On the rationality of quadric surface bundles
Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 97-121.

For any standard quadric surface bundle over 2 , we show that the locus of rational fibres is dense in the moduli space.

Pour tout faisceau de surface quadrique standard sur 2 , nous montrons que le lieu des fibres rationnelles est dense dans l’espace des modules.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3399
Classification: 14E08,  14D07,  13H10,  14J35,  14M25
Keywords: Hodge loci, rationality problem, quadric surface bundles
Paulsen, Matthias 1

1 Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 D-30167 Hannover (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2021__71_1_97_0,
     author = {Paulsen, Matthias},
     title = {On the rationality of quadric surface bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {97--121},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {1},
     year = {2021},
     doi = {10.5802/aif.3399},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3399/}
}
TY  - JOUR
AU  - Paulsen, Matthias
TI  - On the rationality of quadric surface bundles
JO  - Annales de l'Institut Fourier
PY  - 2021
DA  - 2021///
SP  - 97
EP  - 121
VL  - 71
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3399/
UR  - https://doi.org/10.5802/aif.3399
DO  - 10.5802/aif.3399
LA  - en
ID  - AIF_2021__71_1_97_0
ER  - 
%0 Journal Article
%A Paulsen, Matthias
%T On the rationality of quadric surface bundles
%J Annales de l'Institut Fourier
%D 2021
%P 97-121
%V 71
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3399
%R 10.5802/aif.3399
%G en
%F AIF_2021__71_1_97_0
Paulsen, Matthias. On the rationality of quadric surface bundles. Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 97-121. doi : 10.5802/aif.3399. https://aif.centre-mersenne.org/articles/10.5802/aif.3399/

[1] Auel, Asher; Böhning, Christian; Pirutka, Alena Stable rationality of quadric and cubic surface bundle fourfolds, Eur. J. Math., Volume 4 (2018) no. 3, pp. 1-29 | MR | Zbl

[2] Batyrev, Victor; Cox, David On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J., Volume 75 (1994) no. 2, pp. 293-338 | MR | Zbl

[3] Benoist, Olivier Sums of three squares and Noether–Lefschetz loci, Compos. Math., Volume 154 (2018) no. 5, pp. 1048-1065 | DOI | MR | Zbl

[4] Carlson, James; Griffiths, Phillip Infinitesimal variations of Hodge structure and the global Torelli problem, Algebraic Geometry Angers 1979 (1980), pp. 51-76 | Zbl

[5] Ciliberto, Ciro; Harris, Joe; Miranda, Rick General Components of the Noether–Lefschetz locus and their density in the space of all surfaces, Math. Ann., Volume 282 (1988) no. 4, pp. 667-680 | DOI | MR | Zbl

[6] Ciliberto, Ciro; Lopez, Angelo Felice On the existence of components of the Noether–Lefschetz locus with given codimension, Manuscr. Math., Volume 73 (1991) no. 4, pp. 341-357 | DOI | MR | Zbl

[7] Colliot-Thélène, Jean-Louis; Pirutka, Alena Hypersurfaces quartiques de dimension 3: non rationalité stable, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016), pp. 371-397 | DOI | Zbl

[8] Colliot-Thélène, Jean-Louis; Voisin, Claire Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J., Volume 161 (2012) no. 5, pp. 735-801 | Zbl

[9] Cox, David; Little, John; Schenck, Hal Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011 | MR | Zbl

[10] Griffiths, Phillip On the periods of certain rational integrals. I and II, Ann. Math., Volume 90 (1969) no. 3, p. 460-495 and 496–541 | DOI | MR | Zbl

[11] Harima, Tadahito; Watanabe, Junzo The finite free extension of Artinian K-algebras with the strong Lefschetz property, Rend. Semin. Mat. Univ. Padova, Volume 110 (2003), pp. 119-146 | Numdam | MR | Zbl

[12] Hassett, Brendan; Kresch, Andrew; Tschinkel, Yuri Stable rationality in smooth families of threefolds (2018) (https://arxiv.org/abs/1802.06107)

[13] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri Intersections of three quadrics in 7 , Celebrating the 50th anniversary of the Journal of Differential Geometry (Surveys in Differential Geometry), Volume 22, International Press, 2018, pp. 259-274 | MR | Zbl

[14] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri Stable rationality of quadric surface bundles over surfaces, Acta Math., Volume 220 (2018) no. 2, pp. 341-365 | DOI | MR | Zbl

[15] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri A very general quartic double fourfold is not stably rational, Algebr. Geom., Volume 6 (2019) no. 1, pp. 64-75 | MR | Zbl

[16] Kim, Sung-Ock Noether–Lefschetz locus for surfaces, Trans. Am. Math. Soc., Volume 324 (1991) no. 1, pp. 369-384 | MR | Zbl

[17] Paulsen, Matthias Density of Noether–Lefschetz loci and rationality of quadric surface bundles, 2018 Master’s thesis, Ludwig-Maximilians-Universität München (Germany)

[18] Schreieder, Stefan Quadric surface bundles over surfaces and stable rationality, Algebra Number Theory, Volume 12 (2018) no. 2, pp. 479-490 | DOI | MR | Zbl

[19] Schreieder, Stefan On the rationality problem for quadric bundles, Duke Math. J., Volume 168 (2019), pp. 187-223 | MR | Zbl

[20] Springer, Tonny Albert Sur les formes quadratiques d’indice zéro, C. R. Math. Acad. Sci. Paris, Volume 234 (1952), pp. 1517-1519 | MR | Zbl

[21] Stanley, Richard Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods, Volume 1 (1980), pp. 168-184 | DOI | MR | Zbl

[22] Voisin, Claire The Griffiths group of a general Calabi–Yau threefold is not finitely generated, Duke Math. J., Volume 102 (2000) no. 1, pp. 151-186 | MR

[23] Voisin, Claire Hodge Theory and Complex Algebraic Geometry II, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2003 no. 77 | Zbl

[24] Voisin, Claire On integral Hodge classes on uniruled or Calabi–Yau threefolds, Moduli spaces and arithmetic geometry (Advanced Studies in Pure Mathematics), Volume 45, Mathematical Society of Japan, 2006, pp. 43-73 | DOI | MR | Zbl

[25] Voisin, Claire (Stable) rationality is not deformation invariant (2015) (https://arxiv.org/abs/1511.03591v3)

[26] Voisin, Claire Unirational threefolds with no universal codimension 2 cycle, Invent. Math., Volume 201 (2015) no. 1, pp. 207-237 | DOI | MR | Zbl

[27] Voisin, Claire Birational invariants and decomposition of the diagonal, Birational geometry of hypersurfaces (Hochenegger, Andreas; Lehn, Manfred; Stellari, Paolo, eds.) (Lecture Notes of the Unione Matematica Italiana), Volume 26, Springer, 2019, pp. 3-71 | DOI | Zbl

[28] Watanabe, Junzo The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Advanced Studies in Pure Mathematics), Volume 11, North-Holland, 1987, pp. 303-312 | DOI | MR | Zbl

Cited by Sources: