construct a Jacobian elliptic fibration, we define a divisor In in Pic X who ... proving the equivalence of the existence of any 2 of the 5 elliptic fibrations is to.
J . M ath. K yoto U niv. (JMKYAZ)
38-3 (1998) 419-438
K3 surfaces with order five automorphisms By K . OGUISO a n d D .-Q . ZHANG
Introduction
L et T be a normal projective algebraic surface over C with at worst quotient singular points (= Kawamata lo g terminal singular p o in ts in th e sense of [Ka, K o ]). T is called a log Enriques surface if the irregularity h 1 ( T, (9T ) -= 0 and if a positive multiple /K T of the canononial Weil divisor K T is linearly equivalent to z e r o . W ithout loss o f generality, we always assume fro m n o w o n th a t a log Enriques surface h a s n o D u V a l singular points (see th e com m ents after [Z1, Proposition 1.3]). T he smallest integer / > 0 satisfying /KT — 0 is called the (global) index of T. It can be proved that / 6 6 (cf. [Z11). Recently, R. Blache [B1] has shown th a t / < 21. H e also studied th e "generalized" log Enriques surfaces where log canonical singular points are allowed. Rational log Enriques surfaces T can be regarded as degenerations of K3 or Enriques surfaces, w hich i n turn played im portant roles i n Enriques-Kodaira's classification th e o ry f o r s u r f a c e s . I n [Al, A . A lexeev [ A l h a s p r o v e d the boundedness of families of these T. In 3-dimensional case, the base surfaces W of elliptically fibred Calabi-Yau threefolds ( P DI : X —4 W w ith D.c 2 (X ) 0 are rational log Enriques surfaces (cf. [0 1 -0 4 1). L e t T b e a lo g Enriques surface of index I. The G alois Z//Z-cover 7E :
I —1
Y := Spec e.o r (D i = 0 C T ( —iKT ) —>T
is called the (global) canonical covering. Clearly, Y is either an abelian surface or a K 3 surface w ith a t w o rst D u V a l singular p o in ts . W e n o te a lso th a t it is unramified over th e sm ooth p a r t T— Sing T. W e say that T is o f Type A m o r D , if Y has a singular point of Dynkin type A„, o r D n ; T is o f actual T y pe ( 0 , A,n ) 0 ( SD ,,) ( 0 E k ) if S in g Y is of type (0 A,n ) ( 0 D n ) t (10Ek)• A round 1989, M . R e id a n d I . Naruki asked the second au th o r about the uniqueness o f rational log Enriques surface to Type D 1 9 . T he determinations of all isomorphism classes of rational log Enriques surfaces T of Type A 1 9 , D 1 9 , A l8 and D i g have been done in [OZ1, 2 1 (see also [R1]). A s a corrolary, the minimal ,
Communicated by K . Ueno, A pril 4, 1997
K Oguiso and D.-Q. Zhang
420
resolutions X d of the canonical covers of such T are isomorphic to the unique K3 surface of Picard number 20 and discriminant d for d = 3 o r 4 . So there are only tw o such Xd• Here we consider the cases A r and D 1 7 . W e will get some new K3 surface other than X d above (cf. M ain T h e o re m 3 ). O ur m ain results a re a s follows: Theorem 1. ( 1 ) T here is no rational log Enriques surface of T y pe D17. (2) Each rational log Enrigues surface o f Type A 17 has index 2 , 3 , 4 o r 5. Remark 2. The isomorphism classes of rational log Enriques surfaces of Type A 17 a n d in d e x 2 , 3 o r 4 a re determined in [Z3, Z4].
Main Theorem 3. ( 1 ) There are, up to isom orphism s, exactly tw o rational log Enrigues surfaces of index 5 and Type A r . These two are given as T(9), T(14) in Ex am ple 2.1, and both of them are of actual Type A17. (2) L e t Y (i) — > T(i) b e the canonical Galois ZI5Z-cover, g(i) : X (i) — > Y (i) the minimal resolution and 4(i) := g(i) (Sing Y (i)) the exceptional divisor, which is of Dy nk in type A17. W rite Gal( Y (i)1T (i))= . T hen the pairs (X (i),) are eguiv ariantly isom orphic to each other and the f ix ed locus (point wise) X (i) i s a disjoint union of 3 smooth rational curves, w hich are contained in 4 ( i) , an d 13 points. M oreov er, rank Pic X (i) = 1 8 and Idet (Pic Xi )1 = 5. -1
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(i)
The pair (X (i), 0 (i)>) above is characterised in the following result, which is so rt o f th e generalisation of Shioda-Inose's pairs in [OZ1]. -