Arf invariants of real algebraic curves

Symmetric plane curves of degree 7 : pseudo-holomorphic and algebraic classifications

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M-symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudoholomorphic curves of…

平面代数曲线补集的处理分解和Kirby图

Handle decompositions and Kirby diagrams for the complement of plane algebraic curves

本文基于辫形monodromy概念，研究平面代数曲线补集的显式处理分解和Kirby图，提出改进的技巧以获得相关结构。

Topological rigidity of algebraic $¶_3$-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

Algebraic models and arithmetic geometry of Teichmüller curves in genus two

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichmüller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.

Knotting of Algebraic Curves in CP2

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve and having abelian fundamental group of the complement $Cp^2-F$. This gives an answer to Problem 4.110 of Kirby's list.

Signatures of real algebraic curves via plumbing diagrams

We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and signatures of graph links.

Morse theory for plane algebraic curves

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's signature and Levine--Tristram signatures we obtain some informations about possible singularities of a curve C in terms of its topology.

Real algebraic curves on real del Pezzo surfaces

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is known for more general ambient surfaces. We take a step forward in the study of topological types classification of real algebraic curves on non-toric surfaces focusing on real del Pezzo surfaces of degree $1$ and $2$ with multi-components real part. We use degeneration methods a…

Twisted Alexander polynomials of Plane Algebraic Curves

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the classical case, this gives a geometrical interpretation of Libgober's divisibility Theorem. We calculate twisted polynomials for some algebraic curves and show how they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degeneration…

Choreography of divisors on algebraic curves

For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.

Algebraic approximations of fibrations in abelian varieties over a curve

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

Families of algebraic varieties and towers of algebraic curves over finite fields

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.

代数簇的合冲与代数曲线的对称积

Syzygies of algebraic varieties through symmetric products of algebraic curves

本文综述了利用代数曲线对称积的几何方法证明的关于代数簇合冲的若干近期结果，包括亏格猜想、割线簇与切发展曲面的合冲、格林猜想及渐近合冲。

Topology of plane algebraic curves: the algebraic approach

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

Co-fibered products of algebraic curves

We give examples of failure of the existence of co-fibered products in the category of algebraic curves.

An Algebraic Analysis of Conchoids to Algebraic Curves

We study conchoids to algebraic curve from the perspective of algebraic geometry, analyzing their main algebraic properties. We introduce the formal definition of conchoid of an algebraic curve by means of incidence diagrams. We prove that, with the exception of a circle centered at the focus and taking $d$ as its radius, the conchoid is an algebraic curve having at most two irreducible components. In addition, we introduce the notions of special and simple components of a conchoid. Moreover we state that, with the exception of lines passing through the focus, the conchoid always has at least …

具有两个Galois点的代数曲线的后代

Descendants of algebraic curves admitting two Galois points

本文研究代数曲线的Galois点与商曲线的联系，引入具有两个Galois点的后代概念，并证明Fermat曲线的后代均为Fermat曲线，其次数为素数时无后代。

Real algebraic curves, the moment map and amoebas

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications.

Clifford Theorem for real algebraic curves

We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

A Compactness Criterion for Real Plane Algebraic Curves

Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.

A Theory of Duality for Algebraic Curves

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of duality, we prove generalisations of Plucker's formulae for algebraic curves. The results hold for arbitrary characteristic of the base field L, with some occasional exceptions when characteristic(L)=2, which we observe in the course of the paper.

在OSCAR中绘制实平面代数曲线

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。

A note about invariants of algebraic curves

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.

Lipschitz geometry of complex plane algebraic curves

We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.

The join of algebraic curves

Let X,Y be algebraic curves in P^n over C. We give an effective description of the join J(X,Y)\in P^n of X and Y in terms of local parametrizations of X and Y.

A non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence

We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[(-1)*C] in JC modulo algebraic equivalence is torsion.

Algebraic curves with automorphism groups of large prime order

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also give some partial results on $q$-curves for $q = g, g-1$.

On algebraic equivalences among the 27 Abel-Prym curves on a generic abelian 5-fold

This article shows that on a generic principally polarized abelian variety of dimension five the $\mathbb{Q}$-vector space of algebraic equivalences among the 27 Abel-Prym curves has dimension 20.

Tropical and algebraic curves with multiple points

Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.

Generic symmetry defect set of an algebraic curve

Let $X \subset \mathbb{C}^{2n}$ be an $n$-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of $X$. Moreover, we compute its singularities for $X_d$ being a generic curve of degree $d$ in $\mathbb{C}^2$.

