J. Math. Soc. Japan Vol. 53, No. 3, 2001 Galois points on quatic sufaces By Hisao Yoshihaa (Received Nov. 29, 1999) (Revised Ma. 30, 2000) Abstact. Let S be a smooth hypesuface in the pojective thee space and conside a pojection of S fom P A S to a plane H. This pojection induces an extension of elds k S =k H. The point P is called a Galois point if the extension is Galois. We study stuctues of quatic sufaces focusing on Galois points. We will show that the numbe of the Galois points is zeo, one, two, fou o eight and the existence of some ule of distibution of the Galois points. 1. Intoduction. Let k be an algebaically closed eld of chaacteistic zeo. We x it as the gound eld of ou discussion. Let S be a smooth hypesuface of degee d in the pojective thee space P 3 ˆ P 3 k, whee we assume that d b 4. Let K ˆ k S be the ational function eld of S. A sub eld K m is said to be a maximal ational sub eld if it is ational, i.e., a puely tanscendental extension of k, and is not contained in any othe ational sub eld. It seems inteesting to study the stuctue of the extension K=K m. If we know it, we will be able to classify of all the sub elds of K. Because, by Zaiski-Castelnuovo's theoem any sub eld (which is not k) ofk m is ational. So that it is su½cient to study what elds exist between K and K m. Let L be the Galois closue of K=K m, then we need to study the stuctue of the Galois goup Gal L=K m. Fo that eason, the study we have to do st is to nd when the extension is Galois (cf. [6]). Hee the meaning ``when'' is a little ambiguous, it will become clea if we conside the model of K as follows. Fo each point P A S, let p P : S!H be a pojection of S fom P to a plane H. This ational map induces the extension of elds K=k H. We know that the degee of iationality of S is d 1od 2 (cf. [1], [10]), hence k H is a maximal ational sub eld. Clealy the stuctue of this extension does not depend on H, but on P, so that we wite K P instead of k H. Theefoe, the above question is equivalent to say fo which point P A S the extension K=K P becomes Galois. The study following the above method has been done fo cuves of degees 4 and 5 (cf. [6], [7]). 2000 Mathematics Subject Classi cation. Pimay 14J70; Seconday 14J27, 14J28. Key Wods and Phases. Quatic suface, Pojective tansfomation, Galois point, Elliptic suface.
732 H. Yoshihaa Howeve we have to note hee that not all maximal ational sub elds ae obtained as the pojection above, i.e., thee ae many maximal ational sub elds which cannot be obtained fom the pojections. Acknowledgement. The autho expesses his gatitude to M. Takeshi Takahashi fo calculating the numbe of lines on S 8 in Remak 2.8 and nding an example in Example 2.9. 2. Statement of esults. We use the same notation as is used in Section 1. Definition 1. is Galois. A point P A S is called a Galois point if the extension K=K P Let S be the set of lines on S passing though P. Then S 0 ˆ Sn S U P becomes a coveing of U of degee d 1bypP 0 ˆ p Pj S 0, whee U ˆ P 2 nfa nitely many pointsg. Hence P is a Galois point if and only if pp 0 is a Galois coveing. Fist, we want to know the set of Galois points. Theoem 1. Suppose that d b 4. Then the numbe of Galois points is nite. If S is geneal in the class of sufaces with degee d, then the numbe is zeo. Let d ˆ d S denote the numbe of the Galois points. invaiant unde pojective tansfomations of S. Hee we mention a note, which will be often used late. Note that d is Note 2.1. If H P is a geneal plane among the ones passing though P, then S V H P is a smooth cuve. Heeafte we estict ouselves to the case whee d ˆ 4 and we assume that k is the eld of complex numbes. We want to know the exact value d and the place whee the Galois points exist. Fist we nd necessay conditions that a point to be a Galois one. Let X : Y : Z : W be homogeneous coodinates on P 3. Then we have the following standad fom fo the equation of S if S has a Galois point. Theoem 2. If thee exists a Galois point P on S, then S is pojectively equivalent to the suface given by the equation ZW 3 G X; Y; Z ˆ0, whee G is a quatic fom and P ˆ 0 : 0 : 0 : 1. Let T P denote the tangent plane of S at P.
