Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Collections of hypersurfaces containing a curve   We consider the closed locus of $r$-tuples of hypersurfaces in $\mathbb{P}^r$
with positive dimensional intersection, and show in a large range of degrees
that its largest component is the locus of $r$-tuples of hypersurfaces whose
intersection contains a line. We then apply our methods to obtain new results
on the largest components of the locus of hypersurfaces with positive
dimensional singular loci and the locus of smooth hypersurfaces with a higher
dimensional family of lines through a point than expected.

33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th