Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds on the Torsion Subgroups of N\'eron-Severi Group Schemes   Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined
by homogeneous polynomials of degree $\leq d$ over an algebraically closed
field. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$. Let $\mathbf{Pic}^0
X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group
scheme of $X$ is defined by $\mathbf{NS}\, X = (\mathbf{Pic}\,
X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$. We give an explicit upper bound on the
order of the finite group scheme $(\mathbf{NS}\, X)_{{\mathrm{tor}}}$ in terms
of $d$ and $r$. As a corollary, we give an upper bound on the order of the
finite group $\pi^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also
show that the torsion subgroup $(\mathrm{NS}\, X)_{\mathrm{tor}}$ of the
N\'eron--Severi group of $X$ is generated by less than or equal to
$(\mathrm{deg}\, X -1)(\mathrm{deg}\, X - 2)$ elements in various situations.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph