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Constructing Fano 3-folds from cluster varieties of rank 2

Part of: Surfaces and higher-dimensional varieties Arithmetic rings and other special rings

Published online by Cambridge University Press:  09 November 2020

Stephen Coughlan  and
Tom Ducat
Stephen Coughlan
Affiliation:
Lehrstuhl Mathematik VIII, Mathematisches Institut, Universitätsstrasse 30, 95447 Bayreuth, Germany stephen.coughlan@uni-bayreuth.de
Tom Ducat
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK t.ducat19@imperial.ac.uk The Heilbronn Institute for Mathematical Research, Bristol, UK
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Abstract

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.

Keywords

Fano 3-foldscluster varietieskey varieties

MSC classification

Primary: 13F60: Cluster algebras
Secondary: 14J45: Fano varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1873 - 1914
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Copyright
Copyright © The Author(s) 2020

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Footnotes

Current address: Department of Mathematics, Imperial College, South Kensington Campus, Huxley Building, 180 Queen's Gate, London, SW7 2AZ, UK

To Miles Reid on his 70th birthday.

References

Ahmadinezhad, H. and Okada, T., Birationally rigid Pfaffian Fano 3-folds, Algebraic Geom. 5 (2018), 160–199.Google Scholar
Altınok, S., Brown, G. and Reid, M., Fano 3-folds, K3 surfaces and graded rings, in Topology and geometry: commemorating SISTAG (National University of Singapore, 2001), Contemporary Mathematics, vol. 314, ed. A. J. Berrick et al. (American Mathematical Society Providence, RI, 2002), 25–53.Google Scholar
Altınok, S. and Reid, M., Three Fano 3-folds with $|{-}K|$ = empty. Notes, 2pp, http://homepages.warwick.ac.uk/masda/3folds/Ki/Selma4.Google Scholar
Brown, G., Kasprzyk, A. and Qureshi, M. I., Fano 3-folds in $\mathbb {P}^{2}\times \mathbb {P}^{2}$ format, Tom and Jerry, Eur. J. Math. 4 (2018), 51–72.Google Scholar
Brown, G., Kasprzyk, A. and Zhu, L., Gorenstein formats, canonical and Calabi–Yau threefolds, Exp. Math. (2019). doi:10.1080/10586458.2019.1592036.Google Scholar
Brown, G., Kerber, M. and Reid, M., Fano $3$-folds in codimension 4, Tom and Jerry, Part I, Compos. Math. 148 (2012), 1171–1194.10.1112/S0010437X11007226CrossRefGoogle Scholar
Brown, G. and Reid, M., Diptych varieties, II, in Higher dimensional algebraic geometry, Advanced Studies in Pure Mathematics, vol. 74 (Mathematical Society of Japan, Tokyo, 2017), 41–72.Google Scholar
Brown, G. and Suzuki, K., Fano 3-folds with divisible anticanonical class, Manuscripta Math. 123 (2007), 37–51.10.1007/s00229-007-0082-6CrossRefGoogle Scholar
Brown, G. and Suzuki, K., Computing certain Fano 3-folds, Jpn. J. Ind. Appl. Math. 24 (2007), 241–250.Google Scholar
Brown, G. and Zucconi, F., Graded rings of rank 2 Sarkisov links, Nagoya Math. J. 197 (2010), 1–44.Google Scholar
Buckley, A., Reid, M. and Zhou, S., Ice cream and orbifold Riemann–Roch, Izv. Math. 77 (2013), 461–486.10.1070/IM2013v077n03ABEH002644CrossRefGoogle Scholar
Corti, A. and Heuberger, L., Del Pezzo surfaces with $1/3(1,1)$ points, Man. Math. 153 (2017), 71–118.Google Scholar
Corti, A. and Reid, M., Weighted Grassmannians, in Algebraic geometry (Genova, Sep 2001), in memory of Paolo Francia, eds M. Beltrametti and F. Catanese (de Gruyter, Berlin, 2002), 141–163.Google Scholar
Coughlan, S. and Ducat, T., Big tables of Fano 3-folds in cluster formats, sites.google.com/view/stephencoughlan/research/cluster-fanos.Google Scholar
Coughlan, S. and Urzúa, G., On $\mathbb {Z}$/3-Godeaux surfaces, Int. Math. Res. Not. IMRN 2018 (2018), 5609–5637.Google Scholar
Dicks, D., Surfaces with p g=3, K 2=4 and extension-deformation theory, PhD thesis, University of Warwick (1988).Google Scholar
Ducat, T., Mori extractions from singular curves in a smooth 3-fold, PhD thesis, University of Warwick (2015).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie, Publ. Math. l'IHÉS 24 (1965), 5–231.Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.Google Scholar
Gross, M., Hacking, P. and Keel, S., Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168.Google Scholar
Gross, M., Hacking, P. and Keel, S., Moduli of surfaces with an anticanonical cycle, Compos. Math. 151 (2015), 265–291.10.1112/S0010437X14007611CrossRefGoogle Scholar
Brown, G. and Kasprzyk, A., The graded ring database. Online access via http://www.grdb.co.uk/.Google Scholar
Gross, M. and Siebert, B., Intrinsic mirror symmetry and punctured Gromov–Witten invariants, Proc. Sym. Pure Math. 97 (2018), 199–230.10.1090/pspum/097.2/01705CrossRefGoogle Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, trans. M. Reid (Cambridge, University Press, Cambridge, 1989).Google Scholar
Okada, T., Birationally superrigid Fano 3-folds of codimension 4, Algebra Number Theory 14 (2020), 191–212.Google Scholar
Papadakis, S. and Neves, J., A construction of numerical Campedelli surfaces with torsion $\mathbb {Z}/6$, Trans. Amer. Math. Soc. 361 (2009), 4999–5021.Google Scholar
Prokhorov, Y., Fano threefolds of large Fano index and large degree, Sb. Math. 204 (2013), 347–382.10.1070/SM2013v204n03ABEH004304CrossRefGoogle Scholar
Prokhorov, Y., $\mathbb {Q}$-Fano threefolds of index 7, Proc. Steklov Inst. Math. 294 (2016), 139–153.10.1134/S0081543816060092CrossRefGoogle Scholar
Prokhorov, Y. and Reid, M., On $\mathbb {Q}$-Fano threefolds of Fano index 2, in Minimal models and extremal rays (Kyoto 2011), Advanced Studies in Pure Mathematics, vol. 70 (Mathematical Society of Japan, Tokyo, 2016), 397–420.Google Scholar
Reid, M., Parallel unprojection equations for $\mathbb {Z}/3$ Godeaux surfaces, Notes, warwick.ac.uk/ masda/codim4/God3.pdf.Google Scholar
Reid, M., Fun in codimension 4, Notes, warwick.ac.uk/masda/codim4/Fun2.pdf.Google Scholar
Reid, M., Gorenstein in codimension 4 – the general structure theory, in Algebraic geometry in East Asia (Taipei, November 2011), Advanced Studies in Pure Mathematics, vol. 65 (Mathematical Society of Japan, Tokyo, 2015), 201–227.Google Scholar
Grothendieck, A. and Raynaud, M., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Advanced Studies in Pure Mathematics, vol. 2 (North-Holland, Amsterdam, 1968); in French.Google Scholar