Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Is a linear space contained in a variety? - On the number of derivatives
  needed to tell   Let $X^n\subset C^{n+a}$ or $X^n\subset P^{n+a}$ be a patch of an analytic
submanifold of an affine or projective space, let $x\in X$ be a general point,
and let L^k be a linear space of dimension k osculating to order m at x. If m
is large enough, one expects L to be contained in X and thus X contains a
linear space of dimension kthrough almost every point. We show that $L\subset
X$ in the following cases: k=1 and m=n+1; k=n-1, $a\geq 2$, and m=2; $n\geq 4$,
k=n-2 and m=4. We prove these results by first deriving the order of osculation
that generically implies containment and then showing that in these cases
containment must occur.
  If X is a patch of a projective variety, we address the question as to
whether X can be a smooth variety. We show that if there is a P^k through each
point and $codim(X)<\frac{k}{n-k}$ then X cannot be a smooth variety.

1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin