Ulrich line bundles on double planes (Q2035817)

The paper under review mostly studies the existence of Ulrich line bundle on a double cover \(\pi: X \to \mathbb{P}^2\) of the projective plane \(\mathbb{P}^2\) branched along a smooth curve \(B\) of degree \(2s\), in the sense of \textit{M. Aprodu} and \textit{Y. Kim} [Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 63, No. 1, 9--23 (2017; Zbl 1397.14054)], or equivalently, in the sense of \(\pi\)-Ulrich line bundles, cf. [\textit{R. Kulkarni}, \textit{Y. Mustopa}, and \textit{I. Shipman}, J. Algebra 474, 166--179 (2017; Zbl 1368.14028)]. It is well-known that such a double cover is a smooth quadric surface when \(s=1\), and is a del Pezzo surface of degree \(2\) if \(s=2\). In both cases, the structure of the Picard group is well-understood, and in particular, there are Ulrich line bundles. On the other hand, when \(s>2\) and \(B\) is sufficiently general, then \(\operatorname{Pic} (X) \simeq \mathbb{Z}\) and such a surface cannot have an Ulrich line bundle (see also Lemma 3.2 and Theorem 1.4). To clarify the existence even when \(B\) is special, the authors provided equivalence conditions to have an Ulrich line bundle on \(X\) with respect to \(\pi\) in terms of ramified curves and of branch curves. In particular, the existence of Ulrich line bundle is obtained when there is a smooth curve \(C\) of degree \(s\) which is tangent to \(B\) of even order at every point of \(C \cap B\) (see Theorem 1.1). Hence, the non-existence of Ulrich line bundle on \(X\) when \(B\) is general implies that there is no such a curve \(C\) if the given branch curve \(B\) is general enough. One may ask its converse: fix a smooth curve \(C\) of degree \(s\) and ask whether there is a smooth curve \(B\) of degree \(2s\) which is tangent to \(C\) of even order at every point of intersection. The answer is positive: see Proposition 5.3. The authors also studied the non-existence of Ulrich line bundles on a smooth \(d\)-fold cyclic cover of \(\mathbb{P}^n\) when \(d \le n\) and \(n \ge 3\): see Theorem 1.7.