Math. Nachr. 186 (1997), 239-260

On Principally Polarized Adler - van Moerbeke Surfaces

By TOMASZ SZEMBERG of Cracow

(Received Januuy 25, 1995)

Abstract . In the present paper we study Kummer surfaces in P6 arising as projections of a certain family of principally polarized abelian surfacea in Ipe. This family was introduced by ADLEII m d VAN MOERBEKE in connection with some Hamiltonian dynamical systems. In this paper we give explicit equations of the moociated Kummer ~ u r f a c e ~ . This enables us to deacribe their moduli space and to give the full lint of degeneratione.

1. Introduction

In connection with certain Hamiltonian dynamical systems ADLER and VAN Mo- ERBEKE described in [AvMl) birational images of abelian surfaces as complete in- tersections of webs of quadrics in IP'. Their surfaces have a smooth &ne part and a double curve at infinity. They can be divided into two families: one coming up from principally polarized abelian surfaces and the other from surfaces with a (1,2) - polarization. It was BARTH who noticed in [Bl) that Adler-van Moerbeke surfaces fit into the following diagram

A .IP8

1 1 p P K-IPS

where p is the projection from a suitably chosen point. Moreover, he noticed that the mapping cp cornea from the even part of a linear system of the form ID - C k,e, 1 (i.e., a linear subsystem of 101 consisting of divisors vanishing in the points { e i } , to the order ki) , where D defines either a (2,4) or a (4,4)-polarization on A, and ki are some nonnegative integers. In (Bl] it wm also proved that the only abelian surfaces appearing in this context are those obtained by ADLER and VAN MOERBEKE, it explains the title.

1991 Mathematicr Subject Closrificoiion. Primary 14 J 10, 14 J 28; Secondary 14 K 99, 14 M 10. Keycuodr ond phrwca. Kummer ~ u r f a c e ~ , quadrics, moduli.

240 Math. Nachr. 186 (1995)

The m e of (1,2)-polarized surfaces wm studied extensively by BARTH in p2]. The aim of this paper is to complete the picture and give an explicit description of principally polarized Adler - van Moerbeke surfaces and their moduli apace.

It turns out that it is mu& more convenient to study their Kummer ~urfaces. The advantage ia that these ~ u r f a c e ~ have isolated singularities only and there are very efficient methods developed by BAUER in pa] to deal with linear systems on them. Moreover, it is easy to recover their abelian surfaces.

We give explicit quadratic equations of (singular) Kummer surfaces in lP6. This enables us to identify the weighted projective space lP(1,1,1,2) as a compactification of their moduli space and provides subsequently (via equations) a natural geometric interpretation of boundary points.

There are altogether 9 types of surfaces: (i) a Kummer surface with 12 nodes (generic case),

(ii) a conic fibration over a smooth elliptic curve in IP3, (iii) a rational surface, (iv) a sum of two singular quartics, (v) a conic fibration over a smooth conic (with a double structure),

(vi) a conic fibration over an elliptic curve with a node, (vii) the double covering of a smooth quadric (with a double structure), (viii) a Bum of two smooth quadrics (with a double structure), (ix) a configuration of 8 coordinate planes in IPS.

For more preliminary information and the background we refer to [Sl]. An excellent introduction to abelian varieties at large is [LB]. Throughout the paper we work over the field Q: of complex numbers.

2. Equations and the moduli space

2.1. A linear system on A

Let A be a principally polarized abelian surface. If the polarization is irreducible, then A is the Jacobian of a smooth genus 2 curve C and we write A = A c . Let 8 be the image of C in A c under the Abel- Jacobi mapping. From the considerations in (Bl] it follows that the linear system giving a birational morphism A c + lP6 must be of the form L: = 1463 - 2(eo + . . + es)l, where {eo, . . . , es} are halfperiods on Ac which form a group isomorphic to ZZ @ Zz and some (hence any) three of them lie on a translate of 8. Without loss of generality we can assume that eo is the neutral element of the group law on Ac and it lies together with el , ez on 8. Denoting by 0, the translate of 8 by e3-, for a = 0, . . . , 3 we get the following configuration:

Szemberg, Adler - van Moerbeke Surfaces 241

Let e4, . . . ,el6 denote all the others halfperiods of A. Let B1 : & t Ac be the blowing up in eo, . . . , el5 with exceptional divisors Eo, . . . , El5 and Z = BPL, for L the inverse element mapping on Ac. Then we have the following diagram

- El Eo, . . . ,El5 C A c - Ac 3 eo, . . . ,el5

where kc = &/i is the smooth Kummer surface of A and DO, . . . , D15 are (-2)- curves. The linear system C gives rise to the line bundle I?= 0,-(4B1*0 - 2(E0 + . a + E3) ) on 2. This line bundle in turn defines two line bundles M + , M - on k with

i , L s M + @ M -

and a canonical isomorphism

H o ( M + ) Ho(E)+,

where the last space is that of even sections in 2 with respect to T, (cf. [Ba, Prop. 1.11). There is of course also a canonical isomorphism Ho ( 2 ) E Ho (C) which preserves the parity of divisors and we have the following

Propoeition 2.1. Let 33 be a generator of H o ( 0 3 ) and s, its translates generating HO(0i) for i = 0, 1, 2. Then the sections u1 = sisf, 212 = sosl, 2 2 213 = 3033, 2 2 214 = s:s$, US = Stst, = 3;s; fonn a basis of i!fo(L)+.

Proof. This follows immediately from [Sl, Proposition 61. 0

Remark 2.2. Using the canonical isomorphisms mentioned above we consider u1, . . . , 216 also as a basis of Ho(M+) without further notice.

Remark 2.3. A basis of Ho(L) - is described in Lemma 2.14.

2.2. The embedding

This space ca,n be viewed via the mapping Let IP=IP(l, 1,1,2) be the weighted projective space with coordinates (a : fi : 7 ;A) .

J P 3 p - ( A : C r 2 : p 2 : 7 Z : C Y p : a y : P r ) E I P 6 ,

as a cone over the Veronese surface in IP5 (cf. pol] for more information on weighted projective spaces).

The following Proposition establishes a mapping from the moduli space of Kummer surfaces associated to Jacobians of smooth genus 2 curves into IP.

242 Math. Nachr. 186 (1995)

Proposition 2.4. The l i ne bundle M+ defines a birat ional m o v h i s m

VM+ : k C 4 IPS =]P(Ho(hf+))

which contracts the curves D4, . . . , Dls and is an i s o m o v h i s m away from them. There i s a po in t p = (a : p : 7; A) E IP such that the image Xc of q ~ + w a

complete intersect ion of a n e t of quadrics given in the buis u1 , . . . , U6 bg the following symmetric matrices:

s1 = &(PI =

sz = s3 =

Proof. The first past of the proposition and the form of equations is a simple consequence of [Sl, Proposition 8 and 161. The quadrics S2 and Sa intersect in a rational threefold of degree 4. The following mapping gives an explicit isomorphism between IP3 and this threefold:

The pull - back of Sl is for any choose of p E lP a non -trivial equation in lP3. Hence dim Sl n S2 n S3 = 2 and being of degree 8 Xc is a complete intersection in Ips. 0

Definition 2.5. We call the surface XC the intermediate K u m m e r surface.

There is the following straightforward

Corollary 2.6. The correspondence Xc + p E IP stated in the above pmposi- t i on gives a mapping from the moduli space of intermediate K u m m e r surfaces to the pammeter space IP.

Szemberg, Adler - van Moerbeke Surfaces 243

2.3. A 43 Configuration

Let us recall that a key to the above proposition was the observation that each intermediate Kummer surface Xc carries a 43 configuration of lines and conics. More precisely such a configuration consists of

(a) four lines LO, . . . , L3 which are imagea under 4 p ~ + 0: of 80, . . . , 8 3 , respectively, (b) four conics Co, . . . , C3 which are images of &, . . . , E3, respectively, (c) incidence relations which can be visualized in the following way:

The intersection points of the configuration conics and lines we denote by qij = C,nLj, fori , j = 0 , . . . , 3 , i # j .

These 43 configurations are also cut out by quadrics. More precisely we have

Proposition 2.7. Let C be a smooth genus 2 curve. Then there ezists a point (a : p : 7 ) E lP2 such that all quadrics containing the 43 configurntion supported by Xc in Ips form a web spanned by:

I

Proof. See Proposition 15 in [Sl]. 0

In the ul, . . . , basis the points gij can be computed explicitly. In the sequel we

244 Math. Nachr. 186 (1995)

need three of them:

Moreover, we have the following useful

Lemma 2.8. Let 6, denote the proper tmnsfonn of 8, under the blowing up Bl. Then the mapping

((pJ51+ 0 % ) : 6i - La c xc

is 2 : 1 ramified in the s i z intersection points of 6, with the ezceptional divisors E j .

Proof. Compare Lemma 11 in [Sl]. 0

Remark 2.9. The image of the ramification locus described above consists of the

Now we define in IP the following divisors:

1. D, = { ( a : p : 7 ; A ) E I P I a = O } ,

2. Dp = { ( a : , L J : 7 ; A ) E I P I P = O } ,

3. D, = {(a : p : 7 ; A) E P I 7 = 0).

Furthermore, let DX be a divisor in IP given by the equation:

points qJ,, qk,, ql,, where {z, j , k, I } = {0,1,2,3} and three of 12 nodes of Xc.

Equation 2.10.

A' + 2( - 10a2p2 + 1oa2yz - lop272 - a4 - a' - y 4 p 2

+ a* + p8 + 7 8 + 12 (a672 + ,276 - pa2 - asp2 - p672 - 76p7

- 26 (a4p4 + a4y4 + p47') - 244a2/3272 (aZ - p2 + r2)

+ 32(a4p2 + a4y2 + p4a2 - p'r2 - r4a2 - r4p2)A

= 0.

Now we want to show that the mapping defined in Corollary 2.6 has an inverse defined over IPo defined as PO := lP \ (D, U Dp U D, U Dx). To this end we study first the discriminant variety of the web of quadrics defined in Proposition 2.7.

