Complex links and Hilbert-Samuel multiplicities

Journal article

We describe a framework for estimating Hilbert–Samuel multiplicities eXY for pairs of projective varieties X⊂Y from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce X to a point p and Y to a curve C. Next, we establish that epC equals the Euler characteristic (and hence the cardinality) of the complex link of p in C. Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of p in C) to determine this Euler characteristic with high confidence.

Authors: Helmer, M; Nanda, V

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Applied Algebra and Geometry
• Volume: 7
• Issue: 1
• Pages: 29-48
• Publication date: 2023-03-10
• Acceptance date: 2022-11-10
• DOI: 10.1137/22M1475533
• EISSN: 2470-6566
• Language: English
• Keywords: Hilbert–Samuel multiplicity; complex links; intersection multiplicity; stratified Morse theory; FFR