Galois points on quatic sufaces 733 Coollay 2.2. lines. If P is a Galois point, then T P V S consists of fou distinct Let F be the homogeneous de ning equation of S and H F be the Hessian of F. Then we have anothe necessay condition. Poposition 2.3. If P is a Galois point, then H F P ˆ0. Let P be a point on S with P ˆ 0 : 0 : 0 : 1 and put x ˆ X=W; y ˆ Y=W, z ˆ Z=W and f x; y; z ˆF X; Y; Z; W =W 4 ˆ P4 iˆ1 f i, whee f i is a homogeneous pat of f with degee i i ˆ 1; 2; 3; 4. Using these expessions, we have the following citeion that a point to be a Galois point. Poposition 2.4. only if f 2 2 ˆ 3 f 1 f 3. Unde the notation above, a point P is a Galois point if and In the pape [6], we have studied Galois points on quatic cuves. The following poposition is also useful fo checking whethe a point is Galois o not. Poposition 2.5. Suppose that H P is a geneal plane passing though P and let C ˆ S V H P be a quatic cuve. Then a point P is a Galois point of S if and only if it is a Galois point of C. Fo a Galois point P, take thee lines fl i g fom S V T P and conside a diviso D ˆ l 1 l 2 l 3. The ational map associated with the complete linea system jdj gives S a stuctue of a be space, i.e., we have the following. Lemma 2.6. elliptic suface. If thee exists a Galois point on S, then S has a stuctue of an Definition 2. We call the suface with the stuctue de ned in Lemma 2.6 an elliptic suface associated with the Galois point. Note that thee ae fou possibilities fo the choice of the lines S V T P, hence thee ae fou elliptic sufaces associated with the Galois point. Obseving the singula bes of the elliptic suface, we obtain the following. Theoem 3. If S is a quatic suface, then d S ˆ0; 1; 2; 4 o 8. Especially, d S ˆ8 if and only if S is pojectively equivalent to the suface S 8 given by the equation XY 3 ZW 3 X 4 Z 4 ˆ 0. Moeove, the distibution of the Galois points ae illustated as follows, whee the dots indicate Galois points, the lines indicate the lines on S and the boken lines indicate the lines in P 3 but not on S.
734 H. Yoshihaa By Lemma 2.2 thee exist fou lines on S passing though each Galois point, but we omit to illustate hee some of them. Note that the coodinates of the Galois points on S 8 ae 0 : 0 : 0 : 1, 0 : 0 : z : 1, 0 : 0 : z 3 : 1, 0 : 0 : z 5 : 1, 0 : 1 : 0 : 0, z : 1 : 0 : 0, z 3 : 1 : 0 : 0 and z 5 : 1 : 0 : 0, whee z is a pimitive sixth oot of unity. Futhemoe thee exist some ules between Galois points and lines on S, fo the details, see Lemma 3.10. Coollay 2.7. If d S ˆ2 o 8, then S has a stuctue of an elliptic suface whose singula bes ae all of type IV (in the sense of Kodaia's notation in [4]). A quatic suface is a K3 suface, and it is known that the maximum numbe of lines lying on a quatic suface is 64 (cf. [9]). The suface S 8 in Theoem 3 is the most special one among quatic sufaces as we see below. Remak 2.8. The suface S 8 has the following popeties: (a) The numbe of lines on S 8 is 64. (b) The suface S 8 is a singula K3 suface (cf. [2]). Fo each value of d in Theoem 3, thee ae many examples taking the value as follows. Example 2.9. (1) If S is (i) a geneal quatic suface, o (ii) the Femat quatic given by the equation X 4 Y 4 Z 4 W 4 ˆ 0, o (iii) the suface given by the equation X 3 Y Y 3 Z Z 3 W W 3 X ˆ 0, then d S ˆ0. (2) If S is the suface given by the equation (i) ZW 3 G X; Y; Z ˆ0, whee G is a geneal quatic fom, o (ii) ZW 3 X 4 Y 4 YZ 3 ˆ 0, then d S ˆ1.
