Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the numerical dimension of pseudo-effective divisors in positive
  characteristic   Let X be a smooth projective variety over an algebraically closed field of
positive characteristic. We prove that if D is a pseudo-effective R-divisor on
X which is not numerically equivalent to the negative part in its divisorial
Zariski decomposition, then the numerical dimension of D is positive. In
characteristic zero, this was proved by Nakayama using vanishing theorems.

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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an