The realization of hyperelliptic curves through endomorphisms of Kronecker modules

Let K be an algebraically closed field and A the Kronecker algebra over K. A general problem is to study the endomorphism algebras of A-modules M that are extensions of finite-dimensional, torsion-free, rank-one A-modules P, by infinite-dimensional, torsion-free, rank-one A-modules N. Such endomorphism algebras can be studied by means of a quadratic polynomial f(Y) in one variable Y over the rational function field K(X). We call this f(Y) the regulator of the extension. We prove that if the regulator has non-zero discriminant, then is a Noetherian, commutative K-algebra. We also prove that, subject to a regulator with non-zero discriminant, is affine over K if and only if End N is affine, in which case is the coordinate ring of a hyperelliptic curve.