Models of quadric surface bundles (Q2097298)

The author gives a characterization of smooth models of quadric surface bundles over a base of arbitrary dimension. In fact, for a surface base, a similar result is given in [\textit{A. Auel} et al., Doc. Math. Extra Vol., 31--70 (2015; Zbl 1345.14046)]. The content is summarized as following. In Sect. 2, he proves some preliminary results. Then, he gives some reduction steps toward the proof of main result. In Section 4 and Sect. 5, he completes the proof of the main result. In the Appendix, he shows that the construction of involution surface bundles in [\textit{A. Kresch} and \textit{Y. Tschinkel}, Math. Z. 296, No. 3--4, 1081--1100 (2020; Zbl 1450.14005)] may lead to a non-flat model for a base of dimension \(>2\).