Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hyperk\"ahler fourfolds and Kummer surfaces   We show that a Hilbert scheme of conics on a Fano fourfold double cover of
$\mathbb{P}^2\times\mathbb{P}^2$ ramified along a divisor of bidegree $(2,2)$
admits a $\mathbb{P}^1$-fibration with base being a hyper-K\"{a}hler fourfold.
We investigate the geometry of such fourfolds relating them with degenerated
EPW cubes, with elements in the Brauer groups of $K3$ surfaces of degree $2$,
and with Verra threefolds studied in [Ver04]. These hyper-K\"{a}hler fourfolds
admit natural involutions and complete the classification of geometric
realizations of anti-symplectic involutions on hyper-K\"{a}hler $4$-folds of
type $K3^{[2]}$.
  As a consequence we present also three constructions of quartic Kummer
surfaces in $\mathbb{P}^3$: as Lagrangian and symmetric degeneracy loci and as
the base of a fibration of conics in certain threefold quadric bundles over
$\mathbb{P}^1$.

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The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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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: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg