Structure of the outer Galactic disc with Gaia-DR2
  AIMS. We calculate the stellar density using star counts obtained from Gaia
DR2 up to a Galactocentric distance R=20 kpc with a deconvolution technique for
the parallax errors. Then we analyse the density in order to study the
structure of the outer Galactic disc, mainly the warp.
  METHODS. In order to carry out the deconvolution, we used the Lucy inversion
technique for recovering the corrected star counts. We also used the Gaia
luminosity function of stars with $M_G<10$ to extract the stellar density from
the star counts.
  RESULTS. The stellar density maps can be fitted by an exponential disc in the
radial direction $h_r=2.07\pm0.07$ kpc, with a weak dependence on the azimuth,
extended up to 20 kpc without any cut-off. The flare and warp are clearly
visible. The best fit of a symmetrical S-shaped warp gives $z_w= z_\odot+(37\pm
4.2(stat.)-0.91(syst.))$ pc $(R/R_\odot )^{2.42\pm 0.76(stat.) + 0.129 (syst.)}
sin(\phi+9.3\pm 7.37 (stat.) +4.48 (syst.))$ for the whole population. When we
analyse the northern and southern warps separately, we obtain an asymmetry of
an $\sim25\%$ larger amplitude in the north. This result may be influenced by
extinction because the Gaia G band is quite prone to extinction biases.
However, we tested the accuracy of the extinction map we used, which shows that
the extinction is determined very well in the outer disc. Nevertheless, we
recall that we do not know the full extinction error, and neither do we know
the systematic error of the map, which may influence the final result.
  The analysis was also carried out for very luminous stars alone ($M_G<-2$),
which on average represents a younger population. We obtain similar
scale-length values, while the maximum amplitude of the warp is $20-30\%$
larger than with the whole population. The north-south asymmetry is maintained.
