Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stable pairs and log flips   This paper has two parts. In the first part, we review stable pairs and
triples on curves, leading up to Thaddeus' diagram of flips and contractions
starting from the blow-up of projective space along a curve embedded by a
complete linear series of the form K + ample. In the second part, we identify
log canonical divisors which exhibit Thaddeus' flips and contractions as "log"
flips and contractions in the sense of the log-minimal-model program.

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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph