The Dual Variety of a Linear Determinantal Hypersurface

This chapter, although of a marked geometric nature, is firmly grounded on previous chapters. It gives a good grip on the dimension of the dual variety to the determinantal hypersurfaces of many of the linear sections considered in previous chapters. It also gives a notion of the structure of the homogeneous defining ideal of the dual variety in its natural embedding – the latter by no means being a trivial issue as the dual variety is most of the times deficient. By and large, this discussion calls upon the structure of certain ladder ideals, the question of the generating degrees of the defining ideal standing up. As discussed in a previous chapter, an interesting question in general is whether the defining polynomial is a factor of its Hessian determinant with the expected multiplicity (according to Segre).