Degenerating Discriminants

We study the behavior of projective dual varieties and discriminants under flat degenerations.

For Gröbner degenerations, we show that the conormal variety admits a Gröbner degeneration with opposite weights on the dual coordinates.

Using Whitney stratifications and Sabbah's formula, we describe the irreducible components and multiplicities of the special fiber of this degeneration, and hence of the limiting dual hypersurface.

We then extend these results to higher associated hypersurfaces in Grassmannians.

As applications, we recover classical formulas for hypersurfaces with isolated singularities, analyze degenerations of generic complete intersections and reciprocal linear spaces, and relate the theory to mixed discriminants of parametrized polynomial systems.