On the Hilbert Function of Intersections of a Hypersurface with General Reducible Curves

Abstract

Let $W \subset \mathbb P^n, n \ge 3$, be a degree k hypersurface. Consider a "general" nodal union of $d$ lines $L_1, \dots L_d$ with $L_i \cap L_j \neq \emptyset$ if and only if $|i - j| \le 1$ (here called a degree $d$ bamboo). We study the Hilbert function of the set $Y \cap W$ with cardinality $k \deg(Y)$ and prove that it is the expected one (with a few classified exceptions $(n,k,d)$) when $W$ is either a quadric hypersurface of rank at least $2$ or $n = 3$ and $W$ is an integral cubic surface.