Meirav Amram, Mina Teicher
In this article, we compute the braid monodromy of two algebraic curves defined over R. These two curves are of complex level not bigger than 6, and they are unions of lines and conics. We use two different techniques for computing their braid monodromies. These results will be applied to computations of fundamental groups of their complements in C^2 and C PP^2.
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We describe various properties of Hirzebruch surfaces and related constructions: degenerations, braid monodromy, Galois covers and their Chern classes.
This paper is the first in a series of three papers concerning the surface T times T. In this paper we study the degeneration of T times T and the regeneration of its degenerated object. We study the braid monodromy and its regeneration.
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。
In this manuscript we prove that if two cuspidal plane curves have equivalent braid monodromy factorizations, then they are smoothly isotopic in the plane. As a consequence of this and the Chisini conjecture, we obtain that if two discriminant curves (or branch curves in other terminology) of generic projections (to the plane) of surfaces of general type imbedded in a projective space by means of a multiple canonical class have equivalent braid monodromy factorizations, then the surfaces are diffeomorphic (if we consider them as real 4-folds).
We introduce braid monodromy for the discriminant hypersurface in versal unfoldings of hypersurface singularities. Our objective is then to compute this invariant for singularities of Brieskorn Pham type: First we consider the unfolding by linear monomials in some detail and relate it to the versal unfolding. Next we present the A_n case in our framework. The two main chapters then provide the result in case of two variables and the inductive step to all higher dimensions. We finish with a presentation of the fundamental group of the discriminant complement.
We define a monodromy homomorphism for irreducible families of regular elliptic fibrations which takes values in the mapping class group of a punctured sphere. We compute the monodromy for elliptic fibrations only which contain no singular fibres of types other than I_1 and I*_0. There are maximal groups, which can be the monodromy groups of algebraic, resp. differentiable families of elliptic surfaces, and we give an algebraic criterion for the equality of both groups which we can check to apply in case the number of I_1 fibres is at most 6.
The braid monodromy factorization of the branch curve of a surface of general type is known to be an invariant that completely determines the diffeomorphism type of the surface. Calculating this factorization is of high technical complexity; computing the braid monodromy factorization of branch curves of surfaces uncovers new facts and invariants of the surfaces. Since finding the branch curve of a surface is very difficult, we degenerate the surface into a union of planes. Thus, we can find the braid monodromy of the branch curve of the degenerated surface, which is a union of lines. The rege…
This paper is the second in a series. The first one describes pillow degenerations of a $K3$ surface with genus $g$. In this paper we study the $(2,2)$-pillow degeneration of a non-prime $K3$ surface and the braid monodromy of the branch curve of the surface with respect to a generic projection onto $\C¶^2$. In future papers we study the fundamental group of the complement of the branch curve and the fundamental group of the Galois cover of the surface with respect to this generic projection.
We prove that there is an infinite sequence of pairs of plane cuspidal curves $C_{m,1}$ and $C_{m,2}$, such that the pairs $(\Bbb CP^2, C_{m,1})$ and $(\Bbb CP^2, C_{m,2})$ are diffeomorphic, but $C_{m,1}$ and $C_{m,2}$ have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of "Dif=Def" and "Dif=Iso" problems for plane irreducible cuspidal curves. In our examples, $C_{m,1}$ and $C_{m,2}$ are complex conjugated.
In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.
We generalize the Moishezon Teicher algorithm that was suggested for the computation of the braid monodromy of an almost real curve. The new algorithm suits a larger family of curves, and enables the computation of braid monodromy not only of caspidal curves, but of general algebraic curves, with some non simple singularities. Moreover, it works also when in the fiber the curve admits any number of imaginary points. We also provide two examples of how to use the generalized algorithm.
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement ab…
On Positive Braids, Monodromy Groups and Framings
本文为每个正辫子构造辫子单值群，推广孤立平面曲线奇点的几何单值群，并在闭包为纽结时将其等同于带框架映射类群，从而得到良定义的纽结不变量，进而证明不可约奇点的几何单值群由亏格和Arf不变量决定。
We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is given along with full classification of the images of two intersecting lines under degree 2 rational transformation. We also establish a connection between degree 2 rational transformations and the local bra…
To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated ``braided wiring diagram.'' The ensuing presentation of the fundamental group of the complement is shown to be Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that th…
In this work we describe a method to reconstruct the braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting, since it combines two techniques, namely, the reconstruction of a highly non-generic braid monodromy with a systematic method to go from a non-generic to a generic braid monodromy. This "generification" method is independent from Kummer covers and can be applied in more general circumstances since non generic braid monodromies appear more naturally and are oftentimes much easier to compute. Explicit examples are computed using these techniques.
Topology of complex plane curves: braid monodromy, local and global problems
研究复平面代数曲线的嵌入拓扑，探讨其历史发展中的局部和全局问题，结合辫群理论分析曲线结构。
We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.
Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Viète map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $\mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $π_1(\mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of $\mathcal{C}_d$ with a collection of coordinate hyperplanes in $\mathbb{C}^{d+1}$, the image of the map $\mathscr…
We introduce and develop a language of semigroups over the braid groups for a study of braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application we give a new proof of Orevkov's theorem on realization of a bmf over a disc by algebraic curves and show that the complexity of such a realization can not be bounded in terms of the types of the factors of the bmf. Besides, we prove that the type of a bmf is distinguishing Hurwitz curves with singularities of inseparable types up to $H$-isotopy and $J$-holomorphic cuspidal curves in $\C P^2$ up to s…
We establish a braid monodromy representation of functions satisfying an algebraic equation containing three terms (trinomial equation). We follow global analytic continuation of the roots to a trinomial algebraic equation that are expressed by Mellin-Barnes integral representations. We depict braids that arise from the monodromy around all branching points. The global braid monodromy is described in terms of rational twists of strands that yield a classical Artin braid representation of algebraic functions. As a corollary, we get a precise description of the Galois group of a trinomial algebr…
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and get a 3-cuspidal affine quartic curve whose braid monodromy we compute. This calculation of the local braid monodromy is a step towards the determination of global braid monodromies, e.g. for the (a,b,c) surfaces previously considered by the authors. In the revision we fill a ga…
Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
本文通过分支覆盖构造和辫子单值性方法，直接计算Xiao亏格二Lefschetz纤维化的几何单值性分解，得到类型$(4,3)$的有序正分解及七个因子的Artin坐标。
Let G be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus many problems on algebraic curves require the computation of braid orbits. In this paper we describe an implementation of this computation. We discuss several applications, including the classification of irreducible families of indecomposable rational functions with exceptional monodromy group.
Stratified braid groups: monodromy
本文通过将对数导数关联的无限面积平移曲面引入新方法，确定了首一无平方因子复多项式空间分层在辫群中的单值性，并揭示其由水平叶状结构诱导的圆盘框架所支配。
Bidouble covers $π: S \mapsto Q$ of the quadric Q are parametrized by connected families depending on four positive integers a,b,c,d. In the special case where b=d we call them abc-surfaces. Such a Galois covering $π$ admits a small perturbation yielding a general 4-tuple covering of Q with branch curve $\De$, and a natural Lefschetz fibration obtained from a small perturbation of the composition of $ π$ with the first projection. We prove a more general result implying that the braid monodromy factorization corresponding to $\De$ determines the three integers a,b,c in the case of abc-surfaces…
Unimodular bilinear lattices, automorphism groups, vanishing cycles, monodromy groups, distinguished bases, braid group actions and moduli spaces from upper triangular matrices
本专著从对角线为1的整数上三角矩阵出发，系统研究了秩2和3情形下导出的幺模双线性格、单值群、消没循环、辫群轨道及复流形等结构，并探讨了孤立超曲面奇点情形。
Computation of the fundamental group of the complement in the complex plane of the branch curve S , of a generic projection of the Veronese surface to the plane is presented. This paper is a continuation of our previous papers: Braid Group Technique I - IV. In I and II we developed algorithms to compute braid monodromy of a brunch curve, provided there exist a degeneration of the surface to union of planes in a configuration where the associated branch curve is partial to a line arrangement dual to generic. In III we constructed a degeneration of the Veronese surface of order 3 to union of pla…
In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branc…
Given a singular surface X, one can extract information on it by investigating the fundamental group $π_1(X - Sing_X)$. However, calculation of this group is non-trivial, but it can be simplified if a certain invariant of the branch curve of X - called the braid monodromy factorization - is known. This paper shows, taking the Cayley cubic as an example, how this fundamental group can be computed by using braid monodromy techniques and their liftings. This is one of the first examples that uses these techniques to calculate this sort of fundamental group.
Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C^2 or in CP^2. In this article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are almost-solvable. That is - they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces.
A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi-Yau threefolds.
Torsion in the Braid Monodromy of Elliptic Fibrations
本文研究了椭圆纤维化的辫子单项群，通过将其共轭类与$\operatorname{SL}_2$-特征簇上的特殊点相联系，分类了其中的 $n$ 阶元素，并证明了该群中不存在 $n-1$ 或 $n-2$ 阶元素。
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …
Finite braid group orbits on $SL_2$-character varieties
本文分类了带穿孔2-球面映射类群作用下具有有限轨道的Zariski-稠密$SL_2$表示，证明其均为拉回型或由有限复反射群经中间卷积生成，并据此分类了射影直线上具无限阶局部单值性的几何起源秩2局部系统。
Two-step nilpotent monodromy of local systems on special varieties
研究显示，特殊变种上的局部系统单值群是虚拟 nilpotent 的，且阶不超过2，改进了Cadorel等人的定理，并发展了局部系统的变形理论。
We develop a new algebraic theory of positive braids and conjugacy classes in the braid group $B_3$. We use our theory to establish a complete classification of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ with only $A_n$-singularities for $n\geq 1$ (an $A_n$-singularity is locally modelled by the equation $z^2 = w^n$) independent of Gromov's theory of pseudoholomorphic curves. We show that if $C$ and $C'$ are degree three symplectic curves in $\mathbb{CP}^2$ with the same numbers of $A_n$-singularities for each $n\geq 1$, then $C$ is isotopic to $C'$. Furthermore, our …
The pure braid group Γof a quadruply-punctured Riemann sphere acts on the SL(2,C)-moduli M of the representation variety of such sphere. The points in M are classified into Γ-orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite Γ-orbits in M. These examples relate to explicit immersions of constant mean curvature surfaces.
