대한수학회지 (Journal of the Korean Mathematical Society)

  • 제60권4호
  • /
  • Pages.695-743
  • /
  • 2023
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

COMPLEX REFLECTION GROUPS AND K3 SURFACES II. THE GROUPS G29, G30 AND G31

  • 투고 : 2022.01.09
  • 심사 : 2023.03.23
  • 발행 : 2023.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We study some K3 surfaces obtained as minimal resolutions of quotients of subgroups of special reflection groups. Some of these were already studied in a previous paper by W. Barth and the second author. We give here an easy proof that these are K3 surfaces, give equations in weighted projective space and describe their geometry.

키워드

과제정보

The first author is partly supported by the ANR: project No. ANR-16-CE40-0010-01 (GeRepMod) and ANR-18-CE40-0024-02 (CATORE). The second author is partly supported by the ANR: project No. ANR-20-CE40-0026-01 (SMAGP).

참고문헌

  1. W. P. Barth and A. Sarti, Polyhedral groups and pencils of K3-surfaces with maximal Picard number, Asian J. Math. 7 (2003), no. 4, 519-538. https://doi.org/10.4310/AJM.2003.v7.n4.a5 
  2. S. Boissiere and A. Sarti, Counting lines on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 1, 39-52.  https://doi.org/10.2422/2036-2145.2007.1.03
  3. C. Bonnafe, Magma codes for "Some singular curves and surfaces arising from invariants of complex reflection groups", preprint, 2018. https://hal.science/hal-01897587 
  4. C. Bonnafe, Some singular curves and surfaces arising from invariants of complex reflection groups, Exp. Math. 30 (2021), no. 3, 429-440. https://doi.org/10.1080/10586458.2018.1555778 
  5. C. Bonnafe and A. Sarti, Complex reflection groups and K3 surfaces I, Epijournal Geom. Algebrique 5 (2021), Art. 3, 26 pp. https://doi.org/10.46298/epiga.2021.volume5.6573 
  6. C. Bonnafe and A. Sarti, K3 surfaces with maximal finite automorphism groups containing M20, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 2, 711-730.  https://doi.org/10.5802/aif.3411
  7. C. Bonnafe and A. Sarti, Complex reflection groups and K3 surfaces III. The group G28 = W(F4), in preparation. 
  8. W. Bosma, J. J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. https://doi.org/10.1006/jsco.1996.0125 
  9. N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5, 6, Actualites Scientifiques et Industrielles 1337, Hermann, Paris, 1968. 
  10. A. I. Degtyarev, 800 conics on a smooth quartic surface, J. Pure Appl. Algebra 226 (2022), no. 10, Paper No. 107077, 5 pp. https://doi.org/10.1016/j.jpaa.2022.107077 
  11. A.-S. Elsenhans and J. Jahnel, The Picard group of a K3 surface and its reduction modulo p, Algebra Number Theory 5 (2011), no. 8, 1027-1040. https://doi.org/10.2140/ant.2011.5.1027 
  12. D. Festi and D. C. Veniani, Counting elliptic fibrations on K3 surfaces, preprint, 2021. 
  13. H. Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in ℙ3, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 545-560. 
  14. G. Lusztig, Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623-653. https://doi.org/10.2307/1999228 
  15. I. Marin and J. C. M. Michel, Automorphisms of complex reflection groups, Represent. Theory 14 (2010), 747-788. https://doi.org/10.1090/S1088-4165-2010-00380-5 
  16. J. C. M. Michel, The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308-336. https://doi.org/10.1016/j.jalgebra.2015.03.031 
  17. S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183-221. https://doi.org/10.1007/BF01394352 
  18. B. Naskrecki, Explicit equations of 800 conics on a Barth-Bauer quartic, arxiv: 2108.13402. 
  19. N. O. Nygaard and A. Ogus, Tate's conjecture for K3 surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461-507. https://doi.org/10.2307/1971327 
  20. A. Sarti, Pencils of symmetric surfaces in ℙ3, J. Algebra 246 (2001), no. 1, 429-452. https://doi.org/10.1006/jabr.2001.8953 
  21. A. Sarti, Transcendental lattices of some K3-surfaces, Math. Nachr. 281 (2008), no. 7, 1031-1046. https://doi.org/10.1002/mana.200510657 
  22. M. Schutt, Elliptic fibrations of some extremal K3 surfaces, Rocky Mountain J. Math. 37 (2007), no. 2, 609-652. https://doi.org/10.1216/rmjm/1181068770 
  23. M. Schutt and T. Shioda, Mordell-Weil lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 70, Springer, Singapore, 2019. https://doi.org/10.1007/978-981-32-9301-4 
  24. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304. https://doi.org/10.4153/cjm-1954-028-3 
  25. I. Shimada, On elliptic K3 surfaces, Michigan Math. J. 47 (2000), no. 3, 423-446. https://doi.org/10.1307/mmj/1030132587 
  26. H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z. 189 (1985), no. 4, 507-513. https://doi.org/10.1007/BF01168156 
  27. U. Thiel, Champ: a Cherednik algebra Magma package, LMS J. Comput. Math. 18 (2015), no. 1, 266-307. https://doi.org/10.1112/S1461157015000054 
  28. R. M. van Luijk, Quartic K3 surfaces without nontrivial automorphisms, Math. Res. Lett. 13 (2006), no. 2-3, 423-439. https://doi.org/10.4310/MRL.2006.v13.n3.a7 
  29. R. M. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), no. 1, 1-15. https://doi.org/10.2140/ant.2007.1.1 
  30. G. Xiao, Galois covers between K3 surfaces, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 73-88. https://doi.org/10.5802/aif.1507