On the Determinantal Representations of Singular Plane Curves

DMITRY KERNER AND VICTOR VINNIKOV Abstract. Let M be a d×d matrix whose entries are linear forms in the homogeneous coordinates of P2. Then M is called a determinantal representation of the curve {det(M) = 0}. Such representations are well studied for smooth curves. We study determinantal representations of curves with arbitrary singularities (mostly reduced). The kernel of M defines a torsion free sheaf on the curve. We classify torsion free sheaves arising as kernels of determinantal representations and study their properties. Further, we study the local version of the problem: families of square matrices, depending on two parameters. In particular we give various criteria of local decomposability of M into a direct sum of determinantal representations.