Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Explicit Estimates for the Number of Rational Points of Singular
  Complete Intersections over a Finite Field   Let $V\subset\mathbb{P}^n(\overline{F}_{\hskip-0.7mm q})$ be a complete
intersection defined over a finite field $F_{\hskip-0.7mm q}$ of dimension $r$
and singular locus of dimension at most $0\le s\le r-2$. We obtain an explicit
version of the Hooley--Katz estimate $||V(F_{\hskip-0.7mm
q})|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_{\hskip-0.7mm q})|$ denotes
the number of $F_{\hskip-0.7mm q}$-rational points of $V$ and
$p_r:=|\mathbb{P}^r(F_{\hskip-0.7mm q})|$. Our estimate improves all the
previous estimates in several important cases. Our approach relies on tools of
classical algebraic geometry. A crucial ingredient is a new effective version
of the Bertini smoothness theorem, namely an explicit upper bound of the degree
of a proper Zariski closed subset of $(\mathbb
P^{n})^{s+1}(\overline{F}_{\hskip-0.7mm q})$ which contains all the singular
linear sections of $V$ of codimension $s+1$.

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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom