Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Refined intersection products and limiting linear subspaces of
  hypersurfaces   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.

8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex