Zariski pairs on sextics I

On deformations of singular plane sextics

We study complex plane projective sextic curves with simple singularities up to equisingular deformations. It is shown that two such curves are deformation equivalent if and only if the corresponding pairs are diffeomorphic. A way to enumerate all deformation classes is outlined, and a few examples are considered, including classical Zariski pairs.

Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers.

Nodal curves with a contact-conic and Zariski pairs

In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

Apparent contours of nonsingular real cubic surfaces

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.

Classification of sextics of torus type

The second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextics of torus type, without assuming the tameness of the sextics. We show that there exists 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.

On arithmetic Zariski pairs in degree 6

We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.

三次曲面上的Zariski对

Zariski pairs on cubic surfaces

本文定义了一般三次曲面上线构形的Zariski对概念，并给出了这些Zariski对的完整列表。

Weak Zariski decompositions and log terminal models for generalized polarized pairs

We show that the existence of a birational weak Zariski decomposition for a pseudo-effective generalized polarized lc pair is equivalent to the existence of a generalized polarized log terminal model.

On weak Zariski decompositions and termination of flips

We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.

新的μ-扎里斯基表面奇点对

New $μ$-Zariski pairs of surface singularities

本文提出一个非勒-约蒙丁类型的μ-扎里斯基表面奇点对，挑战了以往所有已知例子的假设。

On log minimal models and Zariski decompositions II

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama-Zariski decomposition with nef positive part, or that $K_X+B$ is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$ where $(X,B)$ is a usual projective pair and $P$ is nef, and study the birational geometry of such pairs.

牛顿加权-Lê-Yomdin多项式与曲面奇点的μ-Zariski对

Newton weighted-Lê-Yomdin polynomials and $μ$-Zariski pairs of surface singularities

本文研究牛顿加权-Lê-Yomdin多项式定义的曲面奇点，并利用该子类提出构造新μ-Zariski对的方法。

通过基本群得到的7次和8次圆锥-直线构型的Zariski对

Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

通过基本群和Zariski-van Kampen定理，发现新的8次圆锥-直线构型Zariski对，并验证了2014年提出的两个7次Zariski对。

On Zariski's pairs of m-th canonical discriminant curves

In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i \to P^2$, $i=1, 2$, (ii) $S_1$ and $S_2$ are homeomorphic surfaces of general type, (iii) $f_{i,m}$ is given by linear three-dimensional subsystem of the mth canonical class of $S_i$.

反Zariski对

Anti-Zariski pairs

提出反Zariski对概念，即同胚但属于等奇异族不同分支的平面曲线对，并给出例子及相关讨论。

On the semiampleness of the positive part of CKM Zariski decompositions

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.

Torsion divisors of plane curves and Zariski pairs

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the smooth component was implicitly considered. We formulate this relation clearly and give a criterion for distinguishing the embedded topology in terms of torsion classes. Furthermore, we give a method of systematically constructing examples of curves where our criterion is applicable, and give new examples of Zariski tuples.

Elliptic dihedral covers in dimension 2, geometry of sections of elliptic surfaces, and Zariski pairs for line-conic arrangements

In this article, examples of Zariski pairs $(B_1, B_2)$ satisfying the following condition are given: (i) $°B_1 = °B_2 = 7$. (ii) Irreducible components of $B_i$ $(i = 1, 2)$ are lines and conics. (iii) Singularities of $B_i$ $(i = 1, 2)$ are nodes, tacnodes and ordinary triple points. In order to construct $B_i$ ($i = 1, 2$), we make use of geometry of sections of rational elliptic surfaces and their group structure. Dihedral covers play important roles to distinguish the topology of $({\mathbb P}^2, B_i)$ $(i = 1, 2)$.

Zeta-function and $μ^*$-Zariski pairs of surfaces

A Zariski pair of surfaces is a pair of complex polynomial functions in $\mathbb{C}^3$ which is obtained from a classical Zariski pair of projective curves $f_0(z_1,z_2,z_3)=0$ and $f_1(z_1,z_2,z_3)=0$ of degree $d$ in $\mathbb{P}^2$ by adding a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine surfaces of $\mathbb{C}^3$ -- defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ -- have an isolated singularity at the origin and the same zeta-function for the monodromy associated with their Milnor fibrations (so, in particular, $g_0$ and $g_…

