SYMMETRY DEFECT OF ALGEBRAIC VARIETIES

Author(s)

Janeczko, S. ^{[1, 2]} ; Jelonek, Z. ^[1] ; Ruas, M. A. S. ^[3]

Affiliation

^[1] Polish Acad Sci, Inst Matemat, PL-00956 Warsaw - Poland

^[2] Warsaw Univ Sci & Technol, Wydzial Matemat & Nauk Informacyjnych, PL-00661 Warsaw - Poland

^[3] Univ Sao Paulo, ICMC, Dept Matemat, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: ASIAN JOURNAL OF MATHEMATICS; v. 18, n. 3, p. 525-544, 2014.
• Web of Science Citations: 0

Abstract

Let X, Y subset of k(m) (k = R, C) be smooth manifolds. We investigate the central symmetry of the configuration of X and Y. For p is an element of k(m) we introduce a number mu(p) of pairs of points x is an element of X and y is an element of Y such that p is the center of the interval (xy) over bar. We show that if X, Y (including the case X = Y) are algebraic manifolds in a general position, then there is a closed (semi-algebraic) set B subset of k(m), called symmetry defect set of the X and Y configuration, such that the function mu is locally constant and not identically zero outside B. If k = C, we estimate the number mu (in fact we compute it in many cases) and show that the symmetry defect is an algebraic hypersurface and consequently the function mu is constant and positive outside B. We also show that in the generic case the topological type of the symmetry defect set of a plane curve is constant, i.e. the symmetry defect sets for two generic curves of the same degree are homeomorphic (by the same method we can prove similar statement for any irreducible family of smooth varieties Z(n) subset of C-2n). Moreover, for k = R, we estimate the number of connected components of the set U = k(m) \textbackslash{} B. In the last section we give an algorithm to compute the symmetry defect set for complex smooth affine varieties in general position. (AU)

• FAPESP's process: 08/54222-6 - Singularities, geometry and differential equations
• Grantee: Maria Aparecida Soares Ruas
• Support type: Research Projects - Thematic Grants