Galois points on quatic sufaces 735 (3) Suppose that S is given by the equation XY 3 ZW 3 H X; Z ˆ0, whee H X; Z ˆP4 iˆ0 c ix i Z 4 i, c 0 c 4 0 0. Then, (a) if at least one of c i i ˆ 1; 2; 3 is not zeo, then d S ˆ2, (b) if c 1 ˆ c 2 ˆ c 3 ˆ 0, then d S ˆ8. (4) If S is the suface given by the equation (i) ZW 3 Z 4 H X; Y ˆ0, whee H is a geneal quatic fom, o (ii) ZW 3 X 4 Y 4 Z 4 ˆ 0, then d S ˆ4. Remak 2.10. Although thee ae many lines on the Femat quatic, indeed thee ae 48 pieces lines on it, thee exists no Galois point. 3. Poofs and some othe esults. Fist we pove Theoem 1. Let P be a Galois point and s be an element of Gal K=K P. Then s induces a biational tansfomation of S ove k H G k P 2, which tuns out an isomophism, since S is the minimal model of the eld K. We claim that s is a estiction of a pojective tansfomation of P 3. This assetion is a well known fact in the case whee d b 5, so that we pove it when d ˆ 4. Let H P be a plane passing though P. If it is geneal, then C ˆ S V H P is a smooth quatic cuve by Note 2.1. Let l P be a line in H P passing though P. If l P is geneal, then C V l P consists of fou distinct points fp; P 1 ; P 2 ; P 3 g. By de nition s induces a pemutaiton of the set fp 1 ; P 2 ; P 3 g. Hence we infe that s C ˆC, especially we have that s P ˆP. This implies that s f A H 0 S; O S H P if f A H 0 S; O S H P. Thus s induces an element of Aut H 0 S; O S H P. Since H 0 S; O S H P G H 0 P 3 ; O H P, s is a estiction of a pojective tansfomation of P 3. We denote it by M s A PGL 4; k. Definition 3. We call s an automophism belonging to the Galois point P and M s the epesentation of s. Let L S denote the set of automophisms of S induced by the pojective tansfomations which leave S invaiant. Suppose that s and s 0 ae automophisms belonging to Galois points P and P 0 espectively. Then, it is easy to see that s 0 s 0 if P and P 0 ae distinct points, hence M s 0 M s 0. Thus we infe eadily Theoem 1 fom the following lemma (cf. [5]). Lemma 3.1. The goup L S has a nite ode if d b 3. If S is geneic, then L S consists of only an identity element. Next we investigate the stuctue of the coveing pp 0 : S 0! U. Let C be the disciminant detemined by the pojection p P. Let us expess C explicitly using a suitable a½ne coodinates as follows. Fist we take homogeneous coodinates X : Y : Z : W on P 3 satisfying the following conditions:
736 H. Yoshihaa (1) P ˆ 0 : 0 : 0 : 1 (2) The plane given by Z ˆ 0 is the tangent plane of S at P. (3) The plane given by X ˆ 0 is not a tangent plane at any point of S. (4) The numbe of lines passing though P and touching S at S V fw ˆ 0g is nite. (5) The line given by the equations X ˆ Y ˆ 0 does not touch S. We use the notation in the pevious sections and conside the pojection p P esticted to the a½ne pat W 0 0. Let m be the blowing up of A 3 ˆ P 3 nfw ˆ 0g with cente P. Then in an a½ne pat, m can be expessed as m x; s; t ˆ x; sx; tx. Since the stuctue of the extension K=K P does not depend on the choice of planes H, we may assume that p P x; sx; tx ˆ s; t. Thus ~p P ˆ p p m maps x; s; t to s; t. The extension of elds is not changed if we take ~p P instead of p P. The de ning equation (of the a½ne pat) of the pope tansfom of S is f? x; s; t ˆ f x; sx; tx x A k x; s; tš: Let c ˆ c s; t be the disciminant of f? x; s; t with espect to x. Then C is obtained by homogenizing c and we have that deg C ˆ deg c by the choice of coodinates (1) @ (5). Let I R X; Y denote the intesection numbe of X and Y at R and let X; Y ˆPR I R X; Y. We will conside the intesection numbes on P 2, P 3 o S, and use the same notation. Lemma 3.2. deg C ˆ d 2 d 2. Poof. Let G be the diviso of C on P 2 and G; l be the intesection numbe of G and a line l on P 2. Then we have that deg c ˆ G; l. If H ˆ p 1 P l, then C ˆ S V H is a smooth cuve of degee d if l is geneal by Note 2.1. Using Huwitz's theoem, we infe that the degee of the disciminant fo the smooth cuve C is d 2 d 2 (cf. [6]). Heeafte we assume that d ˆ 4. Let P be a Galois point. Then Gal K=K P is the cyclic goup of ode thee and let s be a geneato of it. Lemma 3.3. The subvaiety F s ˆfQ A S j s Q ˆQg contains a cuve. Poof. Let S be the set of fou lines S V T P. Then we have that s S ˆS and p 0 : S 0! U is a tiple Galois coveing. By Lemma 3.2 we have that deg G ˆ 10. Theefoe p 0 is ami ed along p 1 U V G, thus F s contains a cuve. When A ˆ a ij is a diagonal matix of size fou and a ii ˆ a i i ˆ 1; 2; 3; 4, we denote it by a 1 I a 2 I a 3 I a 4. Let M s A PGL 4; k be the epesen-
Galois points on quatic sufaces 737 tation of s. Since s 3 ˆ id, the matix M s is simila to o I o i I o j I 1, whee o is a pimitive cubic oot of 1 and 0 a i a j a 2. By taking a suitable pojective change of coodinates, we may assume that M s is expessed as above. Fom Lemma 3.3 we infe that s must x a hypeplane. This implies that thee eigen values of M s coincide, hence we have that i ˆ j ˆ 0 o i ˆ j ˆ 1. Consequently we may assume that i ˆ j ˆ 1. We expess F as P 4 iˆ0 F iw 4 i whee F i A k X; Y; ZŠ is a homogeneous polynomial of degee i 0 a i a 4. Since s A L S, we have that F s ˆ lf fo some l A knf0g. Whence we can conclude easily that F has an expession as F 1 W 3 F 4. Since F 1 0 0, this fom can be tansfomed to the standad one by a pojective tansfomation. Thus we complete the poof of Theoem 2. Suppose that P ˆ 0 : 0 : 0 : 1 is a Galois point. Then the equation of S can be given by ZW 3 G X; Y; Z ˆ0. The equation of the tangent plane T P is Z ˆ 0. Since S is smooth, the fom G X; Y; 0 has no multiple facto, this poves Coollay 2.2. The poof of Poposition 2.3 is easy fom the following lemma. Lemma 3.4 (x7, [8]). Let f be the estiction of f to the a½ne tangent plane of S at P. Then the Taylo expansion of f at P stats with a nondegeneate quadatic fom if and only if H F P 0 0. Next we pove Poposition 2.4. If P is a Galois point, then making use of Theoem 2, we easily obtain f 2 2 ˆ 3 f 1 f 3. Convesely we assume this elation. As we have de ned above, f? can be expessed as f? x; s; t ˆ f 4 1; s; t x 3 f 3 1; s; t x 2 f 2 1; s; t x t. Then we have that K ˆ k x; s; t, whee f? x; s; t ˆ 0. Since L P ˆ k x 0 ; x; s; t, whee x 0 is anothe oot of f? x 0 ; s; t ˆ0, we can wite L P ˆ k x; s; t; u, whee u 2 ˆ f 3 f 4 2 4 f 1 f 4 ˆ f 2 2 2 f 1 