2.4. The discriminant variety of the web Q1, . . . ,Q4

We begin by the following change of variables in lP5

Szemberg, Adler - van Moerbeke Surfaces 245

The quadrics 9 1 , . . . , Q 4 defined in Proposition 2.7 gain in the xi coordinates a par- ticularly nice block form

Qi =

Q 2 =

1 0

-1 0

0 Q3 =

0 I 0

0 0

1 0

-1

0 1

0 0

-1 0

The discriminant variety of the web spanned by 9 1 , . . . , Q 4 is given by the equation

This equation defines a surface B of degree 6 in V = P3(q : . - : v 4 ) . Because of the block form of the quadrics Q ( v ) this surface is reducible. In fact B is a sum of three cones K 1 , K 2 , K 3 whose equations are

2 2 2 K 1 :

K 2 :

K 3 :

- 3 ~ r p ~1 + / 9 2 ~ 1 ~ 2 + C Y ' U ~ V ~ + ~ 2 ~ 3 = 0 , - 3 ~ r ~ 7 ' ~ : + Q ~ U ~ V ~ - 7 2 ~ 1 ~ 2 - ~ 2 ~ 4 = 0 ,

- 38 7 ~1 - 7 2 ~ 1 ~ 3 - P 2 v 1 v 4 + ~ 3 ~ 4 = 0 . 2 2 2

Their tops are at the points TI = (0 : 0 : 0 : l), T 2 = (0 : 0 : 1 : 0), T 3 = (0 : 1 : 0 : 0), respectively. The intersection of the cones R = K 1 n K 2 n K3 is the rational normal :urve in V of degree 3. This follows easily from the fact that the tops of the cones lie

246 Math. Nachr. 186 (1995)

in their intersection. The curve R can be easily parameterized as

V1 = p(r'p2 - q 2 ) 1

v2 = - a2P(3r2P - 4) ( r 2 p - 4) 1

v3 = - B2P(3T2P + 4) (?P + a) 1

R = ( for (p:q) E Ip'.

I 214 = q(r4p2 - q2) -

Let now X E, Q: be given. The net of quadrics spanned by 31 = Q 1 + XQ2, SZ = 9 2 - 9 3 , S3 = 9 2 - 9 4 corresponds to the plane F in V given by the equation -Xu1 + wp + vg + v4 = 0.

This plane intersects the curve R in points different from the tops of the cones Tl1T2,T3. Moreover, if X $ Dx then there are three distinct points of intersection T4,Ta1Tal this explains the Equation 2.10 of Dx. In the (p : q) coordinates the points T, are

and the coordinates of T4, Tb, T. are solutions of the equation F n R: Ti = (O:l), T2 = (1:r2), T3 = (1:-r2),

- XP(T4P2 - q 2 ) - a 2 p ( 3 r 2 p - 4) (T2P - 4 )

- B2P(3T2P + 4) ( r 2 p + 4 ) + 4(r4p2 - q2) = 0

Let fi be the double covering of R branched over the pointe TI , . . . , Ts. Then R is of course a smooth genus 2 curve and we have the following

Proposition 2.11. Let p=(a : /3 : 7 ; A ) E ]PO and let &k(a : /3 : 7) be the smooth genus 2 curve defined above. Let 2, be a surface in E" defined by the quadrics

s1 = 31b) =

- s2 = , 33 =

1

1 -

-

Szemberg, Adler - van Moerbeke Surfaces

Then 2, = Xz, where X,- is the intermediate Kummer surface of k.

247

Proof. Fkom the construction of the intermediate Kummer surface Xc it follows that C is isomorphic to the double covering of any of the configuration lines branched over the points qij and the singularities of Xc lying on the configuration line, cf. Lemma 2.8. Hence it is enough to establish an isomorphism h between R and say L3

where yl, ~ 2 , 9 3 are those of nodes of Xc which lie on the line L3. Since explicit coordinates of qi j ’s are known the following

Such that h(T1) = q13 , h(T2) = 403, h(T3) = q23r h(T4) = YIY h(T5) = Y21 h(T6) = %5r

~3 = ((0: 7s : 0: -7t : o : a s + P t ) for (s : t ) E PI} c p5

is easily seen to be a parameterization of L3. Let us use the already given before parameterization of R. It is easy to check that the mapping h : P1(s : t ) - P’(p : q) defined by h(s : t ) := ( a s + Pt : ay2s - Pr2t ) is the required isomorphism.

The mapping h can be geometrically described in the following way. Quadrics Q ( v ) for w 6 R are singular of rank at most 3. Hence the singular locus of each & ( v ) contains a linear subspace of dimension at least 2. The point h-’ (v) is just the point

0 of intersection of L3 with the singular locus of Q(v).

Corollary 2.12. There is a 1 : 1 correspondence between intermediate Kummer surfaces and the points p E PO.

Proof. This is a consequence of the above Proposition and Proposition 2.4. 0

In particular, P can be viewed as a natural compactification of the moduli space of intermediate Kummer surfaces. We can look at this space also in the following way. By fl we denote the plane fl = Pz( (a : P : 7)). Let us consider the following mapping

q : R 3 w ---) q(w)=

a2 0 -as -ap cry ay

-aP -4 P2 0 Pr -Pr -ag -aP 0 P2 -Pr Pr

-a7 Pr -Pr r2 0 a7 -a7 -Pr Pr 0 r2

0 a2 -aP -aP -a? -a?

E P(HOOp(2)).

Then q(w) is the quadric Q1 defined in the Proposition 2.7. Let us observe that the image q(n) in IP(H00p6(2)) is nothing other but the Yeron~rrurfjux. Now, by the fixed 43 configuration (hence fixed w ) the intermediate Kummer surfaces are given as the intersection of two fixed quadrics S2, S3 with the third one S1 which lies on the line through q(w) and 92. If w varies these lines fill a 3-dimensional cone in IP(HOOp(2)) over the Veronese surface q(R) with the top in Q2. But such a cone is the weighted projective 3-space P(1,1,1,2) as already observed. Thus we come back to our previous interpretation.

