  We use numerical N-body hydrodynamical simulations with varying PopIII
stellar models to investigate the possibility of detecting first star
signatures with observations of high-redshift damped Ly$\alpha$ absorbers
(DLAs). The simulations include atomic and molecular cooling, star formation,
energy feedback and metal spreading due to the evolution of stars with a range
of masses and metallicities. Different initial mass functions (IMFs) and
corresponding metal-dependent yields and lifetimes are adopted to model
primordial stellar populations. The DLAs in the simulations are selected
according to either the local gas temperature (temperature selected) or the
host mass (mass selected). We find that 3\% (40\%) of mass (temperature)
selected high-$z$ ($z\ge5.5$) DLAs retain signatures of pollution from PopIII
stars, independently from the first star model. Such DLAs have low halo mass
($<10^{9.6}\,\rm M_{\odot}$), metallicity ($<10^{-3}\,\rm Z_{\odot}$) and star
formation rate ($<10^{-1.5}\,\rm M_{\odot}\,yr^{-1}$). { Metal abundance ratios
of DLAs imprinted in the spectra of QSO} can be useful tools to infer the
properties of the polluting stellar generation and to constrain the first star
mass ranges. Comparing the abundance ratios derived from our simulations to
those observed in DLAs at $z\ge5$, we find that most of these DLAs are
consistent within errors with PopII stars dominated enrichment and strongly
disfavor the pollution pattern of very massive first stars (i.e. 100~$\rm
M_{\odot}$-500~$\rm M_{\odot}$). However, some of them could still result from
the pollution of first stars in the mass range [0.1, 100]~$\rm M_{\odot}$. In
particular, we find that the abundance ratios from SDSS J1202+3235 are
consistent with those expected from PopIII enrichment dominated by massive (but
not extreme) first stars.
