  We present numerical results for the location of the chiral critical line at
finite temperature and zero and non-zero baryon density for QCD with N_f=2+1
flavours of staggered fermions on lattices with temporal extent N_t=4. For
degenerate quark masses, we compare our results obtained with the exact RHMC
algorithm with earlier, inexact R-algorithm results and find a reduction of 25%
in the critical quark mass, for which the first order phase transition changes
to a smooth crossover. Extending our analysis to non-degenerate quark masses,
we map out the chiral critical line up to the neighbourhood of the physical
point, which we confirm to be in the crossover region. Our data are consistent
with a tricritical point at a strange quark mass of ~500 MeV. Finally, we
investigate the shift of the critical line with finite baryon density, by
simulating with an imaginary chemical potential for which there is no sign
problem. We observe this shift to be very small or, conversely, the critical
endpoint \mu^c(m_{u,d},m_s) to be extremely quark mass sensitive. Moreover, the
sign of this shift is opposite to standard expectations. If confirmed on a
finer lattice, it implies the absence of a critical endpoint or phase
transition for chemical potentials \mu_B < 500 MeV. We thus argue that finer
lattices are required to settle even the qualitative features of the QCD phase
diagram.
