Galois embedding of algebraic variety and its application to abelian surface

Galois Embedding of Algebraic Variety and its Application to Abelian Surface.

HISAOYOSHIHARA(*)

ABSTRACT- We define a Galois embedding of a projective varietyVandgive a cri- terion whether an embedding is Galois or not. Then we consider several re- presentations of the Galois group. Following the methoddevelopedin the first half, we consider the structure of an abelian surface with the Galois embedding in the latter half. We give a complete list of all possible groups andshow that the abelian surface is isogenous to the square of an elliptic curve.

1. Introduction.

The main purpose of this article is to present a new viewpoint for the study of projective varieties. Letkbe the groundfieldof our discussion, we assume it to be the fieldof complex numbers, however most results hold also for an algebraically closedfieldof characteristic zero. Let V be a nonsingular projective algebraic variety of dimensionnwith a very ample divisor D; we denote this by a pair (V;D). Let f ˆf_D:V,!P^N be the embedding of V associatedwith the complete linear system jDj, where N‡1ˆdimH⁰(V;O(D)). Suppose that W is a linear subvariety of P^N satisfying dim WˆN n 1 andW\f(V)ˆ ;. Consider the projection p_W with the center W, p_W:P^{N >}W₀, where W₀ is an n-dimensional linear subvariety not meetingW. The compositionpˆp_Wfis a surjective morphism fromV toW0Pⁿ.

Let Kˆk(V) and K0ˆk(W0) be the function fields of V andW0 re- spectively. The covering map p induces a finite extension of fields p:K₀ ,!K of degreedˆdegf(V)ˆDⁿ, which is the self-intersection

(*) Indirizzo dell'A.: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan

E-mail:yosihara@math.sc.niigata-u.ac.jp

2000 Mathematics Subject Classification number : 14N10, 14J99, 14K99

number ofD. It is easy to see that the structure of this extension does not depend on the choice ofW0 but only onW, hence we denote byKW the Galois closure of this extension andbyG_WˆGal(K_W=K₀) the Galois group of K_W=K₀. Note that G_W is isomorphic to the monodromy group of the coveringp:V !W₀, see [4].

DEFINITION1.1. In the above situation we callGWthe Galois group at W. If the extensionK=K0is Galois, we callfandWa Galois embedding and a Galois subspace for the embedding respectively.

DEFINITION1.2. A nonsingular projective algebraic varietyVis saidto have a Galois embedding if there exist a very ample divisorDsatisfying that the embedding associated withjDjhas a Galois subspace. In this case the pair (V;D) is said to define a Galois embedding.

IfWis the Galois subspace andTis a projective transformation ofP^N, thenT(W) is a Galois subspace of the embeddingTf. Therefore the ex- istence of Galois subspace does not depend on the choice of the basis giving the embedding.

REMARK1.3. For a projective varietyV, by taking a linear subvariety, we can define a Galois subspace and Galois group similarly as above. Suppose thatVis not normally embedded and there exists a linear subvarietyWsuch that the projectionpWinduces a Galois extension of fields . Then, takingDas a hyperplane section ofV in the embedding, we infer readily that (V;D) defines a Galois embedding with the same Galois group in the above sense.

By this remark, for the study of Galois subspaces, it is sufficient to consider the case whereV is normally embedded.

We have studied Galois subspaces and Galois groups for hypersurfaces in [13], [14] and[15] andspace curves in [2] and[17]. The idea introduced above is a generalization of the ones usedin these studies. In what follows, we give a criterion for an embedding to be Galois or not and show the existence of several representations of the Galois group when (V;D) d e- fines a Galois embedding. Second we study the structure of abelian sur- faces and their Galois groups when the surface has Galois embedding.

Hereafter we use the following notation andconvention:

Aut(V) : the automorphism group of a varietyV ha1; ;ami: the subgroup generatedbya1; ;am

Z_m: the cyclic group of orderm

Dm: the dihedral group of order2m (f) : the divisor defined by a functionf jGj: the order of a groupG

: the linear equivalence of divisors 1_m: the unit matrix of sizem

G(k;m) : the Grassmanian parametrizingk-planes in m-dimen- sional projective spaceP^m

em:ˆexp (2p 

p 1

=m) r:ˆe₆

D₁D₂ : the intersection number of two divisorsD₁ andD₂ on a surface

D²: the self-intersection number of a divisorDon a surface [a1;. . .;am] : the diagonal matrix with entriesa1;. . .;am

Acknowledgments. This research was startedwhen the author visited Dipartimento di Matematica, Universita' degli studi di Pavia. He thanks the Department for allowing him to use the facilities. He expresses sincere thanks Professor Pirola who gave him useful suggestions andProfessor Fukuma for giving him useful information. He thanks also the referee for carefully reading the manuscript and giving the suitable suggestions for improvements.

2. Statement of results.

By definition, ifWis the Galois subspace, then each elementsofG_Wis an automorphism of KˆK_W over K₀. Therefore it induces a birational transformation of V over V₀. This implies that G_W can be viewedas a subgroup ofBir(V=W0), the group of birational transformations ofV over W0. Further we can say the following:

REPRESENTATION1. The element of G_W turns out to be regular on V, hence we have a representation

a:GW ,!Aut(V):

(1)

Therefore, if the order ofAut(V) is small, thenV cannot have a Galois embedding. On the other hand, we have examples such that there exist infinitely many distinct Galois embeddings, see Example 4.1.

When (V;D) defines a Galois embedding, we identifyf(V) withV. Let Hbe a hyperplane ofP^NcontainingWandD⁰be the intersection divisor of

V and H. SinceD⁰D ands(D⁰)ˆD⁰, for any s2GW, we see that s induces an automorphism ofH⁰(V;O(D)).

REPRESENTATION2. We have a second representation b:G_W ,!PGL(N;C):

(2)

In the case where W is a Galois subspace we identify s2GW with b(s)2PGL(N;C) hereafter. SinceG_W is a finite subgroup ofAut(V), we can consider the quotient V=G_W andlet p_G be the quotient morphism, p_G:V !V=G_W.

PROPOSITION2.1. If(V; D)defines a Galois embedding with the Galois subspaceW such that the projection ispW :P^{N >}W0, then there exists an isomorphismg:V =G_W !W₀satisfyinggp_Gˆp. Hence the projec- tion p is a finite morphism and the fixed loci of G_W consist of only divisors.

Therefore,pturns out to be a Galois covering in the sense of Namba [7].

Now we present the criterion that (V;D) defines a Galois embedding.

THEOREM2.2. The pair(V;D)defines a Galois embedding if and only if the following conditions hold

1) There exists a subgroupGofAut(V)satisfying thatjGj ˆDⁿ. 2) There exists a G-invariant linear subspaceL ofH⁰(V;O(D))of dimensionn‡1such that, for anys2G, the restrictionsj_Lis a multiple of the identity.

3) The linear systemLhas no base points.

It is easy to see thats2G_Winduces an automorphism ofW, hence we obtain another representation ofG_Was follows. Take a basisff₀;f₁;. . .;f_Ng ofH⁰(V;O(D)) satisfying thatff0;f1;. . .;fngis a basis ofLin Theorem 2.2.

Then we have the representation

b₁(s)ˆ

l_s ...

.. . ...

l_s ...

...

0 ... Ms

0 BB BB BB BB

@

1 CC CC CC CC A :

Since the representation is completely reducible, we get another re- presentation using a direct sum decomposition:

b₂(s)ˆls1n‡1M⁰_s: Thus we can define

g(s)ˆM⁰_s2PGL(N n 1;C):

Thereforesinduces an automorphism onWgiven byM⁰_s. REPRESENTATION3. We get a third representation

g:G_W !PGL(N n 1;C):

(3)

LetG₁andG₂be the kernel andimage ofgrespectively.

THEOREM2.3. We have an exact sequence of groups 1 !G₁ !G !^g G₂ !1;

where G1is a cyclic group.

COROLLARY2.4. IfN ˆn‡1, thenGis a cyclic group.

This implies that the Galois group is cyclic if the codimension of the embedding is one (cf. [15]). Furthermore we have another representation.

REPRESENTATION4. We have a fourth representation d:G_W ,!Aut(C);

(4)

where C is a smooth curve in V given by V\L such that L is a general linear subvariety ofP^Nwith dimension N n‡1containing W.

Note that there may exist several Galois subspaces andGalois groups for one embedding (see, for example [2]). Generally we have the fol- lowing.

PROPOSITION2.5. Suppose that(V; D)defines a Galois embedding and letWi(iˆ1;2)be Galois subspaces such thatW16ˆW2. ThenG16ˆG2in Aut(V), whereGiis the Galois groupatWi.

COROLLARY2.6. IfV is of general type, then there are at most finitely many Galois subspaces for one embedding.

REMARK 2.7. It may happen that there exist infinitely many Galois subspaces for one embedding if the Kodaira dimension ofV is small. For example, if V ˆP¹ anddegDˆ3, i.e., f(V) is a twistedcubic, then the Galois lines form two dimensional locally closed subvariety of the Grass- mannianG(1;3) (cf. [17]).

3. Abelian Surfaces.

We apply the methods developed in the previous sections to the study of abelian varietiesA. In the case where dimAˆ1 anddegDˆ3 or 4, we have studied in detail in [2]. Hereafter we restrict ourselves to the case wheredimA2.

First we consider several necessary conditions that (A;D) defines a Galois embedding.

PROPOSITION3.1. Suppose that A has the Galois embedding with a Galois group G. Let R_p be the ramification divisor for the projection p:A !W0. Then, each component ofRp is a translation of an abelian subvariety of dimension n 1 and Rp(n‡1)D. In particular, Rp is very ample andRpnˆ(n‡1)ⁿjGj.

COROLLARY3.2. Simple abelian varieties do not have Galois embed- dings.

We have further results. To state them, we needto prepare some no- tation. LetG be a subgroup ofAut(A). Thens2Ghas the complex re- presentation ~szˆM(s)z‡t(s), where M(s)2GL(n;C), z2Cⁿ and t(s)2Cⁿ. Fixing the representation, we put G₀ˆ fs2Gj M(s)ˆ1_ng andHˆ fM(s)js2Gg. Then, we have the following exact sequence of groups

1 !G₀ !G !H !1:

Suppose thatAhas a Galois embedding with the Galois groupG. Then, BˆA=G₀is also an abelian variety andHG=G₀is a subgroup ofAut(B).

Moreover, it is easy to see thatB=His isomorphic toA=GW₀. Using the notation above, we have the following.

THEOREM3.3. If A has a Galois embedding, then H is not an abelian group.

COROLLARY 3.4. There exists no abelian embedding of an abelian variety of dimension2, i.e., the Galois embedding whose Galois groupis abelian does not exist.

Note that these phenomena do not appear for elliptic curves. In fact, for an elliptic curve we haveHZm, wheremˆ2;3;4 or 6. Moreover it may happen thatGis an abelian group (cf. [2]). We study the structure ofHin detail in the case of abelian surfaces.

LEMMA 3.5. Let A be an abelian surface. Assume that G is a finite automorphism group of A satisfying that A=G is isomorphic toP²and let p:A !P²be the quotient morphism. Ifdegp10, thenp(l)ˆD is very ample for each line l inP².

Combining Theorem 2.2 andLemma 3.5 together, we infer readily the following.

COROLLARY3.6. Under the same assumption and notation of Lemma 3.5, the pair(A; D)defines a Galois embedding.

Making use of the results of [5], we obtain the following theorem.

THEOREM3.7. If an abelian surface A has the Galois embedding, then H is isomorphic to one of the following:

1) D3

2) D4

3) the semi-direct product of groups: Z₂^j H⁰, where H⁰D₄ or Z_mZ_m (mˆ3;4;6)

To state case(3)more precisely, we put Z₂ˆ haiand H⁰ˆ hb;ci. Then the actions of Z₂on H⁰are as follows:In the former case H⁰D₄, we have abaˆbc²; acaˆc; c⁴ˆ1; b²ˆ1 and bcbˆc ¹. In the latter case H⁰ZmZm, we have abaˆb ¹; acaˆc ¹; b^m ˆc^mˆ1and bcˆcb.

COROLLARY3.8. IfAhas a Galois embedding, then the abelian surface BˆA=G₀is isomorphic toEEfor some elliptic curveE.

REMARK 3.9. Although we have jHj 72, the order jGj can be arbi- trarily large. In fact, for anym2 there exist examples withjGj ˆ8m²in Example 4.5 below.

REMARK3.10. In the case whereW\f(A)6ˆ ;there exist examples of Galois embeddings such thatHis an abelian group, see Remark 4.8 below.

4. Examples.

EXAMPLE 4.1. Let E be the elliptic curve C=V, where Vˆ(1;t) is a periodmatrix such that =t>0. Letaandb be the automorphisms ofE defined bya(z)ˆ zandb(z)ˆz‡1=mrespectively, wherez2Candm is a positive integer2. Let Gbe the subgroup ofAut(E) generatedby a;b. Then Gˆ ha;bi Dm; the dihedral group of order 2m. Let y²ˆ4x³‡px‡qbe the Weierstrass normal form ofEandKˆC(x;y).

Then the fixedfieldof K by G is rationalC(t), where t2C(x). Putting Dˆ(t)₁ ; the divisor of poles oft, we infer readily that degDˆ2mand (E;D) defines a Galois embedding for eachm.

We present several examples of abelian surfaces, to which we can apply Corollary 3.6. In order to check thatA=Gis isomorphic toP², we can use the following lemmas. Using the notation in Section 3, we put H₁ˆ fM(s)2Hj detM(s)ˆ1g andlet F(G) be the set of the fixed points of G, i.e., the set fx2Aj 9s2G; s6ˆid; s(x)ˆxg. Then the lemmas can be statedas follows (cf. [12, Theorem 2.1]):

LEMMA4.2. If H satisfies all of the following conditions, then A=G is rational.

1) H6ˆH1.

2) IfHis cyclic, then no eigenvalue of the generator is1.

3) IfHhas the representation 1 0

0 1

; 1 0

0 1

or

0 1

1 0

; 1 0

0 1

,then F(G)is not a finite set.

Since we have a morphismp:A !A=G, ifA=Gis smooth, thenC²0 for each curveC onA=G. Moreover we can sayC²>0 ifHhas an irre- ducible representation (cf. [11, Corollary 3.3.3]).

LEMMA4.3. If A=G is smooth and rational and H is irreducible, then A=G is isomorphic toP².

Now, let us state examples.

EXAMPLE4.4. LetAbe the abelian surface with the periodmatrix Vˆ 1 r² t tr²

1 r t tr

ˆ 1 r²

1 r

1 0 t 0

0 1 0 t

;

where =t>0. Clearly we have AEE where EˆC=(1;t). Letting z2C²andv_ibe thei-th column vector ofV(1i4), we definet_ito be the translation onAsuch thatt_izˆz‡v_i=m, wheremis an integer2.

Letaandbbe the automorphism ofAsuch that the complex representa- tions are

0 11 0

and r 0

0 r²

respectively. PutG0ˆ ht1;. . .;t4iandGˆ hG0;a;bi. ThenG0 is a normal subgroup ofGandG=G₀D₃. Clearly we havejGj ˆ6m⁴. Looking at the fixedloci ofH, we infer thatA=Gis smooth.

EXAMPLE4.5. LetEbe the elliptic curve in Example 4.1, andletAbe the abelian surface EE. We define several automorphisms on A as follows: leta,bandcbe the homomorphisms whose complex representa- tions are

0 11 0

; 1 0

0 1

; 1 0

0 1

respectively. Lett1andt2be translations onAdefined by t₁zˆz‡^t 1

m; 0

and t₂zˆz‡^t 0; 1 m

;

wherez2C²andmis an integer2. PutG0ˆ ht1;t2iandGˆ hG0;a;bi.

ThenG0is a normal subgroup ofGandG=G0D4. ClearlyjGj ˆ8m². We can make use of Corollary 3.6 andLemma 4.2 to prove that (A;D) define a Galois embedding. In another way we can show this as follows (the proof will be done at the end of the paper): letDand G be the elliptic curves inA defined by

Dˆ z

z j z2E

and G ˆ z

z

jz2E

;

respectively. Clearly D²ˆG²ˆ0 andit is easy to see that DGˆ4.

Put D_iˆt₁ⁱ(D) and G_jˆt₁^j(G), where 1i;jm 1. Note that D_iˆt₂^{m i}(D) and G_jˆt₂^j(G). Then put

DˆD0‡D1‡ ‡Dm 1‡G0‡G1‡ ‡Gm 1;

where we assumeD0ˆDand G0ˆG. We haveD²ˆ8m² andinfer that Dis very ample ifm2. We see from Theorem 2.2 that (A;D) d efines a Galois embedding with Galois group G.

EXAMPLE 4.6. Let Abe the abelian surface C²=V such thatV is the periodmatrix

1 0 i (1‡i)=2 0 1 0 (1‡i)=2

; whereiˆe₄:

Recalling Theorem 3.7, we define the homomorphisms a,bandc, whose complex representations are

01 01

; 0 1

1 0

; i 0

0 i

respectively. Put Gˆ ha;b;ci. Then GZ₂^j D₄ and GAut(A) and A=GP².

EXAMPLE4.7. LetEbe the elliptic curve Ein Example 4.1 such that tˆem; mˆ3; 4 or 6. Let A be the abelian surface EE. We define automorphisms onAas follows: leta,bandcbe the homomorphisms whose complex representations are

0 11 0

; t 0

0 1

; 1 0

0 t

respectively. Let Gˆ ha;b;ci. Clearly we have a²ˆb^mˆc^mˆ1, bcˆcb; caˆab and baˆac, and jGj ˆ2m². Moreover we have GZ₂^j (ZmZm). PutE₁ˆE f0gandE₂ˆ f0g E, where 0 is the zero element ofE, then putDˆm(E₁‡E₂), clearly we haveD²ˆ2m². It is well known thatDis very ample ifm3. We see from Theorem 2.2 that (A;D) defines a Galois embedding whose Galois group is isomorphic toG.

Let us examine the casemˆ3 in a different point of view. SinceEis defined by the Weierstrass normal form y²ˆ4x³‡1, we have that C(A)ˆC(x;y;x⁰;y⁰), wherey⁰²ˆ4x⁰³‡1. The automorphismsa,bandc induce the ones ofC(A) as follows:

(i) a interchangesxandx⁰,yandy⁰. (ii) b(x)ˆr²xandbfixesy;x⁰andy⁰. (iii) c(x⁰)ˆr²x⁰ andc fixesx;yandy⁰.

Therefore, the fixedfieldC(A)^G is C(y‡y⁰;yy⁰), andwe have (y‡y⁰)‡D0 and (yy⁰)‡D0. Embedding by 3(E1‡E2) is the

composition of the embedding EE,!P²P² followedby the Segre embedding P²P²,!P⁸. Using homogeneous coordinates (X;Y;Z) [resp. (X⁰;Y⁰;Z⁰)] satisfying that xˆX=Z; yˆY=Z [resp.

x⁰ˆX⁰=Z⁰;y⁰ˆY⁰=Z⁰], we can express this embedding as f(X;Y;Z;X⁰;Y⁰;Z⁰)ˆ(XX⁰;YX⁰;ZX⁰;XY⁰;. . .;ZZ⁰):

Letting (T₀; ;T₈) be a set of homogeneous coordinates of P⁸, we can express the Galois subspace by T5‡T7ˆT4ˆT8ˆ0.

REMARK4.8. In the situation of the latter half of Example 4.7, letWbe the linear subspace defined byT₅ ˆT₇ˆT₈ˆ0. Consider the projection p_Wwith the centerW. In this casef(A)\Wconsists of nine sets of points.

The projectionpWj_Ainduces the extension of fieldsC(A)=C(W0), which is a Galois extension with the Galois group isomorphic toZ3Z3.

EXAMPLE4.9. Take the elliptic curveEin Example 4.7 such thatmˆ3.

If we take different actions of band c, then we can get different Galois subspaces. To show this, first we recall the result of Galois subspaces (i.e., Galois points) for plane elliptic curves. LetEbe a smooth cubic inP². If there exists a Galois point, thenEis projectively equivalent to the curve defined by Y²Zˆ4X³‡Z³ andit has just three Galois points (X:Y:Z)ˆ(1:0:0);(0: 

p 3

:1) and(0: 

p 3

:1). Then we have three projections p:P² !P¹ given by p(X:Y :Z)ˆ(Y :Z);

(X:Y‡ 

p 3

Z) and(X:Y 

p 3

Z), which yields Galois covering pj_E:E !P¹. By taking these projections, we can findnine Galois sub- spaces for the embedding EEP⁸. For example, if we take p₁(X:Y:Z)ˆ(Y:Z) andp₂(X:Y:Z)ˆ(X:Y‡ 

p 3

Z), then the fixed fieldof C(A) by G is equal to C(y‡u;yu), where uˆ(y⁰‡ 

p 3 )=x⁰.

Hence the Galois subspace is defined by

T1‡T5‡ 

p 3

T8ˆT4‡ 

p 3

T7ˆT2ˆ0 using the coordinates in the last part of Example 4.7.

5. Proofs.

We use the notation in Introduction. First we prove Representation 1.

Each elements2G_Winduces a birational transformation ofVoverW₀, we denote it by the same letter s. In the case where dimVˆ1, s is a morphism, so we assume dim V2. For a point P2V we denote by hW;Pithe linear subvariety spannedbyW andP.

LEMMA5.1. IfhW;Pimeets V at d distinct points, thensis regular at P ands(P)is one of the points inhW;Pi \V.

PROOF. The projection p is a finite morphism near p(P) by the hy- pothesis. Sincesis a birational transformation ofVoverW₀andpis finite nearp(P) andVis smooth (normal is enough), by Zariski's Main Theorem, we see thatsis a morphism (necessarily an isomorphism) from a suitable

open set of the formp ¹(U) containingP. p

IfPdoes not fulfill the assumption of Lemma 5.1, thenp(P)2D_p, where D_pis the divisor of the discriminant ofp. LetU_Pbe a small neighborhoodof P. Then ZPˆUP\p ¹(Dp) is a set of zero points of some holomorphic function inU_P. Letsˆ(s1;. . .;sn) be an expression ofsonU_P. Then each s_i is regular andboundedon U_PnZ_P by Lemma 5.1. By Riemann's Ex- tension Theorem ([3, p. 9]),sis regular atP. Thussis regular onV. Hence it becomes an automorphism ofV.

Next we prove Proposition 2.1. By definition we have a birational mapg:ˆpp_G ¹:V=GW >W0. LetW1be a general hyperplane ofW0

andletV₁ˆp(W₁) be a very ample d ivisor. Thenp_G(V₁) is ample, hence V=G_W is a projective variety. Moreover it is normal. Therefore we have only to prove, by Zariski's Main Theorem, that the total transformg(x) [resp. g ¹(y)], where x2V=GW [resp. y2W0] do not contain curves.

Since p_G is a finite morphism, g(x) contains no curves. Thus, it is suf- ficient to prove that p ¹(x) does not contain a curve. Suppose the con- trary. Then, let G be the curve and W₁ be a hyperplane in W₀ not passing through x. Since p(W₁)D, it is a very ample divisor, hence we have p(W1)\G6ˆ ;, which is a contrad iction. Thus g is an iso- morphism. Since p_G is a finite morphism, so is p. By the theorem of purity of branch loci, due to Zariski, each component of the loci has codimension one.

Now we prove Theorem 2.2. Suppose that (V;D) defines the Galois embedding f :V,!P^N with the Galois subspace W. Then, by definition andRepresentation 1, the first assertion (1) is clear. LetW1be a general hyperplane inW0. ThenV1ˆp(W1) is an irreducible smooth subvariety of codimension one in V by Bertini's theorem, and s(V₁)ˆV₁ for any s2G_W. Take n‡1 independent general hyperplanes W_1i of W₀

(iˆ0;1;. . .;n) andlet f_i be the section of H⁰(V;O(D)) defined by the

pullback of W1i by p. Then the linear subspace L generatedby ff0;f1;. . .;fngsatisfies conditions (2) and (3).

Conversely, we assume that conditions (1), (2) and (3) hold.

Letff0;f1;. . .;fNgbe a basis ofH⁰(V;O(D)), wheref0;f1;. . .;fnare gi- ven in (2). Thenf ˆ(f0;f1;. . .;fN) defines an embeddingV,!P^N.

Let (X₀;X₁;. . .;X_N) be the corresponding set of homogeneous co- ordinates on P^N andlet W be the linear subspace defined by

X₀ˆX₁ˆ. . .ˆX_nˆ0. Since the linear system corresponding toLhas no

base points, we infer thatW\f(V)ˆ ;. Therefore the projection induces a morphismp:V !W0. SinceDis ample,pis surjective. HenceKis a fi- nite extension ofK0:ˆk(f1=f0;f2=f0;. . .;fn=f0) and[K:K0]ˆDⁿ. LetK^G be the fixedfieldofKbyG. Then, sinces(f_i=f₀)ˆf_i=f₀for anys2G, we see that K^GK₀. By the assumption jGj ˆDⁿ we have that K^GˆK₀, henceWturns out to be a Galois subspace. This completes the proof.

The proof of Theorem 2.3 is simple. For any s2G1, define a homo- morphism h:G1 !kˆkn f0g by h(s)ˆls=m_s, where g(s)ˆls 1_n‡1m_s1_{N n}. Sincehis injective andG₁is a finite multiplicative sub- group ofk,G₁is cyclic.

The proof of Representation 4 is done inductively. Suppose that (V;D) defines a Galois embedding and let Gbe a Galois group for some Galois subspace W. Then, take a general hyperplane W1 of W0 andput V₁ˆp(W₁). The divisorV₁has the following properties:

(i) Ifn2, thenV1is a smooth irreducible variety.

(ii) V1D.

(iii) s(V₁)ˆV₁for anys2G.

(iv) V₁=Gis isomorphic toW₁.

Put D₁ˆV₁\H₁, where H₁ is a general hyperplane of P^N. Then (V₁;D₁) defines a Galois embedding with the Galois groupG(cf. Remark 1.3). Iterating the above procedures, we get a sequence of pairs (Vi;Di) such that

(V;D)(V1;D1) (Vn 1;Dn 1):

These pairs satisfy the following properties:

(a) V_iis a smooth subvariety ofV_{i 1}, which is a hyperplane section ofV_{i 1}, whereD_iˆV_i‡1,V ˆV₀ andDˆV₁ (1in 1).

(b) (V_i;D_i) defines a Galois embedding, with the same Galois groupG.

In particular, letting C be the curve V_{n 1}, we get the fourth re- presentation.

Next we prove Proposition 2.5. LethW;Pidenote the linear subvariety spannedbyWanda pointP. Suppose the contrary, i.e.,G₁ˆG₂. Then, we

infer readily thathW1;Pi \V ˆ hW2;Pi \Vfor anyP2V. This implies that hW1\W2;Pi \V ˆ hW1;Pi \V. SinceW16ˆW2, we have W1\W2

6ˆW1, therefore we have dim (W₁\W₂)N n 2. ThusVis containedin a linear subvarietyWe such that dim (p_W¹(Q)\W)e N n 1, whereQis a general point inW₀. Hence we conclude thatVis containedin a hyperplane, which is a contradiction.

We proceedwith the proof of the assertions in Section 3. First we consider Proposition 3.1. By the ramification formula we have K_Ap( (n‡1)W₁)‡R_p, whereK_Ais a canonical divisor ofAandW₁ is a hyperplane in W₀. Since A is an abelian variety, we have K_A0, hence R_p(n‡1)D.

By the way, we note that ifs2G, whereeszˆM(s)z‡t(s), then the matrix partM(s) induces an automorphismbsofAdefined bybszˆM(s)z.

LetR be a reduced part of the irreducible component ofR_p. Then there existss2G(s6ˆid) such thatsj_Rˆid. This implies that some translation ofRis containedin the kernel of the homomorphismbs id, wherebsis the homomorphism defined above. This means that R is a translation of an abelian subvariety. Now the proof of Corollary 3.2 is clear, because p:A !W₀is not an unramifiedcovering.

Now we consider the proof of Theorem 3.3. Referring to [11, Corollary 3.2.2], sinceB=HPⁿ, we see thatHis generatedby reflections. Suppose that His an abelian group. Then it can be generatedby elements whose complex representations are diagonal matricess_iˆ[a_i1;. . .;a_in] such that a_ijˆ1 ifi6ˆjanda_iiis a root of unity and 6ˆ1. Take an elements₁ and consider the homomorphismh:B !Bdefined byh(z)ˆs₁(z) z, where z2Cⁿ. Then h(B) is an elliptic curve E. Since H is assumedto be an abelian group, we have shˆhs, where s is an automorphism on E induced by s. We infer from this that there exists a morphism h:B=H !P¹, which is a contradiction, sinceB=HPⁿ.

In order to prove Lemma 3.5, we make use of the following lemma (cf.

[9], [1]).

LEMMA 5.2. Let (A;D)be a polarized abelian surface, where D is an ample divisor on A with D² 4d‡6. Then D is d-very ample unless there exists an effective divisorDon A such that

DD d 1D²<DD=2<d‡1:

By this lemma it is sufficient to prove that there exists no effective divisorDonAsatisfying thatD²ˆ0 andDDˆ1 or 2. PutDˆP^r

iˆ1m_iC_i,

wherem_iis an integer>0 andC_iis an irreducible curve (1ir). Since CiCj0 for anyi;j, we have thatC_i²ˆ0 andCi\Cj ˆ ;ifi6ˆj. ThusCi

is an elliptic curve andwe infer readily, by taking a translation, thatDis algebraically equivalent to mE, where Eis an elliptic curve and m>0.

SinceDDˆ1 or 2, we getDEˆ1 or 2. In the former case, we consider the morphismg:A !A=E₀, whereE₀is a subabelian variety, which is a translation of E. Then Dwill be a section of g, which is a contradiction, since the genus of Dis not less than 6. In the latter case, by taking a translation ofE, we may assume that it does not belong to the ramification divisor of p. Put D²ˆd, which is an even number, and G_Eˆ

ˆ fs2Gjs(E)ˆEg, which is a subgroup ofG. Letf be the rational map associatedwith jDj. Then f(E) is a rational curve, since DEˆ2. Thus there exists s2G such thats(E)ˆE and s6ˆid, therefore GE6ˆ fidg.

Put G=GEˆ hs1; ;sri, wheres_i2G(1ir). Then r<dandris a factor of d. If d egp(E)ˆd⁰, then p(p(E))d⁰D. We can put p(p(E))ˆE₁‡ ‡E_r, where EˆE₁, E_iˆs_i(E). Sinces(D)ˆD, we haveDE_iˆ2 for 1ir. Therefore we get 2rˆd⁰d, henced⁰ˆ1 and rˆd=2. For 1ir we have E1Ei‡ ‡ErEiˆ2 and Ei2ˆ0.

Since r5, there exists j satisfying that (E1‡E2)Ej ˆ0 and E₁E₂1. This is a contradiction, because, since (E₁‡E₂)²2, the di- visorE₁‡E₂is ample (cf. [6, Ch.4, (5.2)]).

The proof of Corollary 3.8 is as follows. Looking at Theorem 1 in [5], we infer easily that except in the case (4;2)1Bis a product type. Concerning the exceptional case, the periodmatrix of the abelian surfaceBˆC²=V has the expression

V ˆ 1 0 i (1‡i)=2 0 1 0 (1‡i)=2

; where iˆe4:

PutB₀ˆE_iE_i, whereE_iˆC=(1;i). ThenBis isogenous toB₀, henceB is a singular abelian surface, this impliesBF₁F₂, whereF₁andF₂are elliptic curves (cf. [10]). Since the scalar matrix [i;i] acts onB,F₁andF₂ have the automorphism defined by z7!iz, where z2C, hence we have F1F2E_i.

Finally we mention the proof of the last assertion of Example 4.5.

Thanks to Lemma 5.2 we have only to prove that there does not exist an elliptic curveFsatisfying thatDFˆ2 orDFˆ1. IfD_iFˆ0 for some iandfor some elliptic curveF, thenFis a translation ofDi, henceFGjˆ4

(jˆ0;. . .;m 1). This implies DF4m. If D_iF1 and G_jF1,

then we haveDF2m. Therefore we conclude thatDF2m. Clearly

we haveA=GP². Take three general linesli(iˆ0;1;2) onA=Gandletfi

be the pull back of liby p:A !A=G. We infer from Theorem 2.2 that (A;D) defines a Galois embedding.

Finally we raise problems.

PROBLEMS

(1) In the situation of Introduction find the set fW2G(N n 1;N)jG_WS_dg.

In particular, is it true that the codimension of the complement of the set is at least two (cf. [8])?

(2) Suppose that dimLin(V)ˆ0, W is close toW⁰ (in the Grass- manian) andW6ˆW⁰. Then is it true thatKW is not isomorphic toKW⁰? (cf. [16])

(3) For an embedding (V;D) findthe structure of Galois groupG_W for eachW 2G(N n 1;N).

(5) Find all the Galois subspaces for one Galois embedding. Espe- cially findthe rule of arrangements of Galois subspaces (cf. [2], [15]).

(6) Consider the similar subject in the case wheref(V)\W6ˆ ;.

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Manoscritto pervenuto in redazione il 17 settembre 2005