Short sections of some bundles on projective line (Q2122476)

The paper under review studies nondegenerate sections of rank two vector bundles on the projective line \(\mathbb{P}^1_{\mathbb{Z}}\). Given a rank two vector bundle \(\mathcal{E}\) over \(\mathbb{P}^1_{\mathbb{Z}}\) with trivial generical fiber, it is well-known, thanks to Grothendieck's theorem, that \(\mathcal{E}_p\cong\mathcal{O}(-d_p)\oplus\mathcal{O}(d_p)\) with \(d_p\geq 0\) for any closed point \(p\in \mathrm{Spec}(\mathbb{Z})\). A bundle \(\mathcal{E}\) has simple jumps if \(d_p\leq 1\) for any \(p\in \mathrm{Spec}(\mathbb{Z})\) and there exists at least one \(p\) for which \(d_p=1\). On the other hand, a \(d\)-section \(s\) of \(\mathcal{E}\) is a map \(s:\mathcal{O}\rightarrow \mathcal{E}(d)\). The section \(s\) is nondegenerate when it has no zeros on \(\mathbb{P}^1_{\mathbb{Z}}\). By a theorem of \textit{C. C. Hanna} [J. Algebra 52, 322--327 (1978; Zbl 0386.18008)], for any vector bundle \(\mathcal{E}\) there exists \(d\) such that \(\mathcal{E}\) has a nondegenerate section. Therefore, the natural question is to find the minimum \(d\) for a fixed vector bundle (namely, finding ``short sections'' in the authors' terminology). The first author [J. Math. Sci., New York 232, No. 5, 721--731 (2018; Zbl 1454.14097); translation from Zap. Nauchn. Semin. POMI 452, 202--217 (2016)] studied the short sections of vector bundles with simple jumps. In particular, they have a section of degree at most 2. In this paper the authors start the study of short sections for some bundles with height at most 2 (namely, \(d_p\leq 2\) for any \(p\in \mathrm{Spec}(\mathbb{Z})\) and there exists at least one \(p\) for which \(d_p=2\)). They are constructed as the pullback of bundles with simple jumps by the squaring map \(\psi:\mathbb{P}^1_{\mathbb{Z}}\rightarrow\mathbb{P}^1_{\mathbb{Z}}\). Among many other interesting results, they manage to prove (Theorem 3.7.2) that these bundles should have a nondegenerate section of degree at most 3 (while the natural bound given by the construction would be 4).