Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-21T19:47:43.151Z Has data issue: false hasContentIssue false
  • English
  • Français

SMOOTH FANO 4-FOLDS IN GORENSTEIN FORMATS

Part of: Surfaces and higher-dimensional varieties Computational aspects in algebraic geometry Special varieties

Published online by Cambridge University Press:  31 May 2021

MUHAMMAD IMRAN QURESHI Open the ORCID record for MUHAMMAD IMRAN QURESHI [Opens in a new window]
MUHAMMAD IMRAN QURESHI*
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
*
e-mail: imran.qureshi@kfupm.edu.sa
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct some new deformation families of four-dimensional Fano manifolds of index one in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anticanonical embedding in some weighted projective space. They also have relatively smaller anticanonical degree than most other known families of smooth Fano 4-folds.

Keywords

Fano varietyGoresntein format4-fold

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14J35: $4$-folds 14M07: Low codimension problems 14Q15: Higher-dimensional varieties

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 424 - 433
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

I would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum and Minerals for funding this work through the project No. SR191006.

References

Batyrev, V. V., ‘On the classification of toric Fano 4-folds’, J. Math. Sci. 94(1) (1999), 1021–1050.10.1007/BF02367245CrossRefGoogle Scholar
Brown, G., Kasprzyk, A. M. and Qureshi, M. I., ‘Fano 3-folds in ${\mathbb{P}}_2\times {\mathbb{P}}_2$ format, Tom and Jerry’, Eur. J. Math. 4(1) (2018), 51–72.10.1007/s40879-017-0200-2CrossRefGoogle Scholar
Brown, G., Kasprzyk, A. M. and Zhu, L., ‘Gorenstein formats, canonical and Calabi–Yau threefolds’, Exp. Math. (2019), Article ID 1592036, 19 pages.Google Scholar
Coates, T., Galkin, S., Kasprzyk, A. and Strangeway, A., ‘Quantum periods for certain four-dimensional Fano manifolds’, Exp. Math. 29(2) (2020), 183–221.10.1080/10586458.2018.1448018CrossRefGoogle Scholar
Coates, T., Kasprzyk, A. and Prince, T., ‘Four-dimensional Fano toric complete intersections’, Proc. Roy. Soc. Ser. A 471(2175) (2015), Article ID 20140704, 14 pages.Google ScholarPubMed
Coates, T., Kasprzyk, A. and Prince, T., ‘Laurent inversion’, Pure Appl. Math. Q. 15(4) (2019), 1135–1179.10.4310/PAMQ.2019.v15.n4.a5CrossRefGoogle Scholar
Corti, A. and Reid, M., ‘Weighted Grassmannians’, in: Algebraic Geometry (eds. Beltrametti, M. C., Catanese, F., Ciliberto, C., Lanteri, A. and Pedrini, C.) (de Gruyter, Berlin, 2002), 141–163.Google Scholar
Di Natale, C., Fatighenti, E. and Fiorenza, D., ‘Hodge theory and deformations of affine cones of subcanonical projective varieties’, J. Lond. Math. Soc. 96(3) (2017), 524–544.CrossRefGoogle Scholar
Fujita, T., ‘On the structure of polarized manifolds with total deficiency one, I’, J. Math. Soc. Japan 32(4) (1980), 709–725.CrossRefGoogle Scholar
Fujita, T., ‘On the structure of polarized manifolds with total deficiency one, II’, J. Math. Soc. Japan 33(3) (1981), 415–434.10.2969/jmsj/03330415CrossRefGoogle Scholar
Fujita, T., ‘On the structure of polarized manifolds with total deficiency one, III’, J. Math. Soc. Japan 36(1) (1984), 75–89.10.2969/jmsj/03610075CrossRefGoogle Scholar
Fujita, T., Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series, 155 (Cambridge University Press, Cambridge, 1990).10.1017/CBO9780511662638CrossRefGoogle Scholar
Hausen, J., Laface, A. and Mauz, C., ‘On smooth Fano fourfolds of Picard number two’, Preprint, 2021, arXiv:1907.08000.10.4171/RMI/1271CrossRefGoogle Scholar
Ilten, N. O., ‘Versal deformations and local Hilbert schemes’, J. Softw. Algebra Geom. 4 (2012), 12–16.10.2140/jsag.2012.4.12CrossRefGoogle Scholar
Iskovskih, V. A., ‘Fano 3-folds. I’ (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 41(3) (1977), 516–562; English translation Math. USSR–Izvestiya 11(3) (1977), 485–527.Google Scholar
Iskovskih, V. A., ‘Fano 3-folds. II’ (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 42(3) (1978), 506–549; English translation Math. USSR–Izvestiya 12(3) (1978), 469–506.Google Scholar
Iskovskikh, V. A. and Prokhorov, Yu. G., Algebraic Geometry V: Fano Varieties, Encyclopaedia of Mathematical Sciences, 47 (Springer, Berlin, 1999).Google Scholar
Kalashnikov, E., ‘Four-dimensional Fano quiver flag zero loci’, Proc. Roy. Soc. A 475(2205) (2019), Article ID 20180791, 23 pages.Google ScholarPubMed
Kobayashi, S. and Ochiai, T., ‘Characterizations of complex projective spaces and hyperquadrics’, J. Math. Kyoto Univ. 13(1) (1973), 31–47.Google Scholar
Kollár, J., Miyaoka, Y. and Mori, S., ‘Rational connectedness and boundedness of Fano manifolds’, J. Differential Geom. 36(3) (1992), 765–779.10.4310/jdg/1214453188CrossRefGoogle Scholar
Küchle, O., ‘On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians’, Math. Z. 218(1) (1995), 563–575.10.1007/BF02571923CrossRefGoogle Scholar
Küchle, O., ‘Some remarks and problems concerning the geography of Fano 4-folds of index and Picard number one’, Quaest. Math. 20(1) (1997), 45–60.10.1080/16073606.1997.9631853CrossRefGoogle Scholar
Mori, S. and Mukai, S., ‘Classification of Fano 3-folds with ${B}_2\ge 2$ ’, Manuscripta Math. 36(2) (1981/82), 147–162.10.1007/BF01170131CrossRefGoogle Scholar
Mori, S. and Mukai, S., ‘Erratum to “Classification of Fano 3-folds with ${B}_2\ge 2$ ”’, Manuscripta Math. 110(3) (2003), 407.CrossRefGoogle Scholar
Przyjalkowski, V. and Shramov, C., ‘Bounds for smooth Fano weighted complete intersections’, Comm. Number Theory and Physics 14(3) (2020), 511–553.10.4310/CNTP.2020.v14.n3.a3CrossRefGoogle Scholar
Qureshi, M. I., ‘Constructing projective varieties in weighted flag varieties II’, Math. Proc. Cambridge Philos. Soc. 158 (2015), 193–209.10.1017/S0305004114000620CrossRefGoogle Scholar
Qureshi, M. I., ‘Computing isolated orbifolds in weighted flag varieties’, J. Symbolic Comput. 79(2) (2017), 457–474.CrossRefGoogle Scholar
Qureshi, M. I., ‘Polarized 3-folds in a codimension 10 weighted homogeneous ${F}_4$ variety’, J. Geom. Phys. 120 (2017), 52–61.CrossRefGoogle Scholar
Qureshi, M. I. and Szendrői, B., ‘Constructing projective varieties in weighted flag varieties’, Bull. Lond. Math. Soc. 43(2) (2011), 786–798.10.1112/blms/bdr012CrossRefGoogle Scholar
Szendrői, B., ‘On weighted homogeneous varieties’, Unpublished Manuscript, 2005.Google Scholar
Tsuji, H., ‘Deformation invariance of plurigenera’, Nagoya Math. J. 166 (2002), 117–134.10.1017/S002776300000828XCrossRefGoogle Scholar
Wilson, P. M. H., ‘Fano fourfolds of index greater than one’, J. reine angew. Math. 379 (1987), 172–181.Google Scholar
Wisniewski, J. A., ‘Fano 4-folds of index 2 with ${b}_2\ge 2$ . A contribution to Mukai classification’, Bull. Pol. Acad. Sci. Math. 38 (1990), 173–184.Google Scholar