We present a complete list of extremal elliptic $K3$ surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an $ADE$-configuration of smooth rational curves on a $K3$ surface to be trivial (Proposition 4.1 and Theorem 4.3).
References
E. Artal-Bartolo, H. Tokunaga and D. Q. Zhang, Miranda-Persson's problem on extremal elliptic $K3$ surfaces , preprint. http://xxx.lanl.gov/list/math.AG, 9809065. E. Artal-Bartolo, H. Tokunaga and D. Q. Zhang, Miranda-Persson's problem on extremal elliptic $K3$ surfaces , preprint. http://xxx.lanl.gov/list/math.AG, 9809065.
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Second edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer, New York (1993). MR1194619 0785.11036 J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Second edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer, New York (1993). MR1194619 0785.11036
––––, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces , With an appendix by Shigeru Mukai, Duke Math. J., 92 (1998), no. 3, 593–603. MR1620514 10.1215/S0012-7094-98-09217-1 euclid.dmj/1077231678 0958.14025 ––––, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces , With an appendix by Shigeru Mukai, Duke Math. J., 92 (1998), no. 3, 593–603. MR1620514 10.1215/S0012-7094-98-09217-1 euclid.dmj/1077231678 0958.14025
R. Miranda and U. Persson, Mordell-Weil groups of extremal elliptic $K3$ surfaces , Problems in the theory of surfaces and their classification (Cortona, 1988), Sympos. Math., XXXII, Academic Press, London (1991), 167–192. MR1273378 0827.14013 R. Miranda and U. Persson, Mordell-Weil groups of extremal elliptic $K3$ surfaces , Problems in the theory of surfaces and their classification (Cortona, 1988), Sympos. Math., XXXII, Academic Press, London (1991), 167–192. MR1273378 0827.14013
D. R. Morrison, On $K3$ surfaces with large Picard number , Invent. Math., 75 (1984), no. 1, 105–121. MR728142 10.1007/BF01403093 0509.14034 D. R. Morrison, On $K3$ surfaces with large Picard number , Invent. Math., 75 (1984), no. 1, 105–121. MR728142 10.1007/BF01403093 0509.14034
S. Mukai, Finite groups of automorphisms of $K3$ surfaces and the Mathieu group , Invent. Math., 94 (1988), no. 1, 183–221. MR958597 10.1007/BF01394352 0705.14045 S. Mukai, Finite groups of automorphisms of $K3$ surfaces and the Mathieu group , Invent. Math., 94 (1988), no. 1, 183–221. MR958597 10.1007/BF01394352 0705.14045
K. Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell-Weil groups , Japan. J. Math. (N.S.), 22 (1996), no. 2, 293–347. MR1432379 K. Nishiyama, The Jacobian fibrations on some $K3$ surfaces and their Mordell-Weil groups , Japan. J. Math. (N.S.), 22 (1996), no. 2, 293–347. MR1432379
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, 7, Springer, New York (1973). MR344216 0256.12001 J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, 7, Springer, New York (1973). MR344216 0256.12001
T. Shioda and H. Inose, On singular $K3$ surfaces. Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), 119–136. MR441982 0374.14006 T. Shioda and H. Inose, On singular $K3$ surfaces. Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), 119–136. MR441982 0374.14006
A. N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces , Invent. Math., 61 (1980), no. 3, 251–265. MR592693 10.1007/BF01390067 A. N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces , Invent. Math., 61 (1980), no. 3, 251–265. MR592693 10.1007/BF01390067
Q. Ye, On extremal elliptic $K3$ surfaces , preprint. http://xxx.lanl.gov/abs/math.AG, 9901081. MR1757153 0976.14026 Q. Ye, On extremal elliptic $K3$ surfaces , preprint. http://xxx.lanl.gov/abs/math.AG, 9901081. MR1757153 0976.14026