arXiv subjects
Publications and source records attributed to Mina Teicher.
Let T be the complex projective torus, and X the surface CP^1 X T. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-Teicher braid monodromy algorithm and group calculations. We show that pi_1(X_Gal) = Z^10.
We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines in $P^3$ . By using some well known material from algebraic geometry, we open new branches to computational vision. We introduce algebraic curves embedded in $P^3$ as the building blocks from which the tensor of a couple of cameras (projections) can be computed. In the process we address dimensional issues and as a result establish the minimal number of algebraic curves required for the tensor variety to be discrete as a function of their degree and genus. We then establish new results on the reconstruction of an algebraic curves in $P^3$ from multiple projections on projective planes embedded in $P^3$ . We address three different presentations of the curve: (i) definition by a set of equations, for which we show that for a generic configuration, two projections of a curve of degree d defines a curve in $P^3$ with two irreducible components, one of degree d and the other of degree $d(d - 1)$, (ii) the dual presentation in the dual space $P^{3*}$, for which we derive a lower bound for the number of projections necessary for linear reconstruction as a function of the degree and the genus, and (iii) the presentation as an hypersurface of $P^5$, defined by the set of lines in $P^3$ meeting the curve, for which we also derive lower bounds for the number of projections necessary for linear reconstruction as a function of the degree (of the curve). Moreover we show that the latter representation yields a new and efficient algorithm for dealing with mixed configurations of static and moving points in $P^3$.
One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan, and it turns out that the incidence lattice of such arrangements determines the projective fundamental group. We use actions on the set of wiring diagrams, introduced in [GTV] (math.AG/0107178), to classify real arrangements of up to 8 lines. In particular, we show that the incidence lattice of such arrangements determines both the affine and the projective fundamental groups.
Wiring diagrams usually serve as a tool in the study of arrangements of lines and pseudolines. In this paper we go in the opposite direction, using known properties of line arrangements to motivate certain equivalence relations and actions on sets of wiring diagrams, which preserve the incidence lattice and the fundamental groups of the affine and projective complements of the diagrams. These actions are used in [GTV] to classify real arrangements of up to 8 lines and show that in this case, the incidence lattice determines both the affine and the projective fundamental groups.
In this paper we present the Braid Monodromy Type (BMT) of curves and surfaces; past, present and future. The BMT is an invariant that can distinguish between non-isotopic curves; between different families of surfaces of general type; between connected components of moduli space of surfaces and between non symplectmorphic 4-manifolds. BMT is a finer invariant than the Sieberg-Witten invariants. Consider 2 simply connected surfaces of general type with the same Chern classes. It is known that if they are in the same deformation class, they are diffeomorphic to each other. Are there computable invariants distinguishing between these 2 classes? The new invariant, proposed here, is located between the 2 classes. In this paper we shall introduce the new invariant, state the current results and pose an open question.
One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.
We describe various properties of Hirzebruch surfaces and related constructions: degenerations, braid monodromy, Galois covers and their Chern classes.
In this paper we introduce a formula to compute Chern classes of fibered products of algebraic surfaces. For f, a generic projection to CP^2, of an algebraic surface X, we define X_k (for k smaller than degf) to be the k products of X over f minus the big diagonal. For k=degf, X_k is called the Galois cover of f w.r.t. full symmetric group. Let S be the branch curve of f. We give a formula for c_1^2 and c_2 of X_k, in terms of degf, degS, and the number of cusps, nodes and branch points of S. We apply the formula in 2 examples and add a conjecture concerning the spin structure of fibered products of Veronese surfaces.
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of surfaces it is expected that the fundamental group has a polycyclic structure, we define a new invariant that comes from this structure. We compute this invariant for a few examples. Braid monodromy factorizations related to curves is a first step in computing the fundamental group of the complement of the curve, and thus we also indicate the possibility of using braid monodromy factorizations of branch curves as an invariant of a surface.
The main result in this paper is as follows: Let S be the branch curve (in the projective plan) of a generic projection of a Veronese surface. Then the fundamental group of the complement of S is an extension of a solvable group by a symmetric group. A group with the property mentioned above is ``almost solvable'' in the sense that it contains a solvable normal subgroup of finite index. This raises the question for which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projection to the complex plane an extension of a solvable group by a symmetric group?
The group described in this paper appeared while studying fundamental groups of complements of branch curves. It turned out that a certain quotient of the braid group acts on those fundamental groups and studying this action is essential for understanding the structure of the fundamental groups. We describe here the quotient of the Artin braid group by commutators of transversal half-twists and we investigates its group actions. We denote the quotient by B_n~ and refer to the groups which admit an action of B_n~ , as B_n~-groups. We distinguish special elements in B_n~-groups which we call prime elements and we give a criterion for an element to be prime. This criterion will be applied to the study of the structure of fundamental groups of complements of branch curves. The group B_n~ itself turns out to be an extension of a solvable group by a symmetric group.
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 group and fundamental groupsto decompose $\MM_{c_1^2(X),c_2(X)}$ into connected components.
This paper will appear in the Santa Cruz proceedings. An overview of the braid group techniques in the theory of algebraic surfaces, from Zariski to the latest results, is presented. An outline of the Van Kampen algorithm for computing fundamental groups of complements of curves and the modification of Moishezon-Teicher regarding branch curves of generic projections are given. The paper also contains a description of a quotient of the braid group, namely $\tilde B_n$ which plays an important role in the description of fundamental groups of complements of branch curves.It turns out that all such groups are ``almost solvable'' $\tildeB_n$-groups. Finally, possible applications to the study of moduli spaces of surfaces of general type are described and new examples of positive signature spin surfaces whose fundamental groups can be computed using the above algorithm (Galois cover of Hirzebruch surfaces) are presented.
We discuss the applications of fundamental groups (of complements of curves) computations (and possibly the computations of the second homotopy group as a model over it) to the classification of algebraic surface. We prove that the fundamental group of the complement of the branch curve of a generic projection of a Veronese surface to the complex plane is an "almost solvable" group in the sense that it contains a solvable group of finite index and thus we can consider the second fundamental group as model over the first.
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 planes with the desired property and in IV we used I -III to compute the braid monodromy of the associated branch curve. Here we use the Van-Kampen method to compute the fundamental group of the complement from a braid monodromy factorization and some extra properties of the factorization , namely invariant properties, also proven in IV. The group is presented using a certain quotient of the Braid group defined by transversal half-twists.