Abstrakt: |
Let X , Y ⊂ C 2 n - 1 be n-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point x ∈ X to a point y ∈ Y . This set is defined as the image of the map Φ (x , y) = x + y 2. Under geometric conditions on X and Y, we prove that the symmetry defect of X and Y, which is the bifurcation set B(X, Y) of the mapping Φ , is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set B(X, Y) and we present an estimate for its degree. Moreover, for any two n-dimensional strong complete intersections X , Y ⊂ C 2 n - 1 (including the case X = Y ) we introduce a generic symmetry defect set B ~ (X , Y) of X and Y, which is defined up to homeomorphism. The set B ~ (X , Y) is an algebraic variety. Finally we show that in the real case if X, Y are compact, then the set B ~ (X , Y) is a hypersurface and it has only Thom-Boardman singularities. In particular if X is compact, then B ~ (X) is a hypersurface, which has only Thom-Boardman singularities. [ABSTRACT FROM AUTHOR] |