Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relative Prym varieties associated to the double cover of an Enriques
  surface   Given an Enriques surface $T$, its universal K3 cover $f: S\to T$, and a
genus $g$ linear system $|C|$ on $T$, we construct the relative Prym variety
$P_H=\Prym_{v, H}(\D/\CC)$, where $\CC\to |C|$ and $\D\to |f^*C|$ are the
universal families, $v$ is the Mukai vector $(0,[D], 2-2g)$ and $H$ is a
polarization on $S$. The relative Prym variety is a $(2g-2)$-dimensional
possibly singular variety, whose smooth locus is endowed with a hyperk\"ahler
structure. This variety is constructed as the closure of the fixed locus of a
symplectic birational involution defined on the moduli space $M_{v,H}(S)$.
There is a natural Lagrangian fibration $\eta: P_H \to |C|$, that makes the
regular locus of $P_H$ into an integrable system whose general fiber is a
$(g-1)$-dimensional (principally polarized) Prym variety, which in most cases
is not the Jacobian of a curve. We prove that if $|C|$ is a hyperelliptic
linear system, then $P_H$ admits a symplectic resolution which is birational to
a hyperk\"ahler manifold of K3$^{[g-1]}$-type, while if $|C|$ is not
hyperelliptic, then $P_H$ admits no symplectic resolution. We also prove that
any resolution of $P_H$ is simply connected and, when $g$ is odd, any
resolution of $P_H$ has $h^{2,0}$-Hodge number equal to one.

14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas