LAUSR

In this note we classify arithmetically Cohen–Macaulay line bundles on a given complex polarized elliptic ruled surface; for each numerical class, we consider only tensor products with the pullback of a general line bundle on the base elliptic curve. In this context, the condition defining an initialized (or Ulrich) bundle will take a particular form. This will lead to an existence result which we then compare to the already known situation for Ulrich line bundles in the case when $$e > 0$$ e > 0 , by focusing on the condition on the coefficient $$\alpha $$ α of the minimal section in the class of the polarization. The notation employed here is explained in the preliminaries.