Determinantal representations of singular hypersurfaces in Pn

A (global) determinantal representation of projective hypersurface X Pn is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for singular (possibly reducible or non-reduced) hypersurfaces. In particular, we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces. Further, we classify the determinantal representations in terms of the corresponding kernel sheaves on X. Finally, we extend the results to the case of symmetric/self-adjoint representations, with implications to hyperbolic polynomials and the generalized Lax conjecture.