The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
type: Item
instance of: scholarly article
title: The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces
author: Riccardo Moschetti
author: Franco Rota
author: Luca Schaffler
published in: Documenta Mathematica
publication date: 2024-10-02T00:00:00Z
review text: 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…
reviewed by: Giacomo Mezzedimi
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14Q10
zbMATH DE Number: 7921217
zbMATH Keywords: Enriques surface
zbMATH Keywords: elliptic fibration
zbMATH Keywords: rational curve
zbMATH Keywords: automorphism
zbMATH Keywords: non-degeneracy invariant
MaRDI profile type: Publication
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DOI: 10.4171/DM/972
description: scientific article; zbMATH DE number 7921217