The singularity of P does not play any role in the sequel. For the sake of complete ness however we give here a brief description of its desingularization. There is some nice geometry involved.

248 Math. Nachr. 186 (1995)

Let T = (O,O,O; 1) be the singular point of IP. We have the following

Proposition 2.13. Let * be the blowing up of IP at T . Then @ is a lP' bundle over R obtained as the pmjectivization of the mnk 2 vector bundle On @ On(2). In particular @ i s smooth.

P roof . Consider the mapping

f : I P 3 ( a : / 3 : 7 ; X ) - ( X : a 2 : B 2 : r 2 : a B : a r : B r ) E P 8 .

Let X be the image of IP under f and let A be the image of R under f. Then X is the cone over A with the top at P = (1 : 0 : 0 : 0 : 0 : 0 : 0). A is contained in the 5-space D = {ZO = 0). Let Blp : IPS - lP8 be the blowing up at P, and let E be the exceptional divisor. By 3 we denote the proper transform of X under the blowing up. Then 2 has clearly the structure of a IP' bundle over = S l p ' ( A ) . Let EX = E n 2. Thus EX is a section of the bundle 2. Since NE+ = OE(-~), we have also by the restriction N E x , z = 0 ~ ~ ( - 1 ) . Since the 5-space D intersects X transvernally in A and P # A, we have that is a section in the bundle 2. Since N~/pa = Oo(l), we have N Q ~ = q ( 1 ) .

Now we have two disjoint sections in a IP' bundle and their normal bundles. This is an analogue of the case of Hirzebruch surfaces, cf. [BPV, V.4). Using the tautological bundle of 2 we can show that 2=lP(OpXJx(l)). Then it follows P=IP(On8On(2))

0

-

because Of f'(OA(1)) = On(2) .

The bundle @ can be identified also in the following way. Let

F : R - G r ( 2 , H 0 ( O p s ( 2 ) ) )

be the mapping assigning to each (a : /3 : 7) a 2 -dimensional vector space spanned by the quadrics Q1, Q 2 . Let W be the disjoint Bum of these spaces. Then W is a r k 2 vector bundle on R and in fact V = On @ On(2) since Q 1 is given by a degree 2 polynomials, whereas Q 2 is constant. The choice of S 1 = Q 1 + X Q 2 corresponds to the choice of a T & 1 subbundle in W. These subbundles are parameterized by IP(W). Thus we get once again the description of the (smooth) moduli space.

2.5. Equations of Adler - van Moerbeke surfaces

Here we want to establish a connection between intermediate Kummer surfaces and

Let so, . . . ,s3 be defined as in Proposition 2.1, then we have the following principally polarized Adler - van Moerbeke surfaces.

Lemma 2.14. The section uo = sos1s2ss is a genemtor of H o ( C ) - . In particular

Proof. The zero divisor of uo is equal to UO = 8 0 + 0 1 + 0 2 + 0 9 and is obviousl~ symmetric. Hence by (LB, Lemma 7.7.11 uo is either even or odd. Since 4 8 is tot all^

hO(L)- = 1.

Szemberg, Adler - van Moerbeke Surfaces 249

symmetric and uo vanishes in each halfperiod to an odd order, it follows that uo is odd.

Now suppose that u is another element of Ho(L) - . Then u vanishes to the order at least 3 in the halfperiods eo, . . . ,eg. In particular the intersection multiplicity of its zero divisor U with 80 is U.80 2 9. On the other hand U.80 = 4 8 . 8 = 8. This implies that 80 is contained in U being irreducible. Applying the same argument we show that UO C U. Lying in the same linear system these divisors must ,be equal, hence u = c a s t . uo and thus ho(L)- = 1.

Remark 2.15. We view in the sequel uo also as an element of Ho(z).

Let p~ denote the morphism

given by the line bundle I?. Its image is a principally polarized Adler - van Moerbeke surface in IP'. It is now easy to write down its equations.

Let SO be the quadric in IP' 3 IP(HO(2)) defined by the equation uf = ~ 1 ~ 2 .

Proposition 2.16. For every pammeter p E PO the variety Yp defined as the intersection of the quadn'cs So, 4, Sz, 5'3 is the Adler-van Moerbeke surfoce YE, wheR is defined as in Proposition 2.11.

Proof. The quadric SO clearly contains YE. The rest follows from Proposition 2.11

We conclude this section with the following

and the previous considerations. D

Remark 2.17. 1. The projection from (1 : 0 : 0 : 0 : 0 : 0 : 0) in IP' defines a 2 : 1 covering Yp 4 2, branched over the 43 configuration. The branch locus is contained in the plane at infinity (in IP') and the affine part of Yp is smooth.

2. This defines a 1 : 1 correspondence between surfaces Yp and 2,. In particular IP is the moduli apace of principally polarized Adler - van Moerbeke surfaces.

3. Degenerate Kummer surfaces

In this part of the paper we describe surfaces defined by the quadrics with coefficients in the boundary D, U . . . U Dx of the moduli space IP.

