的单尖有理曲线C是非负的。
Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.
On a new class of rational cuspidal plane curves with two cusps
本文考虑有理尖点平面曲线恰好有两个尖点，其补曲线的对数Kodaira维数为2。我们对这类曲线进行了分类，给出了它们的严格变换通过尖点的最小嵌入分辨率具有最大自相交数的性质。
If the first Betti number of the Milnor fibre of a plane curve singularity is zero, then the defining function is equivalent to $x^r$.
Logarithmic vector fields along smooth plane cubic curves
本文研究了射影平面上沿光滑三次曲线的对数向量场的层，证明了一个Dolgachev-Kapranov意义下的Torelli型定理.
A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described.
We find a recursive formula for the motivic Milnor fiber of an irreducible plane curve, using the notions of a truncation and derived curve. We then apply natural transformations to obtain a similar recursion for the Hodge-theoretic spectrum.
On Orevkov's rational cuspidal plane curves
本文考虑有理尖点平面曲线，其尖点恰好是补曲线的Kodaira维数为2的对数。我们对这类曲线进行了分类，并给出了它们的严格变换通过尖点的最小嵌入分解具有最大自相交数的性质。我们表明，由分类给出的曲线与Orevkov构造的曲线相吻合。
We classify plane curves $\mathcal{C}$ possessing two Galois points $P_1$ and $P_2 \in \mathbb{P}^2 \setminus \mathcal{C}$ such that the associated Galois groups $G_{P_1}$ and $G_{P_2}$ generate the semidirect product $G_{P_1}\rtimes G_{P_2}$. New examples of plane curves with two Galois points are also presented.
The group law for the Jacobi variety of plane curves
通过用点群代替点，用代数曲线代替直线，将三次曲线的弦和切线群律推广到n>4次平面曲线的Jacobi簇.曲线是非奇异的或具有简单的奇异性。在n= 4，5，6的情况下，我们与n=3的情况有密切的相似之处.我们使用Groebner基地描述一个算法。
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