  We present a renormalization group (RG) analysis of a fermionic "hot spot"
model of interacting electrons on the square lattice. We truncate the Fermi
surface excitations to linearly dispersing quasiparticles in the vicinity of
eight hot spots on the Fermi surface, with each hot spot separated from another
by the wavevector $(\pi, \pi)$. This motivated by the importance of these Fermi
surface locations to the onset of antiferromagnetic order; however, we allow
for all possible quartic interactions between the fermions, and also for all
possible ordering instabilities. We compute the RG equations for our model,
which depend on whether the hot spots are perfectly nested or not, and relate
our results to earlier models. We also compute the RG flow of the relevant
order parameters for both Hubbard and $J$, $V$ interactions, and present our
results for the dominant instabilities in the nested and non-nested cases. In
particular, we find that non-nested hot spots with $J$, $V$ interactions have
competing singlet $d_{x^2-y^2}$ superconducting and $d$-form factor
incommensurate density wave instabilities. We also investigate the enhancement
of incommensurate density waves near experimentally observed wavevectors, and
find dominant $d$-form factor enhancement for a range of couplings.