3.1. Conic flbrations

Here we study degenerate Kummer surfaces defined by the points in the divisors D,, Dp, D,. This leads to the type (ii) as described in the Introduction. Because of the symmetry it is enough to describe surfaces in one of these divisors, say D,.

250 Math. Nachr. 186 (1995)

Let us define the following curves in the weighted projective plane Da

~ , p = { ( O : p : 7 ; X ) E D a I p = O } = D a n D g ,

Dav - - { ( O : ~ : ~ ; A ) E D , ( ~ = O } = D a n & ,

D ~ x = { ( O : @ : ' y ; X ) E D a (X+02+72=O}, D,+X = ((0 : : 7; A) E Da I -A + p2 + r2 + 487 = 0) , DiA = ((0 : p : ?;A) E Da I -A +pZ +r2 -487 = 0) s

The last 3 curvea are the irreducible components of Da n DA. This follows straight- forward from Equation 2.10. NOW let p E Da \ (Dap U Da, U DzA U D,+X U D i A ) and 2, be 88 usual the variety

defined by the equations

sl(p) : p 2 ( u 3 + 214) + 72(% + 'h3) + 2&(W - u4)(uS - u6) = 0 y

sz : u 1 u 2 - u 3 u 4 = 0 ,

s 3 : u 3 u 4 - 1152L6 = 0.

We observe that the points q,j defined in the section 2.3. fall now together and we have

pol = 402 = 412 = 403 = 413 = (0 : 1 : 0 : 0 : O : 0) , -3 = 420 = Q Z ~ zz 430 = 431 = (1 :O:O:O:O:O) ,

q01 = ( 0 : 0 : 7 : O : o : p ) , 410 = ( 0 : 0 : 0 : 7 : p : O ) ,

423 = ( O : O : O : - 7 : O : p ) , 432 = (O:O:-7:0:/3:O).

Let us define L, as the line joining the points qk, for k # i and fixed a = 0, . . . ,3. The curves Ci defined now as C, = Fi n 2, for a = 0, . . . ,3 turn out to be lines, too. These altogether 8 lines on the surface 2, form the following configuration (cf. al80

the picture on the page 5).

Szemberg, Adler - van Moerbeke Surfaces 251

The dotted points are the singularities of 2,. Four of them are as before ordinary nodes and two other p o l , p z 3 are elliptic singularities. Resolving them we get two isomorphic elliptic curves. They may be also identified as the image of the surface 2, under the projection XL : IPS -+ IP3 from the line L through pol and p23. More precisely we have the following

Proposition 3.1. LetpED,\(D,pUD,,UD~xUD,+,UD,x) and 2, = S 1 ( p ) n S 2 n S 3 .

Let TL be the projection defined above. Then the curve E := TL(Z,) is an elliptic curve and the fibers (TL Iz,>-'(e) fo re E E are smooth conics except in 4 w e s in which the fiber splits in the sum of two lines C,, Lj for ( i , j ) = (0, l), (1,0), (2,3), (3,2).

Proof. Since the line L is a coordinate line in IPS, the equations of E can be easily obtained by the elimination of the variables 211, u2 from the equations defining 2,. This an easy procedure due to the form of these equations. In IP3 = lP3(u3, . . . ,u6)

we get

w1 : p2('fh3 + 214) + T2('u5 + 216) + 2&(213 - 214)(U5 - 216) + 2A213214 = 0 { w 2 : u3214 - u!jug = 0

It is well known that the base locus of the pencil sW1 + tWz, (s : t ) E IP' is a smooth elliptic curve E = W1 f l W2 in P3 if and only if there are 4 distinct cones (quadrics of rank 3) in the pencil. Their cross-ratio determines then E. Applying this criterion in our case we can easily see that the exceptional values of the parameters @, 7, A lie on the curve DP@ U D,, U D t x U D:x u DZA. It remains to check the statement that (Z,, ?TL lz,) is a conic fibration over E. This follows by a degree argument. The singular fibers appear if u3u4 = 0. This gives 4 points ~ 0 1 , 4 2 3 , 410, 432 on E and hence 4 singular fibers. 0

3.2. Elliptic flbrations

In this paragraph we study surfaces in lP6 corresponding to points in the divisor DA. The key tool here is the double covering trick applied to the discriminant variety B of the net of quadrics spanned by Sl(p), Sz, 5'3. The outcoming surfaces are of the ty e ui) as described in the Introduction. ke$m ly the description of B. In this context we prefer to work with the variables z, intro uce on the page 6. Let us recalI that in these variables the quadrics 5'1, Sz, 5'3 have the following form

I- I I 1

&(PI =

252

- 1

0 -1

-1 0

1 -

Math. Nachr. 186 (1995)

1 0

s3 =

It is more convenient to replace also S1 with 21 = &(p) = Sl(p) + P2S2 + r2S3 =

Now we put 2 2 = SZ, 4 ~ 2 2 2 + w3Z3 for (w1 : plane curve of degree 6. reducible. In fact B spli

Z3 = S3 and consider the net of quadrics Z(w) = wlZl + w2 : w3) E W E P2. The discriminant B of this net is a Because of the block form of the quadrics Z(w) this curve is

.ts into 3 conics B1, B2, B3 given by the equations (a2 + p2 + 7 2 + X)Wl + w2 + w3 -2apw1 3 = o

-2apw1 -w2 B1 : det

(a2 - p2 - 7 2 - X)Wl - w2 - w3 2ayw1

-2cr7w1 -w3 ] = o

2p2w1 + wz 2P7Wl ] = o [ 287Wl 2T2W1 + w3

BZ : det

8 3 : det

Computing the matrix form of the conics B1, B2, B3 it is easy to check that the conics B1, B2, B3 are smooth as soon as we take the parameter p away of DaUDpUDy.

