On the rationality of quadric surface bundles
Recently, in the study of the question whether rationality is deformation-invariant in smooth families, people identified smooth projective varieties \(X\) where the locus of rational fibers in dense in the parameter space of \(X\). Among them, there are smooth quadric surface bundle of type \((0,2,2,4)\) found by Voisin (see Proposition 25 in [\textit{S. Schreieder}, Duke Math. J. 168, No. 2, 187--223 (2019; Zbl 1409.14030)]) and smooth complex \((2,2)\)-hypersurface in \(\mathbb{P}^2\times \mathbb{P}^3\) (see [\textit{B. Hassett} et al., Acta Math. 220, No. 2, 341--365 (2018; Zbl 1420.14115)]). The latter example is birational to a quadric surface bundle of type \((2,2,2,2)\). In this article, let \(d_0,d_1,d_2,d_3\geq 0\) be integers of the same parity, the author showed that any smooth quadric surfaces bundle over \(\mathbb{P}^2\) of type \((d_0,d_1,d_2,d_3)\) satisfies the density property, i.e. the locus of rational fibers is dense in the parameter space. Following Voisin's approach, there is a density criterion via the surjectivity of an infinitesimial period map (see also Proposition 6 in [\textit{B. Hassett} et al., Acta Math. 220, No. 2, 341--365 (2018; Zbl 1420.14115)]). Using the toric characterization of quadric surface bundles, it is further reduced to an explicit problem about polynomials. To solve this problem, the author introduced a new method based on the strong Lefschetz property of certain complete intersections proven in [\textit{T. Harima} and \textit{J. Watanabe}, Rend. Semin. Mat. Univ. Padova 110, 119--146 (2003; Zbl 1150.13305)].
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