Complex Links and Hilbert-Samuel Multiplicities

Abstract

We describe a framework for estimating Hilbert-Samuel multiplicities $e_{X}Y$ 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 $e_{p}C$ 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.

16 pages, 7 figures. This is a major revision. There is a problem with the Lefschetz theorem from Sec 6 of v1: it applies not to the usual complex linking space but to a projective version thereof (which has a very different topology in general). We have removed the old Sec 6 and replaced it with a new theorem on inferring multiplicities from point samples with high confidence