The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces (Q6613386)

Let \(Y\) be a complex Enriques surface. It was classically known that \(Y\) admits an elliptic pencil of the form \(|2F|\), where \(F\in \mathrm{Pic}(Y)\) is a \textit{half-fiber}, i.e. a primitive nef isotropic vector. This led \textit{F. R. Cossec} and \textit{I. V. Dolgachev} [Enriques surfaces. I. Boston, MA etc.: Birkhäuser Verlag (1989; Zbl 0665.14017)] to introduce the \textit{non-degeneracy invariant} \(\mathrm{nd}(Y)\) of \(Y\), defined as the maximum \(m\) such that there exist half-fibers \(F_1,\ldots,F_m\) with \(F_i.F_j = 1\) for every \(i\ne j\). It is proved by \textit{G. Martin} et al. [``Enriques surfaces of non-degeneracy 3'', Preprint, \url{arXiv:2203.08000}] that \(\mathrm{nd}(Y)\ge 4\), and the rank of \(\mathrm{Pic}(Y)\) gives the easy bound \(\mathrm{nd}(Y) \le 10\). Understanding the non-degeneracy invariant of an Enriques surface \(Y\) gives some insight on the geometry of \(Y\) and of its projective models. The non-degeneracy invariant of Enriques surfaces with finite automorphism group was computed by \textit{I. V. Dolgachev} and \textit{S. Kondō} [Enriques surfaces. II. (2024), \url{http://www.math.lsa.umich.edu/~idolga/EnriquesTwo.pdf}] and more recently by \textit{R. Moschetti} et al. [Exp. Math. 33, No. 3, 400--421 (2024; Zbl 07921116)], while the non-degeneracy invariant of a general Enriques surface is \(10\).\N\NThe present paper aims at computing the non-degeneracy invariant of the Enriques surfaces in the \(155\) families studied by \textit{S. Brandhorst} and \textit{I. Shimada} [Found. Comput. Math. 22, No. 5, 1463--1512 (2022; Zbl 1511.14058)]. The authors show that the non-degeneracy invariant is \(10\) for most families, and it is \(4\) for the family studied by \textit{W. Barth} and \textit{C. Peters} [Invent. Math. 73, 383--411 (1983; Zbl 0518.14023)]. The proof is obtained via computer, using the code developed by the same authors in [Exp. Math. 33, No. 3, 400--421 (2024; Zbl 07921116)]. The main idea is to produce a large list of smooth rational curves on the Enriques surface \(Y\) at hand via the knowledge of the automorphism group of \(Y\) computed by Brandhorst and Shimada, and to find many reducible fibers of elliptic pencils supported on the union of such curves, in order to obtain a lower bound for \(\mathrm{nd}(Y)\).