  The space of degree d single-variable monic and centered complex polynomial
vector fields can be decomposed into loci in which the vector fields have the
same topological structure. We analyze the geometric structure of these loci
and describe some bifurcations, in particular, it is proved that new homoclinic
separatrices can form under small perturbation. By an example, we show that
this decomposition of parameter space by combinatorial data is not a cell
decomposition. The appendix to this article, joint work with Tan Lei, shows
that landing separatrices are stable under small perturbation of the vector
field if the multiplicities of the equilibrium points are preserved.
