We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $Res(f, g) = 0 \subset \mathbb P^d \times \mathbb P^e"$ where the corresponding degree $d, e$ polynomials have a common root. In particular, I am interested in doing this in all characteristics and ideally, I would like to show that there exists a resolution where the exceptional divisors have an affine stratification (or something close to it).
More generally, I would really like to do this for the general problem of the resultant locus in $\mathbb P^{n_1}\times\dots\mathbb P^{n_r}$ which corresponds to any two of the $r$ polynomials having a common root.
The degree one case of the general problem is solved by exactly the Fulton-Macpherson compactification of configuration space and I suspect the higher degree version will be some combination of this with the case corresponding to having just two polynomials in some way.
This locus also happens to be a determinantal locus and I know there is a lot known about that subject but I know none of it!