$P$-平凡MMP、Zariski分解与广义配对的极小模型

$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs

本研究建立$P$-平凡MMP理论，证明满足特定Nakayama-Zariski分解及数值有效条件的广义klt配对存在极小模型，并推广至三维广义lc配对，拓展了相关领域的经典结论。

关于势三元组的极小模型程序与Zariski分解

On minimal model program and Zariski decomposition of potential triples

本文研究了由对与伪有效除子构成的势三元组的性质，证明若除子允许双有理Zariski分解则可关联广义对结构并运行广义MMP，同时证明了pklt对在特定分解条件下极小模型的存在性。

On $ μ$-Zariski pairs of links

The notion of Zariski pairs for projective curves in $\mathbb P^2$ is known since the pioneer paper of Zariski \cite{Zariski} and for further development, we refer the reference in \cite{Bartolo}.In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski ) pair of curves $C=\{f(x,y,z)=0\}$ and $C'=\{g(x,y,z)=0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. Th…

相对Nakayama-Zariski分解与广义对的最小模型

Relative Nakayama-Zariski decomposition and minimal models of generalized pairs

本文证明了相对Nakayama-Zariski分解的基本性质，并将其应用于研究lc广义对，证明了由丰富除子极化的lc广义对的log极小模型或Mori纤维空间的存在性。

反典范极小模型与 Zariski 分解

Anticanonical minimal models and Zariski decomposition

本文将 Birkar-Hu 关于 lc 配对极小模型存在性的 Zariski 分解判据平行推广到 pklt 配对，证明了反典范除子容许双有理 Zariski 分解时反典范极小模型存在。

A note on Zariski pairs

A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present a method to construct certain type of Zariski pairs, and make infinite series of examples.

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective $\R$-Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension $d-1$, a lc pair $(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a weak Zariski decomposition$/Z$.

通过算法和计算分类检测锥线排列中的Zariski对

Detecting Zariski Pairs by Algorithms and Computational Classification in Conic Line Arrangements

本文通过算法和计算分类方法，提出了一种检测锥线排列中Zariski对的新方法，利用组合条件和基本群计算进行系统分类。

Nakayama-Zariski 分解与翻转的终止性

Nakayama-Zariski decomposition and the termination of flips

本文证明在伪有效射影对中，假设关于Nakayama-Zariski分解在极小模型程序操作下行为的自然猜想，一次翻转序列的终止性蕴含所有翻转的终止性。

含唯一圆锥曲线的圆锥曲线-直线排列的Zariski对

Zariski pairs of conic-line arrangements with a unique conic

本研究提出两对9次、含唯一圆锥曲线的圆锥曲线-直线排列构成的Zariski对，通过连通数区分二者的拓扑结构。

弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何学 via 理性椭圆面和扎里斯基对

Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs

研究弱接触圆锥与具有2个节点和1个尖点的不可约四次曲线的几何结构，通过理性椭圆面和扎里斯基对构造度数为7和8的扎里斯基对。

镜像六次双平面纤维化的三维簇

Threefolds Fibred by Mirror Sextic Double Planes

本文系统研究以镜像六次双平面K3曲面为纤维的三维簇，提出广义函数与同调不变量，构造Weierstrass形式及极小形式，并分析其奇异纤维、典范除子和Betti数。

Trisections on Certain Rational Elliptic Surfaces and Families of Zariski Pairs Degenerating to the same Conic-line Arrangement

In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties explicitly. As a result we are able to construct new examples of Zariski pairs. Especially, we show the existence of a family of Zariski pairs that degenerate to the same conic-line arrangement.

The link of {f(x,y)+z^n=0} and Zariski's Conjecture

We consider suspension hypersurface singularities of type g=f(x,y)+z^n, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated h…

Some open questions on arithmetic Zariski pairs

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose complements are homeomorphic, but whose embeddings in $\mathbb{P}^2$ are not. Most of the known invariants used to detect Zariski pairs depend on the étale fundamental group. In the case of Galois-conjugate curves, their étale fundamental groups coincide. Braid monodromy factorizat…

A note on splitting numbers for Galois covers and $π_1$-equivalent Zariski $k$-plets

In this paper, we introduce \textit{splitting numbers} of subvarieties in a smooth variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. By splitting numbers, we give a necessary and sufficient condition for two plane curves of type $(b,m)$ to be topologically equivalent as pairs of the complex projective plane and plane curves, where a plane curve of type $(b,m)$ is an arrangement of two smooth plane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, we prove that there are $π_1$-equivalent Zariski $k$-plets for any $k\geq2…

On the Lipman-Zariski conjecture for logarithmic vector fields on log canonical pairs

We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic $1$-forms) is locally free. We prove that in this case the following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor D\rfloor $ is snc. If $(X,D)$ is lc or the logarithmic $1$-forms are locally generated by closed forms, then $(X,\lfloor D\rfloor)$ is toroidal.