代数曲线的利普希茨嵌入

Lipschitz embeddings of algebraic curves

研究实代数曲线利普希茨分类，证明其内部与\(\mathbb{R}^3\)中具LNE连通分量的外部分类等价，找到实仿射代数曲线的LNE模型，还指出复范畴不成立此结论。

Injective linear series of algebraic curves on quadrics

We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

Signatures of algebraic curves via numerical algebraic geometry

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the compl…

The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve

A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every d>=8 showing that the Fermat …

Asymptotic Behavior of an Implicit Algebraic Plane Curve

In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.

Algebraic structures on the topology of moduli spaces of curves and maps

We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.

Algebraic curves admitting non-collinear Galois points

A criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves is presented. A new example of a plane curve with non-collinear Galois points as an application is described. Furthermore, a new characterization of the Fermat curve in terms of non-collinear Galois points is presented.

Very special divisors on 4-gonal real algebraic curves

Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.

On the equivariant algebraic Jacobian for curves of genus two

We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.

Algebraic cycles on the Jacobian of a curve with a $g^r_d$

We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.

Flash Geometry of Algebraic Curves

In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of asymptotic degenerations, a particular class of degenerations of plane projective algebraic curves. In particular, we consider the geometrical technique in relation to the Severi problem of degenerating nodal curves to lines in general position, and correct Severi's original proof of his conjecture.

结点代数曲线上主$\Bbb C^*$-丛的模栈的上同调

Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves

研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。

无穷远处爆破球面同胚下实代数曲线的分类

Classification of real algebraic curves under blow-spherical homeomorphisms at infinity

该研究针对无穷远处爆破球面同胚下的实代数曲线，提出了带标准形式的完整分类结果。

Curve selection lemma in infinite dimensional algebraic geometry and arc spaces

We prove a Curve Selection Lemma valid in infinite-dimensional algebraic geometry and apply it, together with a structure theorem of Grinberg-Kahzdan and Drinfeld, to obtain a Curve Selection Lemma in arc spaces.

Applications of Level Curves to Some Problems on Algebraic Surfaces

In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some topological features of surfaces (reality, compactness, connectivity) and to the problem of plotting.

Improvement on Parameters of Algebraic-Geometry Codes from Hermitian Curves

Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in [9,10].

具有固定j-不变量的热带与代数椭圆平面曲线

Tropical and algebraic elliptic plane curves with fixed j-invariant

该研究通过热带计数特定椭圆曲线，结合对应定理，给出代数计数的另一种证明，拓展了枚举几何的相关方法。

A birational embedding of an algebraic curve into a projective plane with two Galois points

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.

Algebraic curves admitting the same Galois closure for two projections

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. As an application, it is proved that the Hermitian curve in positive characteristic coincides with the Galois closures of projections of some plane curve from some two non-uniform points.

On large automorphism groups of algebraic curves in positive characteristic

In his investigation on large $K$-automorphism groups of an algebraic curve, Stichtenoth obtained an upper bound on the order of the first ramification group of an algebraic curve $\cX$ defined over an algebraically closed field of characteristic $p$. Stichtenoth's bound has raised the problem of classifying all $\K$-automorphism groups $G$ of $\cX$ with the following property: There is a point $P\in \cX$ for which \begin{equation} |G_P^{(1)}|>\frac{p}{p-1}g. \end{equation} Such a classification is obtained here by proving Theorem 1.3

最小的代数空间填充曲线在射影线的积上

Minimal algebraic space filling curves on the product of projective lines

本文研究了射影线积上最小次数的光滑代数空间填充曲线，证明了存在大量例子并扩展了Homma和Kim的结果。

On automorphisms of algebraic curves

An irreducible, algebraic curve $\mathcal X_g$ of genus $g\geq 2$ defined over an algebraically closed field $k$ of characteristic $\mbox{char } \, k = p \geq 0$, has finite automorphism group $\mbox{Aut} (\mathcal X_g)$. In this paper we describe methods of determining the list of groups $\mbox{Aut} (\mathcal X_g)$ for a fixed $g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible.

Calculating cohomology groups of moduli spaces of curves via algebraic geometry

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

Algebraic curves with collinear Galois points

A criterion for the existence of a birational embedding into a projective plane with three collinear Galois points for algebraic curves is presented. The extendability of an automorphism induced by a Galois point to a linear transformation of the projective plane is also discussed, under the assumption that two Galois points exist.