On the rationality of quadric surface bundles
type: Item
instance of: scholarly article
title: On the rationality of quadric surface bundles
zbMATH Open document ID: 1469.14029
author: Matthias Paulsen
publication date: 2021-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1811.05271
review text: 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)].
reviewed by: Ruijie Yang
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14J35
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 7362099
zbMATH Keywords: rationality problem
zbMATH Keywords: Hodge loci
zbMATH Keywords: quadric surface bundles
MaRDI profile type: Publication
OpenAlex ID: W4289293141
arXiv ID: 1811.05271
cites work: Stable rationality of quadric and cubic surface bundle fourfolds
cites work: On the Hodge structure of projective hypersurfaces in toric varieties
cites work: Sums of three squares and Noether–Lefschetz loci
cites work: Q3936833
cites work: General components of the Noether-Lefschetz locus and their density in the space of all surfaces
cites work: On the existence of components of the Noether-Lefschetz locus with given codimension
cites work: Three-dimensional quartic threefolds: stable non-rationality
cites work: Unramified cohomology and the entire Hodge conjecture
cites work: Toric varieties
cites work: On the periods of certain rational integrals. I, II
cites work: The finite free extension of Artinian \(K\)-algebras with the strong Lefschetz property
cites work: Stable rationality and conic bundles
cites work: Intersections of three quadrics in \(\mathbb{P}^7\)
cites work: Stable rationality of quadric surface bundles over surfaces
cites work: A very general quartic double fourfold is not stably rational
cites work: Noether-Lefschetz Locus for Surfaces
cites work: Quadric surface bundles over surfaces and stable rationality
cites work: On the rationality problem for quadric bundles
cites work: Q5811220
cites work: Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property
cites work: The Griffiths group of a general Calabi-Yau threefold is not finitely generated
cites work: Q4420474
cites work: Q3438053
cites work: Unirational threefolds with no universal codimension \(2\) cycle
cites work: Birational Invariants and Decomposition of the Diagonal
cites work: Q3792779
DOI: 10.5802/AIF.3399
published in: Annales de l’institut Fourier
description: scientific article