whitneyStratify(I)
For a affine or projective complex variety $X$ this command computes a Whitney stratification WS where WS#i is a list of strata closures in dimension $i$; for a prime ideal $P$ in WS#i the associated open (connected) strata is given by $V(P)-Z$ where $Z$ is the union of the varieties defined by the entries of WS#(i-1). We demonstrate the method for the affine Whitney umbrella and its projective closure below.
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Now the projective version.
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Another projective example.
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Note that as with the Whitney umbrella, simply taking successive singular loci will not yield the correct stratification, in particular one would miss the two points defined by the second entry of V#0.
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The object whitneyStratify is a method function.