f 2 3 f 2 1 4 f 1 f 3 : Thus, if f 2 2 ˆ 3 f 1 f 3,thenu 2 becomes a complete squae in k x; s; t. Hence K=K P is a Galois extension. The poof of Poposition 2.5 may be clea if we conside the banching diviso. The point P is a Galois point if and only if the diviso G is two times of some diviso. Since H P is a geneal plane passing though P and the disciminant of the pojection of C fom P to a line is obtained by esticting C, the assetion may be clea. When P is a Galois point, let P 3 iˆ0 l i be the fou lines S V T P and put D ˆ l 1 l 2 l 3. The complete linea system jdj is obtained as follows. Conside the set H ˆfH l j H l I l 0 ; H l is a planeg. Then S V H l can be witten as a diviso l 0 C l, whee C l is a cuve. Taking o the xed pat l 0
738 H. Yoshihaa fom the linea system fs V H l j H l A Hg, we obtain a base points fee linea system, which coincides with jdj. Thus we obtain the following lemma. Lemma 3.5. We have that D 2 ˆ 0, D; l 0 ˆ3, dim H 0 S; O D ˆ 2 and the complete linea system jdj has no base point. Consequently we obtain an elliptic suface f ˆ f jdj : S! P 1 in Lemma 2.6. Now we poceed with the poof of Theoem 3. The elliptic bation f : S! P 1 has a singula be D, which is of type IV. Lemma 3.6. The automophism s peseves each be of f, i.e., s F a ˆF a, whee F a ˆ f 1 a fo a A P 1. Especially, a smooth be is an elliptic cuve with an automophism of ode thee. Poof. Since s is detemined fom the pojection, we have that s l 0 C l ˆ l 0 C l and s l 0 ˆl 0, whee S V H l ˆ C l l 0 as above. Hence we obtain s C l ˆC l. Note that l 0 =hsi is the base cuve P 1 of the elliptic bation. Thus f j l0 : l 0! P 1 is a tiple Galois coveing with two banching points which ae xed points of sj l0, i.e., l 0 is a tiple section of f. Hence we infe the following. Lemma 3.7. The elliptic bation f : S! P 1 has at most one singula be D 0 besides D satisfying that D 0 is of type IV and D 0 V l 0 consists of one point. Especially, if l is a line on S, then the numbe of Galois points on l is at most two. Lemma 3.8. If P 0 is anothe Galois point and s 0 is an automophism belonging to P 0, then s P 0 is also a Galois point and ss 0 s 1 is an automophism belonging to s P 0. Poof. Put s 00 ˆ ss 0 s 1 and s P 0 ˆP 00. Suppose that l is a line passing though P 00 and I Q S; l b 2 fo some point Q A S. Then we have that I Q S; l ˆ I s Q S; l 0, whee l 0 ˆ s 1 l. Since l 0 passes though the Galois point P 0, 1 we have that I Q S; l b 3, this means that P 00 is a Galois point. Since s 0 is an automophism belonging to P 0, we have that s 0 l 0 ˆl 0. Hence we have that s 00 l ˆl, this implies that s 00 is an automophism belonging to P 00. Lemma 3.9. Suppose that P and P 0 ae two Galois points and the line l passing though these points does not lie on S. Then in Lemma 3.8 we have that s P 0 0 P 0, hence thee exist two moe Galois points s P 0 and s 2 P 0. Poof. In case I P l; S b 2, the line l is contained in T P S, hence it lies on S by Lemma 2.2. Theefoe we have that I P l; S ˆ1. Suppose that s P 0 ˆP 0. Then we have that I P 0 l; S ˆ3. By the same easoning as above we have that l must lie on S, which is a contadiction.
