The Determinantal Representations and the Joint Characteristic Functions in the Case of Real Smooth Cubics

Abstract

Since the inner product (the pairing) (10.152–10.153) on the line bundle corresponding to a selfadjoint determinantal representation of a real smooth plane curve C of degree n and genus g = (n - 1)(n - 2)/2 is not given by a straightforward analytic formula, Theorem 10.5.7 is not quite an analog of Theorem 10.2.1. Furthermore, it is hard to envisage for a mapping of line bundles on C a factorization theorem such as Theorems 10.2.2–10.2.3. However, the line bundle corresponding to a selfadjoint determinantal representation of C is isomorphic, up to a certain fixed twist, to a unitary flat line bundle on C corresponding to some point in the Jacobian variety of C. Under this isomorphism the pairing (10.152–10.153) can be expressed analytically (in general, in terms of theta functions), and the joint characteristic function can be identified with a (scalar) multivalued multiplicative function on C, with some multipliers of absolute value 1, satisfying certain expansivity conditions. This way we arrive to the notion of the normalized joint characteristic function, which is the correct analog for commutative (two-operator) vessels of the notion of the characteristic function of a single-operator colligation. Furthermore, the output determinantal representation of a commutative vessel can be computed explicitly from the input determinantal representation and the normalized joint characteristic function — a fact of central importance for the construction of triangular models.