  We quantify the impact of massive neutrinos on the statistics of low density
regions in the intergalactic medium (IGM) as probed by the Lyman-alpha forest
at redshifts z=2.2--4. Based on mock but realistic quasar (QSO) spectra
extracted from hydrodynamic simulations with cold dark matter, baryons and
neutrinos, we find that the probability distribution of weak Lyman-alpha
absorption features, as sampled by Lyman-alpha flux regions at high
transmissivity, is strongly affected by the presence of massive neutrinos. We
show that systematic errors affecting the Lyman-alpha forest reduce but do not
erase the neutrino signal. Using the Fisher matrix formalism, we conclude that
the sum of the neutrino masses can be measured, using the method proposed in
this paper, with a precision smaller than 0.4 eV using a catalog of 200 high
resolution (S/N~100) QSO spectra. This number reduces to 0.27 eV by making use
of reasonable priors in the other parameters that also affect the statistics of
the high transitivity regions of the Lyman-alpha forest. The constraints
obtained with this method can be combined with independent bounds from the CMB,
large scale structures and measurements of the matter power spectrum from the
Lyman-alpha forest to produce tighter upper limits on the sum of the masses of
the neutrinos.
