Determinantal representations of singular hypersurfaces in P^n

TL;DR

This paper investigates determinantal representations of singular hypersurfaces in projective space, providing criteria for decomposability, classification via kernel sheaves, and extensions to symmetric cases with implications for hyperbolic polynomials.

Contribution

It introduces decomposability criteria for reducible hypersurfaces, classifies representations through kernel sheaves, and extends results to symmetric/self-adjoint cases related to hyperbolic polynomials.

Findings

Decomposability criteria for reducible hypersurfaces

Classification of representations via kernel sheaves

Extension to symmetric/self-adjoint representations

Abstract

A (global) determinantal representation of hypersurface in P^n 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 generalized Lax conjecture.