Galois points on quatic sufaces 739 Suppose that d ˆ d S b 2 and take anothe Galois point P 0. Then one of the following cases takes place. (i) Thee exists a unique i satisfying l i C P 0 i ˆ 0; 1; 2; 3, o (ii) Thee does not exist i satisfying l i C P 0. In the case (i) we may assume that i ˆ 0 and conside the elliptic be space f : S! P 1 associated with the Galois point P with the singula be l 1 l 2 l 3. By Lemma 2.2, S V T P 0 can be expessed as P 3 iˆ0 l i 0, whee l i 0 i ˆ 0; 1; 2; 3 is a line on S. Since thee exist just fou lines on S passing though P 0, we may assume that l 0 ˆ l0 0. Thus D0 ˆ l1 0 l 2 0 l 3 0 is a singula be of f, especially D V D0 ˆ q. On the contay in the case (ii), put l ˆ T P V T P 0. Since the degee of S is fou, we infe that l does not lie on S, and P B l and P 0 B l. Let X be the set consisting of the Galois points and the lines on S passing though at least one Galois point. Combining the esults obtained above, we conclude the following popeties of distibution of the Galois points. Lemma 3.10. The set X has the following popeties. (P1) Fo each point of X, thee exist fou lines of X passing though it. (P2) Fo any two points P and P 0 of X, the con guation of lines ae illustated as follows, whee a line indicates a line ofxand a boken line indicates a line in P 3 but not belonging to X.
740 H. Yoshihaa (P3) Fo each line l of X, thee exist one o two points of X lying on l. (P4) Fo each point P ofx, thee exists an automophism s of S belonging to P, which has the following popeties: (a) s P ˆP. (b) s has an ode thee. (c) s induces a pemutation of elements in X. (d) If P 3 iˆ0 l i is the lines passing though P, i.e., it is T P V S, then s l i ˆl i 0 a i a 3 and sj li is an automophism of l i with an ode thee and xes two points. (P5) If thee does not exist a line ofxpassing though two points P and P 0 of X, then thee exist two moe points ofx. These fou points ae collinea, but the line passing though them is not an element of X. In the case whee d b 2, thee ae two lines on S not meeting each othe as we have seen above (P2). Refeing to Poposition 1 in [10], we obtain the following. Coollay 3.11. If d S b 2, then the degee of iationality of S is two. Let us pove Theoem 3 by examining the following cases sepaately. (1) Fo each line l of X, thee exists just one point of X on l. (2) Thee exists a line l ofxon which thee exist two points P and P 0 ofx. Take one point P of X and conside the associated elliptic suface f : S! P 1. Hee we assume that k is the eld of complex numbes. Then the topological Eule chaacteistic of S is 24. In the case (1), fo each point Q 0P ofx, if we choose a suitable a line l fom the ieducible components of S V T Q, then l meets l 0 and does not meet D by (P2). That is, l is contained in a singula be of f. Suppose that d b 5. Then fou Galois points P ˆ P 1 ; P 2 ; P 3 ˆ s P 2 and P 4 ˆ s P 3 ae collinea and suppose that Q ˆ Q 1 is anothe Galois point. Then we can nd two moe Galois points Q 2 ˆ s Q and Q 3 ˆ s Q 2. Next we conside the automophism s 