Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Negative curves on algebraic surfaces   We study curves of negative self-intersection on algebraic surfaces. We
obtain results for smooth complex projective surfaces X on the number of
reduced, irreducible curves C of negative self-intersection C^2. The only known
examples of surfaces for which C^2 is not bounded below are in positive
characteristic, and the general expectation is that no examples can arise over
the complex numbers. Indeed, we show that the idea underlying the examples in
positive characteristic cannot produce examples over the complex number field.
The previous version of this paper claimed to give a counterexample to the
Bounded Negativity Conjecture. The idea of the counterexample was to use Hecke
translates of a smooth Shimura curve in order to create an infinite sequence of
curves violating the Bounded Negativity Conjecture. To this end we applied
Hirzebruch Proportionality to all Hecke translates, simultaneously
desingularized by a version of Jaffee's Lemma which exists in the literature
but which turns out to be false. Indeed, in the new version of the paper, we
show that only finitely many Hecke translates of a special subvariety of a
Hilbert modular surface remain smooth. This new result is based on work done
jointly with Xavier Roulleau, who has been added as an author. The other
results in the original posting of this paper remain unchanged.

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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg