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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!

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    $\begingroup$ The resultant varieties have a natural conical stratification (by the length of the base locus of the system of polynomials). There is an associated resolution algorithm to produce a minimal wonderful resolution in the article "Making conical compactifications wonderful" by MacPherson and Procesi (following earlier work by de Concini and Procesi). $\endgroup$ Feb 7, 2023 at 12:51
  • $\begingroup$ Thank you! This is great. After a short skim, I believe their procedure works unmodified in any characteristic - is that right? And if so, do you know if there is a reference for the positive characteristic version since that is my real interest? I would prefer to cite something rather than simply say "the construction can be modified to work in positive characteristic without difficulty". $\endgroup$
    – Asvin
    Feb 7, 2023 at 16:31
  • $\begingroup$ Hi Jason, Sorry to come back to this after so long. But I couldn't actually figure out how to prove that the resultant varieties have a conical stratification (even in characteristic $0$). Could you explain more or point me to a reference? $\endgroup$
    – Asvin
    Mar 26, 2023 at 3:49

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