2 belonging to P 2. Then we can nd new Galois points s 2 Q 1, s2 2 Q 1, s 2 Q 2, s2 2 Q 2, s 2 Q 3 and s2 2 Q 3. Next we conside the automophism belonging to the Galois point P 3. Then we can nd new Galois points, etc. In this way, continuing these pocesses, we will be able to nd moe than 24 pieces of Galois points. This contadicts to the Eule chaacteistic of S. Thus in this case d ˆ 1 o 4. In the case (2), st we pove the following. Lemma 3.12. Suppose that thee exists a line l on S satisfying that thee exist two Galois points on l. Then the de ning equation of S can be given by the equation XY 3 ZW 3 H X; Z ˆ0, whee H X; Z is a quatic fom. The
Galois points on quatic sufaces 741 coodinates of Galois points ae 0 : 0 : 0 : 1 and 0 : 1 : 0 : 0 and the equations of the line ae given by X ˆ Z ˆ 0. Especially each singula be of the elliptic suface associated with the Galois points is of type IV. Poof. By Theoem 2 we have the standad fom ZW 3 G X; Y; Z. Since G X; Y; 0 factos into fou distinct linea foms, we can tansfom G X; Y; 0 to X G 3 X; Y, whee G 3 is a cubic fom. We may assume that two Galois points P and P 0 lie on the line given by the equations l 0 : X ˆ Z ˆ 0. Since the automophism s belonging to P xes the point at in nity W ˆ 0, we see that P ˆ 0 : 0 : 0 : 1 and P 0 ˆ 0 : 1 : 0 : 0 by Lemma 3.7. The cuve C ˆ S V fw ˆ 0g is a smooth quatic cuve given by the equation G X; Y; Z ˆ0on the plane W ˆ 0. Since P 0 lies on the plane given by W ˆ 0, the point P 0 is also a Galois point of the quatic cuve C. This assetion can be poved by simila agument of the poof of Poposition 2.5. By the way, G can be witten as P 4 iˆ1 G i X; Z Y 4 i, hence G X; Y; Z =Y 4 ˆ g x; z ˆP4 iˆ1 g i x; z. It is easy to see that g 1 x; 0 0 0, hence we can tansfom g to the expession whose linea pat is x. So that we may assume that g 1 ˆ x. The simila assetion to Poposition 2.4 holds tue fo quatic cuves, i.e., we have that g2 2 ˆ 3xf 3. This implies that g 2 and g 3 ae divisible by x, fom which we infe that, by taking pojective tansfomations, G can be expessed as XY 3 H X; Z. Since each be of f is obtained by cutting S by ax bz ˆ 0, we infe easily the last assetion. Claim 1. Thee is no Galois point not lying on T P U T P 0 V S. Poof. Suppose the contay. Then, let Q be such a point. By the popety (P5) thee exist thee points Q ˆ Q 1 ; Q 2 ˆ s Q 1 and Q 3 ˆ s Q 2, which ae collinea. Coesponding to each point Q i i ˆ 1; 2; 3, thee exists a line m i meeting l 0 and does not meet D by (P2). By Lemma 3.12 these thee lines make a singula be of type IV. Moeove, take an automophism s 0 belonging to P 0. Consideing s 0 Q i and s 0 2 Q i iˆ1; 2; 3 and using the popety (P3) and Lemma 3.12, we obtain a singula be containing m 1 m 2 m 3, which cannot appea as a be of any elliptic be space (cf. [4]). This is a contadiction. Theefoe we conclude that d a 8 in view of (P3). Claim 2. In the case (2) we have that d ˆ 2, 5 o 8. Poof. In case d b 3, we use (P5) and Claim 1. As we see fom the illustation below, we conclude that d ˆ 5 o 8.
