On the singular structure of graph hypersurfaces

I show that the singular loci of graph hypersurfaces correspond set-theoretically to their rank loci. The proof holds for all configuration hypersurfaces and depends only on linear algebra. To make the conclusion for the second graph hypersurface, I prove that the second graph polynomial is a configuration polynomial. The result indicates that there may be a fruitful interplay between the current research in graph hypersurfaces and Stratified Morse Theory.

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Published 1 January 2010