Complex Links and Hilbert-Samuel Multiplicities

TL;DR

This paper introduces a novel framework for estimating Hilbert-Samuel multiplicities of projective variety pairs from finite point samples, linking algebraic invariants to topological data analysis.

Contribution

It establishes invariance of multiplicities under hyperplane sections and connects them to Euler characteristics of complex links, providing a sampling-based estimation method.

Findings

Proves invariance of multiplicities under hyperplane sections.

Relates multiplicities to Euler characteristics of complex links.

Provides bounds on sample size for high-confidence estimation.

Abstract

We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ 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 $e_pC$ 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.