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We present a new method to directly map the neutral-hydrogen distributionduring the reionization epoch and to constrain the emission properties of thehighest-redshift quasars (QSOs). As a tracer of HI, we propose to use theLy-alpha radiation produced by quasar ionization fronts (I-fronts) that expandin the partially ionized intergalactic medium (IGM) before reionization iscomplete. These Ly-alpha photons are mainly generated by collisionalexcitations of hydrogen atoms in the boundary of the rapidly expanding HIIregion. The observable signal is produced by the part of the I-front that liesbehind the QSO with respect to the observer. Combining two radiative transfermodels (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 theionizing QSO and of the surrounding medium. We find that the expected signal isobservable as a single (broad) emission line with a characteristic width of100-200 km/s. The expected SB produced at redshift z~6.5 within a fully neutralregion (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 decreasesproportionally to $(1+z)^2$ for a given QSO age. QSOs with harder spectra mayproduce a significantly brighter emission at early phases. The signal may coverup to a few hundred square arcmin on the sky and should be already detectablewith current facilities by means of moderate/high resolution spectroscopy. Thedetection of this Ly-alpha emission can shed new light on the reionizationhistory, the age and the emission properties of the highest-redshift QSOs.(abridged)

Mapping Neutral Hydrogen During Reionization with the Ly-alpha Emission from Quasar Ionization Fronts