  We present a new method to directly map the neutral-hydrogen distribution
during the reionization epoch and to constrain the emission properties of the
highest-redshift quasars (QSOs). As a tracer of HI, we propose to use the
Ly-alpha radiation produced by quasar ionization fronts (I-fronts) that expand
in the partially ionized intergalactic medium (IGM) before reionization is
complete. These Ly-alpha photons are mainly generated by collisional
excitations of hydrogen atoms in the boundary of the rapidly expanding HII
region. The observable signal is produced by the part of the I-front that lies
behind the QSO with respect to the observer. Combining two radiative transfer
models (one for the QSO ionizing radiation and one for the Ly-alpha photons),
we estimate the expected Ly-alpha spectral shape and surface brightness (SB)
for a large number of configurations where we varied both the properties of the
ionizing QSO and of the surrounding medium. We find that the expected signal is
observable as a single (broad) emission line with a characteristic width of
100-200 km/s. The expected SB produced at redshift z~6.5 within a fully neutral
region (at mean density) by a typical QSO I-front lies in the range
$10^{-21}-10^{-20}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ and decreases
proportionally to $(1+z)^2$ for a given QSO age. QSOs with harder spectra may
produce a significantly brighter emission at early phases. The signal may cover
up to a few hundred square arcmin on the sky and should be already detectable
with current facilities by means of moderate/high resolution spectroscopy. The
detection of this Ly-alpha emission can shed new light on the reionization
history, the age and the emission properties of the highest-redshift QSOs.
(abridged)
