  The signature left in quasar spectra by the presence of neutral hydrogen in
the Universe allows one to constrain the sum of the neutrino masses with
improved sensitivity, with respect to laboratory experiments, and may shed a
new light on the neutrino mass hierarchy and on the absolute mass scale of
neutrinos. Constraints on cosmological parameters and on the dark energy
equation of state can also be derived, from a joint parameter estimation
procedure. However, this requires a detailed modeling of the line-of-sight
power spectrum of the transmitted flux in the Lyman-Alpha (LyA) forest on
scales ranging from a few to hundreds of Mpcs, which in turns demands the
inclusion and careful treatment of cosmological neutrinos. To this end, we
present here a suite of state-of-the-art hydrodynamical simulations with cold
dark matter, baryons and massive neutrinos, specifically targeted for modeling
the low-density regions of the IGM as probed by the LyA forest at
high-redshift. The simulations span volumes ranging from (25 Mpc/h)^3 to (100
Mpc/h)^3, and are made using either 3 X 192^3~21 millions or 3 X 768^3~1.4
billion particles. The resolution of the various runs can be further enhanced,
so that we can reach the equivalent of 3 X 3072^3~87 billion particles in a
(100 Mpc/h)^3 box size. The chosen cosmological parameters are compatible with
the latest Planck (2013) results, although we also explore the effect of slight
variations in the main cosmological and astrophysical parameters. We adopt a
particle-type implementation of massive neutrinos, and consider three
degenerate species having masses M_nu =0.1, 0.2, 0.3, 0.4 and 0.8 eV,
respectively. We improve on previous studies in several ways, in particular
with updated routines for IGM radiative cooling and heating processes, and
initial conditions based on 2LPT rather than the Zeldovich approximation.
