Ulrich bundles on non-special surfaces with $$p_g=0$$ p g = 0 and $$q=1$$ q = 1

Abstract

Let S be a surface with \(p_g(S)=0\), \(q(S)=1\) and endowed with a very ample line bundle \({\mathcal {O}}_S(h)\) such that \(h^1\big (S,{\mathcal {O}}_S(h)\big )=0\). We show that such an S supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p. Moreover, we show that S supports stable Ulrich bundles of rank 2 if the genus of the general element in \(\vert h\vert \) is at least 2.