Geometric Oversight

In this chapter we give an overview of the main geometric players related to a projective hypersurface, among them the polar map and image, the gradient ideal and the Hessian matrix and its determinant. A discussion is enticed about the rank of the Jacobian matrix of a set of forms in a polynomial ring over a field, and a characteristic-free proof is given of the rank of the Grassmann Jacobian (i.e., the Jacobian matrix of the maximal minors of a generic matrix over a field). Complete coverage delivers the main properties of the determinant of the generic square matrix and the generic symmetric matrix over a field of characteristic \(\neq 2\) . Some consideration is given to the question as to when a projective hypersurface is defined by the determinant of a matrix of linear entries and how the algebraic features of this matrix as the ones in the book may reflect back into nontrivial traits of the hypersurface.