Braid Monodromy Type and Rational Transformations of Plane Algebraic Curves

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 braid monodromy of the image at the intersecting point of two lines. Moreover, we present an example of two birationally isomorphic curves with the same braid monodromy type and non diffeomorphic real parts. Introduction The braid monodromy is a powerful tool in the study of algebraic surfaces and curves. There exists several algorithms for computing braid monodromy for many types of curves. Usually one considers algebraic curves up to birational isomorphisms, therefore it is natural to consider the effect that a rational transformation has on the braid monodromy of a curve. Recently a new algorithm for computing the explicit image of a given algebraic curve under rational transformation was obtained. Hence, we consider the combination of these two techniques and study the braid monodromy of the image of a curve under a rational transformation. In particular, it is interesting to study the braid monodromy of an algebraic curve under a rational transformation which resolves the curve’s singularities. In this paper we lay out the basics of the technique as follows: In Chapter 1 we recall the notions and definitions of braid group, half-twists and braid monodromy. In Chapter 2 we present explicit formulas for the image of a complex line under a given rational transformation. We establish the connection between a rational transformation and the local braid monodromy of the image of the intersection point of two intersecting lines under this rational transformation. We present a full classification of the global braid monodromy for the image of two intersecting lines under degree 2 rational transformation In Chapter 3. Chapter 4 explains a new technique which allows to find the image of curve of any degree under any rational transformation, and we give an example of degree 3. We conclude with the This work is part of the first author Ph.D Thesis in Bar-Ilan university. Partially supported by EU-network HPRN-CT-2009-00099(EAGER) , (The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany), the Israel Science Foundation grant # 8008/02-3 (Excellency Center ”Group Theoretic Methods in the Study of Algebraic Varieties”). The authors wish to thank Prof. Victor Vinnikov for helpful advices. 1 2 KAPLAN, SHAPIRO AND TEICHER computation of the global braid monodromy for the image of de-singularized curve of degree 4. 1. Braid group preliminaries In this chapter we recall the definition of the braid group, some of its important elements and the braid monodromy. Readers who are interested in braid group could find more information in [1, 4, 5]. For information about braid monodromy we suggest readers to consult [11, 12]. 1.1. The braid group. Definition 1.1. Artin’s braid group Bn is the group generated by {σ1, ..., σn−1} subjected to the relations σiσj = σjσi where |i− j| ≥ 2, σiσi+1σi = σi+1σiσi+1 for all i = 1, ..., n− 2. We distinguish some important elements in the braid group Bn which are called half-twists. Half-twists are actually the elements of the conjugacy class of any of the generators σi (it is known that all generators of the braid group are conjugated to one another). Since the easiest way to describe and work with half-twists is based on a topological equivalent definition for the braid group, we bring it here. Let D be a closed disc, and K = {k1, · · · , kn} a finite set such that K ⊂ int(D). Definition 1.2. Let B be the group of all diffeomorphisms β of D such that β(K) = K, β|∂D = Id|∂D. For β1, β2 ∈ B we say that β1 is equivalent to β2 if β1 and β2 induce the same automorphism of π1(D \ K,u), where u is a point on ∂D. The quotient of B by this equivalence relation is called the braid group Bn[D,K] (n = |K|). The elements of Bn[D,K] are called braids. Now, let D,K, u be as above. Let a, b be two points of K. We denote Ka,b = K \ {a, b}. Let σ be a simple path in D \ (∂D ∪Ka,b) connecting a with b. Choose a small regular neighborhood U of σ and an orientation preserving diffeomorphism f : R → C such that f(σ) = [−1, 1], f(U) = {z ∈ C | |z| < 2}. Let α(x), 0 ≤ x be a real smooth monotone function such that: α(x) = { 1, 0 ≤ x ≤ 32 0, 2 ≤ x Define a diffeomorphism h : C → C as follows: for z = re ∈ C let h(z) = re For the set {z ∈ C | 2 ≤ |z|}, h(z) = Id, and for the set {z ∈ C | |z| ≤ 3 2}, h(z) is a rotation by 180 in the positive direction. Considering (f ◦ h ◦ f)|D (we will compose from left to right) we get a diffeomorphism of D which switches a and b and is the identity on D \U . Thus it defines an element of Bn[D,K]. The diffeomorphism (f ◦ h ◦ f)|D defined above induces an automorphism on π1(D \K,u), that switches the position of two generators of π1(D \K,u), as can be seen Figure 1. Definition 1.3. Let H(σ) be the braid defined by (f ◦ h ◦ f)|D. We call H(σ) the positive half-twist defined by σ. The connection between the topological definition of the half-twists and the geometrical braid can be seen in Figure 2. BRAID MONODROMY TYPE AND RATIONAL TRANSFORMATIONS 3 Figure 1. The switch of two generators of π1(D \K,u) induced by a half-twist. k 1 k 3 k 2