Let Di = DX \ (Dax U D ~ x U D-yx).

Lemma 3.2. For a point p E Di there are distinct points P I , P2 E W such that

B l n B 2 = PlUP2UR1,

Szemberg, Ader -van Moerbeke Surfaces 253

B ~ ~ B ~ = p l u P z u R z ,

B ~ ~ B ~ = P ~ U ~ U R ~ ,

#with R1 = (0 : 1 : -l), RZ = (0 : 0 : 1) and R3 = (0 : 1 : 0).

Proof. To verify our assertion we have to solve three pairs of quadratic equations n the variables w1, 202, WQ. We do this restricting first to the line w1 = 0 and then :onsidering the equations in the &ne part 201 = 1. The first step gives the three !xplicit points in the lemma. The second gives for any pair of the conics the following !quation

033 + ( - a2 - p2 + x + 377 W: + ( - 6a27' + 2r21 + 2p2r2 + - 8a2r4 = 0.

t can be easily checked that the discriminant of this equation is up to a constant the m e as the discriminant of the equation defining D A . Hence this equation has for E Dx one double root, say PI . We omit here the simple calculations showing that a

0 riple root is not possible.

Corollary 3.3. The intersection B1 nBzf1E13 consists of 2 points P I , Fz. Moreover, n one of these points, sag PI, the conics have a common tangent line, whereas in the ither point, PZ , they intersect tmnsversally.

We proceed with the foIlowing simpIe

Remark 3.4. Let Q be a smooth quadric in P5. Then there are two rulings of ,lanes on Q. Two planes in the same ruling either coincide or intersect in a point. 'wo planes in the opposite rulings are either disjoint or intersect in a line.

For the above Remark we refer to [R] and [Hr, Prop. 22.8). We need also the juowing fact concerning pencils of quadrics in IP' .

Lemma 3.5. Let Q,Q' be smooth quadrics an lP5 and suppose that in the pencil Q + tQ' there are six distinct cones. Then there is no plane contained in the base sw of th is pencil.

Proof. Suppose that F is a plane in X = Q n Q'. Then F is also contained in all mes in the net. But there is only one ruling on the quadric of rank 5 in Ips and I1 planes in the ruling go through the top of the cone. Of course the only planes mtdned in the cone are members of the ruling. Hence F contains the tops of all 6 me8 in the pencil. Equivalently these tops are coplanar, this is a contradiction. 0

In the sequel we need the following

Lemma 3.6. Let Q be a smooth quadric in E" and let F be a k -space ( F S Pk) mtained an Q with k < [v] . Then the [v] -spaces P such that F C P C Q aw t a 1 : 1 correspondence with the [v] -spaces contained in a smooth quadric in t N - l k - z , i. e., with the subspaces of maximal dimension contained an such a quadric.

254 Math. Nachr. 185 (1995)

Proof. Let A : Q: N+l -+ P N be the natural mapping, We identify the quadric_Q with a quadratic form in Q: N+l. There is a (k + 1) -space F in Q: N+l such that A ( F ) = F. Let -

F' = { g E Q : N + ' I g Q f = O f o r a l l f E ~ } .

We have of course C @' and dim$ = N - k. Let us consider the quotient space V := @'/F. There is an induced quadratic form Q v on V, which has also the maximal rank. This is now clear that the maximal subspaces (generators in Reid's terminology) correspond 1 : 1 to the maximal, hence [v] -dimensional spaces in Q containing F. 0

Corollary 3.7. Let Q be a smooth quadric in IPS and L a line in Q. Then there ore two planes F1 , F2 contained in Q such that L = F' r l Fa. Mowover, if F is a plane such that L C F C Q then F = F1 or F = F2.

= 4. Hence V = ZL/z is 2-dimensional and Qv defines a smooth quadric in IP1 , hence 2 points.

We note that by the Remark 3.4 the planes F', F2 belong to the different rulings on Q. 0

Now we are in a position to state the main proposition of this paragraph. Let V,

Proof. Using the notation from the Lemma 3.6 we have dim E = 2 and dim

denote the double covering of W branched over B = B@), then the following holds

Proposition 3.8. For a pammeter p E 0: the surfaces 2, and V, are bimtional.

Proof. If w E W \ B then the quadric Z ( w ) is smooth. Hence there are 2 rulings Rx(w), R ~ ( w ) of PZ's on Z(w). These two rulings can be viewed as an unbranched double covering of W \ B realized in the set of rulings of all smooth quadrics in IPS. Of course, it is enough to establish a birational mapping between the set of rulings of smooth quadrics in the net W and the set of points of the surface 2,.

We begin by defining the inverse construction. Let x E 2, be a point away of L. Then x , L span a plane F,. We want to show that F, is a ruling plane of one of the quadrics in the net. It suffices to prove that the following mapping

restr : W 3 Z ( w ) -+ (Z(w)lF,) E H0(O~=(2))

has a nontrivial kernel. Let W, be the image of restr. Since L,x are contained in the basis locus of W , we have in fact W, c Ho(Z, .ZL . OF, (2)), where I,, ZL denote the ideal sheaf of the point x, respectively the line L in F,. But

ho(Zz.ZL.0F+(2)) = ho(Zz.OFm(1)) = 3 - 1 = 2.

Hence W, is a pencil and there is w, E W such that Z(w,)IF, = 0 or equivalently F, c Z(w,). Moreover, for a generic 2 E 2, the quadric Z(w,) is smooth. If R(w,) denotes this ruling of Z(w,) whose element F, is, we define W1(x) := R(w,).

Our strategy for the rest of the proof is to construct a. Let X1 # Xz be two bed smooth quadrics in the net W. We fix one of the configuration lines L on 2,. Let

Szemberg, Adler-van Moerbeke Surfaces 255

w E W \ B be chosen in such a way that X I , X2 and Z(w) form a basis of the net. (w must be just picked away of the line through Xl ,X2 in W.) According to the Corollary 3.7 there are planes F,!, E Rl(w) and FZ E &(w) such that L = Fi n Fi.

Claim. The planes F:, F: are not contained in X I , X2. According to the Lemma 3.5 we are done if the pencils spanned by Z(w) , XI and

Z(w) , X2 are general, i.e., contain 6 distinct cones. This is the w e if w is taken away of the tangents through Xi to Bj for all i = 1, 2 and j = 1, 2, 3, and away of the lines joining X i with the intersection points of the conics B,. All these conditions are satisfied in W in codimension 1, i. e., away of a divisor.

fiom the claim it follows that XI, XZ intersect 8’; in the conics GZ, GE, respec- tively, for i = 1, 2. Since L C XI rl X Z f l Z(w), the conics GZ split in a sum of lines GZ = LU L$, for i, j = 1, 2.

Claim. L # LZ. This is clear by the construction of By a similar argument the lines L$ , LE do not coincide, hence being coplanar they

intersect in a point xf E 2,. Again it is an easy check to verify that this point does not lie on L in general. We define @(R.,(w)) := EL.

Since we identified already @-’, it follows that @ is generically 1 : 1, hence birational. 0

The above proposition enables us to identify the surfice 2,.

Proposition 3.9. For p E D i the surface 2, w national.

Proof. By the Proposition 3.8 it is equivalent to show that V,, the double covering of W branched over BI@), B2, B3 is rational. To this end we consider the minimal resolution of V,. Let us note that the lines through the point PI in W form an elliptic fibration on V,. This follows easily from the fact that the ramification divisor intersects these lines generically in four points: in PI (which we take with multiplicity 1) and three more intersection points with the conics Big i = 1, 2, 3. Hence there is abo an elliptic fibration on over the exceptional divisor of the resolution in PI. The following lines are candidates to give rise to a singular fiber on &:

(a) the tritaagent T to B at P I , (b) the line T’ through PI and P2, (c) three lines T, through PI and R,, for i = 1, 2,3. It turns out that the first one is a regular fiber since the singularity of B in PI is an

elliptic one. The other singularities are P2 of type D4 and R1, Rz, R3 of type A1 . It follows (cf. [MP, Table 1.11) that the Euler characteristic of V, is equal to

e(V’) = e(T‘) + e(T1) + e(T2) + e(T3) = 6 + 2 + 2 + 2 = 12.

is the minimal resolution, it follows from the Enriques- Kodaira classification is either rational or an Enriques surface. The second possibility is excluded

Fact. Every elliptic fibration f : Y + IP’ of an Enriques surface Y has exactly two

Since that by the following well known

multiple fibers.

256 Math. Nachr. 186 (1995)

Our fibration has no multiple fibers. 0

3.3. Degenerations in codimension two

Here we want to give a description of surfaces in IPS which correspond to the points in IP which lie on the intersections of the divisors Da, Dg, D, and Dx. In fact, as already pointed out in the Section 3.1. due to the symmetry it is enough to study intersections of one of the divisors Da, Dg, D,, say Da, with all the other ones.

We use the notation introduced in the Section 3.1 and introduce the following points

3.3.1. Surfaces on Dap and Da,

By the symmetry it is again enough to consider only one of the curves, say D,p. We get the type (iv) from the Introduction.

Proposition 3.10. Let p E Dap \ {Xl,X2,X3). Then the surface 2, has two irreducible components ZP+ , 2;. They are both of d e g m 4 and are contained in distinct 4-planes. The intersection ZP+ n 2; consists of 4 coordinate lines which in turn intersect in the singular locus of each of the quartics.

Proof. The equations of Z,, are

From the last equation we get easily equations of the two 4-spaces:

V* : y2Us + (A f J 3 - 7 ) u 5 = 0 .

Now the equations of Zpf C V* are

For the intersection 2: n 2; = LI U L2 U CO U C3 we have the following picture (cf. also figure on the page 12)

Szemberg, Adler - van Moerbeke Surfaces 257

Each of the quartics Zpf is singular in the 4 coordinate points pol, pas, po3, p12. 0

Remark 3.11. The quartics Z* are rational. More precisely they are images of 1p2 under a subsystem of op2(3), the parametrization being:

u1 = - ( X f d w - ) z 2 & 212 = 72z2y, u3 = - ( X * J W ) z 2 z ,

u4 = 72xg2, 115 = T2zyz, 'Il6 = -(A f d w ) z v z for (z : y : z) E I P ~ .

