Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with $\mathbb{P}^2\times \mathbb{P}^2$-fibrations
DOI:
https://doi.org/10.7146/math.scand.a-158550Abstract
In our previous research, we constructed the affine varieties $\Sigma _{\mathbb {A}}^{13}$ and $\Pi _{\mathbb {A}}^{14}$ whose partial projectivizations admit $\mathbb {P}^{2}\times \mathbb {P}^{2}$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth ℚ-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp. 8) classes in the Graded Ring Database, as weighted complete intersections in weighted projectivizations of $\Sigma _{\mathbb {A}}^{13}$ (resp. $\Pi _{\mathbb {A}}^{14}$ or its cone). We also show that a general member of the anticanonical linear system of a general prime ℚ-Fano $3$-fold constructed in this way is a quasi-smooth $K3$ surface with at worst Du Val singularities.
References
Altınok, S., Graded rings corresponding to polarised $K3$ surfaces and ℚ-Fano $3$-folds, Ph.D. thesis, University of Warwick, 1998, available at https://homepages.warwick.ac.uk/staff/Miles.Reid/doctors/Selma/.
Altınok, S., Brown, G., and Reid, M., Fano $3$-folds, $K3$ surfaces and graded rings, Topology and geometry: commemorating SISTAG, 25–53, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002. https://doi.org/10.1090/conm/314/05420
Altınok, S., Brown, G., Iano-Fletcher, A., Reid, M., and Suzuki, K., The graded ring database, Online database, available at http://www.grdb.co.uk/forms/fano3.
Bosma, W., Cannon, J., and Playoust, C., Magma Algebra System, School of Mathematics and Statistics, University of Sydney (1997), available at http://magma.maths.usyd.edu.au.
Brown, G., Kerber, M., and Reid, M., Fano $3$-folds in codimension $4$, Tom and Jerry. Part I, Compos. Math. 148 (2012), no. 4, 1171–1194. https://doi.org/10.1112/S0010437X11007226
Chen, J.-J., Chen, J. A., and Chen, M., On quasismooth weighted complete intersections, J. Algebraic Geom. 20 (2011), no. 2, 239–262. https://doi.org/10.1090/S1056-3911-10-00542-4
Coughlan S., and Ducat, T., Constructing Fano $3$-folds from cluster varieties of rank $2$, Compos. Math. 156 (2020), no. 9, 1873–1914. https://doi.org/10.1112/s0010437x20007368
Goto, S., and Watanabe, K., On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213. https://doi.org/10.2969/jmsj/03020179
Fano, G., Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche, Comment. Math. Helv. 14 (1942), 202–211. https://doi.org/10.1007/BF02565618
Gusheltextasciiacute , N. P., Fano varieties of genus $6$, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1159–1174, 1343.
Iano-Fletcher, A., Working with weighted complete intersections, Explicit birational geometry of 3-folds, 101–173, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000.
Kleiman, S., Bertini and his two fundamental theorems, Rend. Circ. Mat. Palermo (2) Suppl. No. 55 (1998), 9–37.
Kollár J., and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662560
Okada, T., Birationally superrigid Fano $3$-folds of codimension $4$, Algebra Number Theory 14 (2020), no. 1, 191–212. https://doi.org/10.2140/ant.2020.14.191
Reid, M., Fun in codimension $4$, preprint. https://mreid.warwick.ac.uk/codim4/Fun2.pdf
Takagi, H., On classification of ℚ-Fano $3$-folds of Gorenstein index $2$. I, II, Nagoya Math. J. 167 (2002), 117–155, 157–216. https://doi.org/10.1017/S0027763000025460
Takagi, H., Key varieties for prime ℚ-Fano threefolds related with $mathbb P^2times mathbb P^2$-fibrations. Part II, arXiv:2111.14328v1. https://doi.org/10.48550/arXiv.2111.14328
Takagi, H., Classification of ℚ-Fano $3$-folds of Gorenstein index $2$ via key varieties constructed from projective bundles, Internat. J. Math. 34 (2023), no. 10, Paper No. 2350060, 47 pp. https://doi.org/10.1142/S0129167X2350060X
Takagi, H., Key varieties for prime ℚ-Fano threefolds defined by Jordan algebras of cubic forms. Part I, arXiv:2403.14991v2. https://doi.org/10.48550/arXiv.2403.14991
Takagi, H., Key varieties for prime ℚ-Fano threefolds defined by Jordan algebras of cubic forms. Part II, arXiv:2403.15991v2. https://doi.org/10.48550/arXiv.2403.15991
Takagi, H., Key varieties for prime ℚ-Fano threefolds defined by Freudenthal triple systems, Geom. Dedicata 218 (2024), no. 5, Paper No. 96, 38 pp. https://doi.org/10.1007/s10711-024-00945-9
Takagi, H., Mathematica code for paper `Constructing prime ℚ-Fano threefolds of codimension four via key varieties related with $P^2times P^2$-fibrations', Notebook file: CodeQFano.nb, available at https://github.com/hiromicitakagi/mathematica-code.
Takagi, H., Key varieties for prime ℚ-Fano threefolds related with $P ^2times P^2$-fibration, preprint.
Takagi, H., Constructing and characterizing prime ℚ-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularities via key varieties, arXiv:2507.13657. https://doi.org/10.48550/arXiv.2507.13657
Taylor, R., Type II unprojections, Fano threefolds and codimension four constructions, PhD thesis, University of Warwick (2020), available at https://wrap.warwick.ac.uk/145363/1/WRAP_Theses_Taylor_2020.pdf.
Wolfram Research, Inc., Mathematica, Version 12, Wolfram Research, Inc., https://www.wolfram.com/mathematica, Champaign, Illinois (2019).
