  We study a model of directed polymers in random environment in dimension
$1+d$, given by a Brownian motion in a Poissonian potential. We study the
effect of the density and the strength of inhomogeneities, respectively the
intensity parameter $\nu$ of the Poisson field and the temperature inverse
$\beta$. Our results are: (i) fine information on the phase diagram, with
quantitative estimates on the critical curve; (ii) pathwise localization at low
temperature and/or large density; (iii) complete localization in a favourite
corridor for large $\nu \beta^2$ and bounded $\beta$.
