arXiv subjects

Michael Friedman

Publications and source records attributed to Michael Friedman.

9 recordsLinked to original sources

arXiv2018

On Left regular bands and real Conic-Line arrangements

An arrangement of curves in the real plane divides it into a collection of faces. In the case of line arrangements, there exists an associative product which gives this collection a structure of a left regular band. A natural question is whether the same is possible for other arrangements. In this paper, we try to answer this question for the simplest generalization of line arrangements, that is, conic--line arrangements. Investigating the different algebraic structures induced by the face poset of a conic--line arrangement, we present two different generalizations for the product and its associated structures: an alternative left regular band and an associative aperiodic semigroup. We also study the structure of sub left regular bands induced by these arrangements. We finish with some chamber counting results for conic--line arrangements.

math.CO↗

arXiv2013

Conjugation-free groups, lower central series and line arrangements

The quotients $G_k/G_{k+1}$ of the lower central series of a finitely presented group $G$ are an important invariant of this group. In this work we investigate the ranks of these quotients in the case of a certain class of conjugation-free groups, which are groups generated by $x_1,...,x_n$, and having only cyclic relations: $$ x_{i_t} x_{i_{t-1}} ... x_{i_1} = x_{i_{t-1}} ... x_{i_1} x_{i_t} = ... = x_{i_1} x_{i_t} ... x_{i_2}.$$ Using tools from group theory and from the theory of line arrangements we explicitly find these ranks, which depend only at the number and length of these cyclic relations. It follows that for these groups the associated graded Lie algebra $gr(G)$ decomposes, in any degree, as a direct product of local components.

math.GR↗

arXiv2013

On the structure of fundamental groups of conic-line arrangements having a cycle in their graph

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we investigate the structure of this fundamental group when the graph of the conic-line arrangement is a unique cycle of length $n$ and the conic passes through all the multiple points of the cycle. We show that if n is odd, then the affine fundamental group is abelian but not conjugation-free. For the even case, if n>4, then using quotients of the lower central series, we show that the fundamental group is not even a direct sum of a free abelian group and free groups.

math.GT↗

arXiv2013

On the structure of conjugation-free fundamental groups of conic-line arrangements

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkable properties: either it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we first give a complete proof to the following key lemma: if we draw a new line through only one intersection point of a given real line arrangement whose fundamental group is conjugation-free, then the fundamental group of the new arrangement is also conjugation-free. Second, we generalize this lemma to the case of conic-line arrangements. Moreover, we prove that once the graph associated to conic-line arrangements (defined slightly different than the corresponding graph for line arrangements) has no cycles, then the fundamental group of its complement has a conjugation-free geometric presentation and in addition can be written as a direct sum of free groups and a free abelian group. Also, we show that if the graph consists of one cycle, and the conic does not pass through all the multiple points corresponding to the vertices of the cycle, then the fundamental group has a conjugation-free geometric presentation as well. For conclusion, we extend the family of real line arrangements having a conjugation-free geometric presentation (for their fundamental group) by defining the notion of a conjugation-free graph. We also extend this notion to certain families of conic-line arrangements.

math.GT↗

arXiv2011

On fundamental groups related to degeneratable surfaces: conjectures and examples

We argue that for a smooth surface S, considered as a ramified cover over the projective plane branched over a nodal-cuspidal curve B one could use the structure of the fundamental group of the complement of the branch curve to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces - a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of the mentioned fundamental group for these surfaces and present two new conjectures on the structure of this group, regarding non-embedded-degeneratable surfaces. We prove two theorems supporting our conjectures, and show that for an empbedding of a product of a projective line with a curve of genus g, the fundamental group of the complement of the branch curve is a quotient of an Artin group associated to the degeneration.

math.AG↗

arXiv2010

On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise. First, what is the_geography_ of branch curves among all nodal-cuspidal curves? Second: what is the_geometry_ of branch curves? In other words, how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic curve with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these two questions, mentioning both simple and some non-trivial results. We continue with study of classical work of Beniamino Segre which gives a complete answer to the second question in the case when S is a smooth surface in a three-dimensional projective space. We give an interpretation of Segre's work in terms of a study of Picard group of 0-cycles on a singular plane curve B. We also review examples of small degree. The Appendix written by Eugenii Shustin shows the existence of many new Zariski pairs of plane curves. We hope to continue this paper with a generalization to the case of smooth surfaces in a projective space of any dimension.

math.AG↗

arXiv2008

On non Fundamental Group Equivalent Surfaces

In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP^2 and a degenerations of the surface into a union of planes - the "pillow" degeneration for the non-prime surface and the "magician" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a projective deformation of each other

math.AG↗

arXiv2007

On fundamental groups related to the Hirzebruch surface F_1

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.

math.AG↗

arXiv2006

The Regeneration Of A 5-Point

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 regeneration of the singularities of the branch curve, studied locally, leads us to find the global braid monodromy factorization of the branch curve of the original surface. So far, only the regeneration of the BMF of 3,4 and 6-point (a singular point which is the intersection of 3 / 4 / 6 planes) were done. In this paper, we fill the gap and find the braid monodromy of the regeneration of a 5-point. This is of great importance to the understanding of the BMT (braid monodromy type) of surfaces. This braid monodromy will be used to find the global braid monodromy factorization of different surfaces; in particular - the monodromy of the branch curve of the Hirzebruch surface $F_{2,(2,2)}$.

math.AG↗