The matroid structure of vectors of the Mordell-Weil lattice and the topology of plane quartics and bitangent lines

In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski $N$-ples of relatively low degree. Namely we show that a Zariski 102-ple of degree 18 exists.

Almost non-degenerate functions and a Zariski pair of links

Let $f(\mathbf z)$ be an analytic function defined in the neighborhood of the origin of $\mathbb C^n$ which have some Newton degenerate faces. We generalize the Varchenko formula for the zeta function of the Milnor fibration of a Newton non-degenerate function $f$ to this case. As an application, we give an example of a pair of hypersurfaces with the same Newton boundary and the same zeta function with different tangent cones.

某个7次圆锥曲线-直线排列的实现空间与一个π₁等价Zariski对

The realization space of a certain conic line arrangement of degree 7 and a $π_1$-equivalent Zariski pair

本文通过构造一个π₁等价Zariski对，确定了具有特定固定组合结构的7次圆锥曲线-直线排列实现空间的连通分支数。

Non-homotopicity of the linking set of algebraic plane curves

The linking set is an invariant of algebraic plane curves introduced by Meilhan and the first author. It has been successfully used to detect several examples of Zariski pairs, i.e. curves with the same combinatorics and different embedding in $\mathds{CP}^2$. Differentiating Shimada's $π_1$-equivalent Zariski pair by the linking set, we prove, in the present paper, that this invariant is not determined by the fundamental group of the curve.

Torsion divisors of plane curves with maximal flexes and Zariski pairs

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper we consider curve arrangements whi…

Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curves

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study …

Splitting curves on a rational ruled surface, the Mordell-Weil groups of hyperelliptic fibrations and Zariski pairs

Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…

关于纤维型曲线的拓扑：仿射结点曲线的一个扎里斯基对

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。

An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the …

7次和8次圆锥-直线构型的强Ziegler对示例

Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8

本文证明了某些具有相同组合结构但嵌入拓扑不同的平面曲线对（Zariski对）同时也是Milnor代数不同构的强Ziegler对。

Even sets of nodes on sextic surfaces

We determine the possible even sets of nodes on sextic surfaces in $\Pn 3$, showing in particular that their cardinalities are exactly the numbers in the set $\{24, 32, 40, 56 \}$. We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence relies on computer calculations. The construction gives a maximal family…

潜在线性极小模型程序的结构

Structure of the Anticanonical Minimal Model Program for Potentially klt Pairs

本文在假设反典范形允许birational Zariski分解下，给出了潜在klt对反典范形极小模型程序存在的另一种证明，并建立了结构定理，表明从潜在klt对出发的任何部分反典范形MMP可提升为非正映射序列。

拟齐次非孤立奇点超曲面的Zariski重数猜想

Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities

本文针对2≤n≤4维拟齐次非孤立奇点超曲面，利用拟齐次正规形推导出由权重和次数决定的重数显式公式，证明重数在拓扑等价下不变，从而证实Zariski重数猜想。

通过Frobenius结构定理比较非阿基米德镜像构造与对数镜像构造

Comparison of non-archimedean and logarithmic mirror constructions via the Frobenius structure theorem

本文证明在X\D光滑仿射且D为nef除子支集的条件下，Gross-Siebert镜像代数上迹形式的结构常数等于Keel-Yu定义的朴素曲线计数，从而推出两类镜像代数相等，并证明Mandel提出的Fano镜像猜想。

Large sets of generating tuples for Lie groups

We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.

Configurations of points and topology of real line arrangements

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties. Using this dual point of view, we construct several examples of co…

Existence of Two View Chiral Reconstructions

A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper we provide a complete classification of k point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the non-emptiness of certain semialgebraic sets. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconst…

辛对数柯尔加雷维度为-∞，仿射被覆性和无 cusps 理性曲线

Symplectic log Kodaira dimension $-\infty$, affine-ruledness and unicuspidal rational curves

本文研究了辛对数柯尔加雷维度为-∞的辛对 (X, ω, D) 的性质，引入了辛仿射被覆性的概念，并探讨了无 cusps 理性曲线在该结构中的作用。