Rational double points on supersingular 𝐾3 surfaces

Shimada, Ichiro

doi:10.1090/S0025-5718-04-01641-2

Rational double points on supersingular $K3$ surfaces
HTML articles powered by AMS MathViewer

by Ichiro Shimada PDF

Math. Comp. 73 (2004), 1989-2017 Request permission

Abstract:

We investigate configurations of rational double points with the total Milnor number $21$ on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular $K3$ surfaces.

References

Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI 10.2307/2372985
Michael Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136. MR 199191, DOI 10.2307/2373050
M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899, DOI 10.24033/asens.1279
M. Artin, Coverings of the rational double points in characteristic $p$, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22. MR 0450263
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
I. Dolgachev and S. Kond\B{o}, A supersingular $K3$ surface in characteristic 2 and the Leech lattice, Int. Math. Res. Not. 1 (2003), 1–23. MR 1935564, DOI 10.1155/S1073792803202038
I. Dolgachev, M. Mendes Lopes, and R. Pardini, Rational surfaces with many nodes, Compositio Math. 132 (2002), no. 3, 349–363. MR 1918136, DOI 10.1023/A:1016540925011
Wolfgang Ebeling, Lattices and codes, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1994. A course partially based on lectures by F. Hirzebruch. MR 1280458, DOI 10.1007/978-3-322-96879-1
Yasuhiro Goto, On the Néron-Severi groups of some $K3$ surfaces, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 305–328. MR 1738864, DOI 10.1090/crmp/024/15
Hiroyuki Ito, The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic $3$, Math. Z. 211 (1992), no. 1, 1–39. MR 1179777, DOI 10.1007/BF02571415
Hiroyuki Ito, The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic $2$, Tohoku Math. J. (2) 46 (1994), no. 2, 221–251. MR 1272880, DOI 10.2748/tmj/1178225759
Hiroyuki Ito, On automorphisms of supersingular $K3$ surfaces, Osaka J. Math. 34 (1997), no. 3, 713–724. MR 1613053
Hiroyuki Ito, On extremal elliptic surfaces in characteristic 2 and 3, Hiroshima Math. J. 32 (2002), no. 2, 179–188. MR 1925896
Shigeyuki Kond\B{o}, Algebraic $K3$ surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989), 1–15. MR 1029967, DOI 10.1017/S0027763000001653
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Viacheslav V. Nikulin, Weil linear systems on singular $K3$ surfaces, Algebraic geometry and analytic geometry (Tokyo, 1990) ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 138–164. MR 1260944
Ken-ichi Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell-Weil groups, Japan. J. Math. (N.S.) 22 (1996), no. 2, 293–347. MR 1432379, DOI 10.4099/math1924.22.293
A. N. Rudakov and I. R. Šafarevič, Supersingular $K3$ surfaces over fields of characteristic $2$, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 4, 848–869 (Russian). MR 508830
A. N. Rudakov and I. R. Shafarevich, Surfaces of type $K3$ over fields of finite characteristic, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 115–207 (Russian). MR 633161
B. Saint-Donat, Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI 10.2307/2373709
Ichiro Shimada, On supercuspidal families of curves on a surface in positive characteristic, Math. Ann. 292 (1992), no. 4, 645–669. MR 1157319, DOI 10.1007/BF01444641
Ichiro Shimada, On elliptic $K3$ surfaces, Michigan Math. J. 47 (2000), no. 3, 423–446. MR 1813537, DOI 10.1307/mmj/1030132587
Ichiro Shimada and De-Qi Zhang, Classification of extremal elliptic $K3$ surfaces and fundamental groups of open $K3$ surfaces, Nagoya Math. J. 161 (2001), 23–54. MR 1820211, DOI 10.1017/S002776300002211X
Tetsuji Shioda, On unirationality of supersingular surfaces, Math. Ann. 225 (1977), no. 2, 155–159. MR 429911, DOI 10.1007/BF01351719
Tetsuji Shioda, Supersingular $K3$ surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 564–591. MR 555718
—, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240.
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Tohsuke Urabe, Dynkin graphs and combinations of singularities on plane sextic curves, Singularities (Iowa City, IA, 1986) Contemp. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1989, pp. 295–316. MR 1000608, DOI 10.1090/conm/090/1000608
Jin-Gen Yang, Sextic curves with simple singularities, Tohoku Math. J. (2) 48 (1996), no. 2, 203–227. MR 1387816, DOI 10.2748/tmj/1178225377
Jin-Gen Yang, Enumeration of combinations of rational double points on quartic surfaces, Singularities and complex geometry (Beijing, 1994) AMS/IP Stud. Adv. Math., vol. 5, Amer. Math. Soc., Providence, RI, 1997, pp. 275–312. MR 1468283, DOI 10.1090/amsip/005/19