Update of the direct detection of dark matter and the role of the nuclear spin

We update our exploration of the MSSM parameter space at the weak scale where new accelerator and cosmological constraints are respected. The dependence of WIMP-nucleon cross sections on parameters of the MSSM, uncertainties of the nucleon structure and other theoretical assumptions like universality and co-annihilation are considered. In particular, we find that the coannihilation does not have a significant effect in our analysis in certain regions which are allowed even with coannihilation. The new cosmological constraint on the relic neutralino density used in the form $0.1 < \Omega_\chi h^2_0<0.3$ does also not significantly affect the regions of allowed neutralino-nucleon cross sections. We notice that for nuclear targets with spin the spin-dependent interaction may determine the lower bound for the direct detection rate when the cross section of the scalar interaction drops below about $10^{-12} $pb.