  Incoherent pion photoproduction on the deuteron is studied in the first
resonance region. The unpolarized cross section, the beam asymmetry, and the
vector and tensor target asymmetries are calculated in the framework of a
diagrammatic approach. Pole diagrams and one-loop diagrams with $NN$ scattering
in the final state are taken into account. An elementary operator for pion
photoproduction on the nucleon is taken in various on-shell forms and
calculated using the SAID and MAID multipole analyses. Model dependence of the
obtained results is discussed in some detail. A comparison with predictions of
other works is given. Although a reasonable description of many available
experimental data on the unpolarized total and differential cross sections and
photon asymmetry has been achieved, in some cases a significant disagreement
between the theory and experiment has been found. Invoking known information on
the reactions $\gamma d\to\pi^0 d$ and $\gamma d\to np$ we predict the total
photoabsorption cross section for deuterium. We find that our values strongly
overestimate experimental data in the vicinity of the $\Delta$ peak.
