Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Twelve points on the projective line, branched covers, and rational
  elliptic fibrations   The following divisors in the space Sym^{12} P^1 of twelve points on P^1 are
actually the same: (A) the possible locus of the twelve nodal fibers in a
rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree
12 binary forms that can be expressed as a cube plus a square; (C) the locus of
the twelve tangents to a smooth plane quartic from a general point of the
plane; (D) the branch locus of a degree 4 map from a hyperelliptic genus 3
curve to P^1; (E) the branch locus of a degree 3 map from a genus 4 curve to
P^1 induced by a theta-characteristic; and several more.
  The corresponding moduli spaces are smooth, but they are not all isomorphic;
some are finite etale covers of others. We describe the web of interconnections
between these spaces, and give monodromy, rationality, and Prym-related
consequences. Enumerative consequences include: (i) the degree of this locus is
3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11
given general points of the base); (ii) if C --> P^1 is a cover as in (D), then
there are 135 different such covers branched at the same points; (iii) the
general set of 12 tangent lines that arises in (C) turns up in 120 essentially
different ways.
  Some parts of this story are well-known, and some other parts were known
classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified
picture is surprisingly intricate and connects many beautiful constructions,
including Recillas' trigonal construction and Shioda's E8-Mordell-Weil lattice.

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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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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: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga