arXiv subjects

M. Teicher

Publications and source records attributed to M. Teicher.

24 records · Page 2Linked to original sources

arXiv2002

Recovering an Algebraic Curve Using its Projections From Different Points. Applications to Static and Dynamic Computational Vision

We study how an irreducible closed algebraic curve X embedded in CP^3 can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP^3 can be recovered modulo some actions of the group of projective transformations of CP^3. For two projections, we show how in a generic situation, a characteristic matrix of the two embeddings can be recovered. We also establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degree and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. Then we consider another problem. N projections, with known projections operators and N >> 1, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in CP^3. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a different point. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve of minimal degree generated by the centers of projection. A second part is devoted to applications to computer vision. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.

math.AG↗

arXiv2001

Graph Theoretic Method for Determining non Hurwitz Equivalence in the Braid Group and Symmetric group

Motivated by the problem of Hurwitz equivalence of $Δ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $Δ^2$ factorizations into $S_n$. We get $1_{S_n}$ factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.

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arXiv2001

Properties of Hurwitz Equivalence in the Braid Group of Order n

In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $Δ_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in \cite{Tz}, using an algebraic approach unlike the proof in \cite{Tz} which is geometric. The application of the result is applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We construct a new invariant based on a Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformations of each other. The precise definition of the new invariant can be found in \cite{KuTe} or \cite{Te}. The main result of this paper will help us to compute the new invariant.

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arXiv2001

Solving the Braid Word Problem Via the Fundamental Group

The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological point of view, and thus using a new computerized representation of some elements of the fundamental group we gave a solution for its word problem. In this paper we will give an algorithm that will make it possible to transform the new presentation into a syntactic presentation. This will make it possible to computerize the group operation to sets of elements of the fundamental group, which are isomorphic to the braid group. More over we will show that it is sufficient enough to look at the syntactic presentation in order to solve the braid word problem, resulting with a better and faster braid word solution.

math.GR↗

arXiv2001

Hurwitz Equivalence in Braid Group B_3

In this paper we prove certain Hurwitz equivalence properties of $B_n$. In particular we prove that for $n=3$ every two Artin's factorizations of $Δ_3 ^2$ of the form $H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}$ (with $i_k, j_k \in \{1,2 \}$) where $\{H_1, H_2 \}, \{F_1, F_2 \}$ are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.

math.AG↗

arXiv1999

Braid Monodromy Factorization and Diffeomorphism Types

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).

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