Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bertini and his two fundamental theorems   After reviewing Bertini's life story, a fascinating drama, we make a critical
examination of the old statements and proofs of Bertini's two fundamental
theorems, the theorem on variable singular points and the theorem on reducible
linear systems. We explain the content of the statements in a way that is
accessible to a nonspecialist, and we develop versions of the old proofs that
are complete and rigorous by current standards. In particular, we prove a new
extension of Bertini's first theorem, which treats variable $r$-fold points for
any $r$.

6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess