Curves arising from Kronecker modules Original Research Article

Let K be an algebraically closed field. A Kronecker module M is a pair of K-vector spaces (S,T) together with a K-bilinear map K2×S→T. The space S is called the domain space of M, while T is called the range space of M. To each power series α in K[[X]] we attach a Kronecker module Pα whose domain and range spaces are denoted by V− and V, respectively. Both V− and V are modules over the endomorphism algebra End Pα of Pα. We show that if End Pα is non-trivial, then the sequence of coefficients of α is defined by a linear or a quadratic recursion. In the quadratic case End Pα is the coordinate ring of an affine curve. An affine curve is called realizable when its coordinate ring is isomorphic to some End Pα. We show that the realizable curves can be constructed, up to birational equivalence, by pairs of non-zero polynomials (p,q) with degq