Refined intersection products and limiting linear subspaces of hypersurfaces

Abstract

Let $X$ be a hypersurface of degree $d$ in $\Bbb P^n$ and $F_X$ be the scheme of $\Bbb P^r$'s contained in $X$. If $X$ is generic, then $F_X$ will have the expected dimension (or empty) and its class in the Chow ring of $G(r+1,n+1)$ is given by the top Chern class of the vector bundle $S^dU^*$, where $U$ is the universal subbundle on the Grassmannian $G(r+1,n+1)$. When we deform a generic $X$ into a degenerate $X_0$, the dimension of $F_X$ can jump. In this case, there is a subscheme $F_{lim}$ of $F_{X_0}$ with the expected dimension which consists of limiting $\Bbb P^r$'s in $X_0$ with respect to a general deformation. The simplest example is the well-known case of $27$ lines in a generic cubic surface. If we degenerate the surface into the union of a plane and a quadric, then there are infinitely many lines in the union. Which $27$ lines are the limiting ones and how many of them are in the plane and how many of them are in the quadric? The goal of this paper is to study $F_{lim}$ in general.