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation theory of higher genus braid groups can be used to produce interesting examples of projective surfaces defined over the field of complex numbers.
In the present work we use the Levelt's valuation theory to describe all monodromy representations that can be realized by Riemann equation. Also we show that if the monodromy of Riemann equation lies in $SL(2,\mathbb{C})$, then such a monodromy can be realized by a more special Riemann-Sturm-Liouville equation as well. After all the criterion for hypergeometric equation to have monodromy in $SL(2,\mathbb{Z})$ is presented.
Every smooth minimal complex algebraic surface of general type, $X$, may be mapped into a moduli space, $\MM_{c_1^2(X), c_2(X)}$, of minimal surfaces of general type, all of which have the same Chern numbers. Using the braid group and braid monodromy,we construct infinitely many new examples of pairs of minimal surfaces of general type which have the same Chern numbers and non-isomorphic fundamental groups. Unlike previous examples, our results include $X$ for which $|π_1(X)|$ is arbitrarily large. Moreover, the surfaces are of positive signature. This supports our goal of using the braid grou…
By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.
On the Alexander Invariants of Trigonal Curves
本文证明辫群 B3 中大多数亏格零子群对亚历山大不变量无贡献，并引入“本原”亏格零子群来完全决定不变量，进而分类特殊子类并重新得到不可约三角曲线二面体覆盖的已知分类。
We give a criterion for the good reduction of semistable $K3$ surfaces over $p$-adic fields using purely $p$-adic methods. We use neither $p$-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or $K3$ surfaces. We achieve our goal by realizing the special fiber $X_s$ of a semistable model $X$ of a $K3$ surface over the $p$-adic field $K$, $X_K$ as a special fiber of a log-family in characteristic $p$ and use an arithmetic version of the Clemens-Schimd exact sequence in order to obtain a Kulikov-Persson-Pinkham classification theorem…
On Oda's problem and special loci
本文指出Oda问题具有栈性质，通过证明单值域相对于具对称性曲线的更精细特殊轨迹数据的独立性，为Oda的预测提供了新证明。
This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding s…
We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $β=γ=0$, $δ={1\over2}$ and $α$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of o…
Families of cyclic curve coverings with maximal monodromy
本文研究了具有最大单值性的循环曲线覆盖族，通过Hodge结构变分的G分解条件，证明了单值性最大性的判据，并探讨了双伽罗华覆盖和三伽罗华覆盖的测地性问题。
In the paper we show that for a normal-crossings degeneration $Z$ over the ring of integers of a local field with $X$ as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product $X \times X$. More precisely, they also define algebraic cycles on the special fibre of a resolution of $Z \times Z$. In the paper, we give an explicit description of these cycles for a degeneration with at worst triple points as singularities. These cycles explain geometrically the presence of poles on specific local f…
We consider an arbitrary polynomial map $f:{\mathbb C}^{n+1}\to {\mathbb C} $ and we study the Alexander invariants of ${\mathbb C}^{n+1}\setminus X$ for any fiber $X$ of $f$. The article has two major messages. First, the most important qualitative properties of the Alexander modules are completely independent of the behaviour of $f$ at infinity, or about the special fibers. Second, all the Alexander invariants of all the fibers of the polynomial $f$ are closely related to the monodromy representation of $f$. In fact, all the torsion parts of the Alexander modules (associated with all the pos…
We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite br…
In the last years a lot of work has been concentrated on the study of the behaviour at infinity of polynomial maps. This behaviour can be very complicated, therefore the main idea was to find special classes of polynomial maps which have, in some sense, nice properties at infinity. In this paper, we completely determine the complex algebraic monodromy at infinity for a special class of polynomial maps (which is complicated enough to show the nature of the general problem).
We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…
We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $\integ/\ell$ monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group $\sp_{2g}(\integ/\ell)$. We prove that the $\integ/\ell$ monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group $\su_{(r,s)}(\integ/\ell\tensor\integ[ζ_3])$.
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。
In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. In the fourth and fifth sections, we specialize to quite special classses of hypergeomtric sheaves. We give relatively "simple" formulas for their trace functions, and a criterion for them to have finite monodromy. In the next section, we prove that three of them have finite monodromy groups.We then give some results on finite complex linear groups. We next use these group theoretic results to show that one of our local systems, of rank $24…
We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$, where $Σ_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimen…
Finite monodromy groups in degenerating families and the stability degree of curves
该研究计算亏格g的曲线与指定秩的半阿贝尔簇的稳定度，证明前者等于Minkowski界，通过退化族有限单值群的 specialization 结果结合素数2处显式构造完成证明。
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