742 H. Yoshihaa We now pove that the suface with d ˆ 5 cannot exist. Let H X; Z be expessed as P 4 iˆ0 a ix 4 i Z i in Lemma 3.12, whee a 0 0 0. Suppose that d b 5. Then R ˆ x : 1 : 0 : 0, whee a 0 x 3 1 ˆ 0, is also a Galois point by Claim 1 and R is on the plane given by W ˆ 0. Putting u ˆ X=Y; v ˆ Z=Y; w ˆ W=Y and h ˆ F=Y 4, we have that h u; v; w ˆu vw 3 h 4 u; v. Moeove, putting u 0 ˆ u x, we have that h 0 u 0 ; v; w ˆh u 0 x; v; w. Hee we make use of Poposition 2.4. Then we obtain that a 1 ˆ a 2 ˆ a 3 ˆ 0. This implies that d ˆ 8. Combining the assetions obtained above, we see that d S ˆ8 if and only if S ˆ S 8. Thus we complete the poof of Theoem 3. Remak 3.13. Suppose that s and s 0 ae the automophisms belonging to Galois points P and P 0 espectively. Then, ss 0 ˆ s 0 s if and only if the line l passing though P and P 0 lies on S. Poof. If l lies on S, then by Lemma 3.12 we may assume that M s ˆ o I o I o I 1 and M s 0 ˆ o i I 1 I o i I o i, whee i ˆ 1 o 2. Especially s and s 0 ae commutative. Convesely, if ss 0 ˆ s 0 s, then M s and M s 0 can be diagonalized simultaneously. Hence in view of Lemma 3.3, s and s 0 have the same pojective epesentation as above. By using the action of M s and M s 0 on the de ning equation of S, we obtain the same de ning equation as in Lemma 3.12, this implies that l lies on S. Remak 3.14. Let G be the goup geneated by the automophisms belonging to the Galois points on S 8. We will show in the fothcoming pape [3] that G has an ode 288 and some othe popeties. We mention the methods to check Example 2.9. By Poposition 2.3 all the Galois points exist on the cuve given by the equations F ˆ H F ˆ0. Poposition 2.5 may be helpful fo checking some example such as the Femat quatic. Next, we use the distibution ule of the Galois points in Lemma 3.10. Using Poposition 2.4, we will be able to nd all the Galois points. Finally we aise poblems. Poblem 3.15. d ˆ 1. (1) Find the degees of iationality fo the sufaces with
Galois points on quatic sufaces 743 (2) Let S ~ be the nonsingula pojective model of the Galois closue of K=K P, whee K is the function eld of a quatic suface S. Suppose that P is not a Galois point. Then is it tue that the Kodaia dimension of S ~ is two? Moeove, nd seveal geometic invaiants of it as we have done in the case of quatic cuves (cf. [6]). (3) Descibe the con guation of X fo the suface S 8. Refeences [ 1 ] R. Cotini, Degee of iationality of smooth suface of P 3, to appea. [ 2 ] H. Inose, On cetain Kumme suface which can be ealized as non-singula quatic sufaces in P 3, J. Fac. Sci. Univ. Tokyo, Sec. IA 23 (1976), 545±560. [ 3 ] M. Kanazawa, T. Takahashi and H. Yoshihaa, The goup geneated by automophisms belonging to Galois points of the quatic suface, to appea. [ 4 ] K. Kodaia, On compact analytic sufaces, II, Ann. of Math., 77 (1963), 563±626. [ 5 ] H. Matsumua and P. Monsky, On the automophisms of hypesufaces, J. Mat. Kyoto Univ., 3 (1964), 347±361. [ 6 ] K. Miua and H. Yoshihaa, Field theoy fo function elds of plane quatic cuves, J. Algeba, 226 (2000) 283±294. [ 7 ] ÐÐÐ, Field theoy fo the function eld of the quintic Femat cuve, Comm. Algeba, 28 (2000), 1979±1988. [ 8 ] M. Reid, ``Undegaduate Algebaic Geomety,'' London Math. Soc. Student Texts 12. [ 9 ] B. Sege, The maximum numbe of lines lying on a quatic sufaces, Oxfod Quately Jounal, 14 (1943) 86±96. [10] H. Yoshihaa, Degee of iationality of an algebaic suface, J. Algeba, 167 (1994), 634± 640. Hisao Yoshihaa Depatment of Mathematics Faculty of Science Niigata Univesity Niigata 950-2181 Japan E-mail: yosihaa@math.sc.niigata-u.ac.jp