2024/04/29 by L. R. G. Dias, Dias, L. R. G., Z. Jelonek +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #Digital Image Processing Techniques #FOS: Mathematics #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2404.18927
openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X, Y ⊂ ℂ2n-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⊂ ℂ2n-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.