3.3.2. Surfaces on DEx

This is the type (v) as in the Introduction.

Lemma 3.12. Let p E Dzx\{X2, X,, X,, X,). Then the equations of 2, a=

Proof. The first two equations are the usual ones. The last one results from inserting in

instead of 2Xu1u2 the expression 2P2u3u4 - 272u5us which is equivalent by the as- 0

From the above Lemma it follows that 2, is a quartic with a double structure and is contained in the 4- space V = (Pu3 - @u4 + 7u5 - Tug = 0). Let Z r d be the reduced underlying variety, then we have the following

2 2 p (213 + 21:) + T2(u; + '1Li) + 2PT(U3 - U4)(u5 - u6) + 2Xulu2 = 0

sumption of the lemma and the first two equations.

Proposition 3.13. The surface Z r d admits a conic fibnation f : Z r d -+ Y over a smooth conic Y. There are exactly 4 singular fibers off, each of them consists of two intersecting lines.

Proof. Projecting from the coordinate line su1 + tu2, (s : t ) E IP1, we get as the image the conic

Y : cyu3u4 - pugu5 + pu4u5 - rug = 0

258 Math. Nachr. 186 (1995)

which is smooth due to our assumptions on p = (0 : p : 7; A). The singular fibers lie 0 aa usual over q01,q23,q10,q32, cf. Section 3.1.

3.3.3. Surfaces on D& and D i A

We consider here only surfaces deacribed by the points on D:A since in the case DOA the description is the same up to a sign in the equations. This leads to the type (vi) .

Proposition 3.14. Letp E D,+,\{X3, Xa, X S } , then the surface Z, admits a conic

Proof. The equations of Z,, are:

SI : 211u2 - 213214 = 0,

fibmtion over an elliptic curve with a double point.

sz : 213214 - 21S216 = 0,

s3 : p2 (21$+21:) + T2(UX+Ui) + 2@'y(213-211)(216-216) + 2(P2+72+4P7)UsU4 = 0.

Let f : Z,, 3 E c IPS = 1p3(213 : - . - : 216) be the restriction of the projection from the line L := (113 = = U6 = 0). The image curve is described by the last two equations. We computed already in the Section 3.1 that the discriminant of the pencil of quadrics defined by sS2 + tS3 for (B : t ) E IP' has a double zero. Here we compute the zeros explicitly:

( 8 : t ) = (7' : 1), (S : t ) = (-(2p2+4&+r2) : 1), (3 : t ) = (-(2@7+r2) : l),

the last one being the double zero. One can easily check that the singular quadrics in the pencil are all simple cones, hence the pencil's Segre symbol is (2,1,1) and its intersection E is an elliptic curve with a double point. Now f is a conic fibration by

0 the very definition and the degree argument. This verifies our statement.

3.4. Isolated degenerations

In this section we describe the remaining surfaces corresponding to the points X I , ..., x7 E I P .

Proposition 3.15. 1. Zx, is a aum of 8 planes which form a coordinate octahedron

2. Zx, for i = 2, 3, 4, 5 has a double structure. The underlying reduced varieties Z;;cld, . , . , Zyt are the double coverings of a smooth quadric Y in IPS bmnched over 2 pairs of lines from each of the rulings of Y (type (viii)). 3. Zx,, Zx, have a double structure. The underlying r e d u d varieties Z;;',", 22:

have two i d u c i b l e components which are just smooth quadrics of dimension 2 inter- secting in a smooth conic (type (vii)).

m e (%).

Proof. 1. This is clear from the equations. 2. In the case of ZX, we have the following equations

Szemberg, Adler -van Moerbeke Surfaces 259

(216 - us)2 = 0 , 211212 - u3u4 = 0 ,

The first one defines a 4-space containing Zx, and imposes the double structure. The second one is a smooth quadric Y in P3 = IP3(u1 : : 214). The last one gives the double covering ramified over (Y n (211 = 0)) U (Y n (212 = 0)). Using the Formula V.22.(9) from [BPV] we get easily the numerid invariants of Zxz:

211212 - 215" = 0 .

x = 1, C?(ZX,) = 4 , ~ 2 ( Z x , ) = 8.

The remaining three cases are completely analogous. 3. Equations defining ZX, are:

211112 - u3U4 = 0, 219214 - = 0, (213 - 214 - 215 + 'I@)2 = 0 .

The last one is the reason of the double structure. The last two together describe a sum of two 3-spaces. This gives rise to the observation that the irreducible components of Zyt are given by the equations

211212 - UgU4 = 0 ,

u 213 = 0,

u4 + us = 0.

0

These components are clearly smooth quadrics and they intersect in the conic param- eterized by ( - a2 : t2 : -st : st : -st : a t ) , where (s : t ) E IP'. This verifies our Meertion, the description of Zx, being analog.

{ i 214 - us = 0

U1U2 - 213214 = 0 u3 - 215 = 0 z;;.," =

Acknowledgements

This work was partidly supported by Daimler-Bent Stiflung project C.9&34 and KBN p n t 2PO9A - 061 - 08.

This paper is 4 compediow version of [SC]. I want to take this opportunity to ezpnss my thanks to Prof. W. BARTH who advised my thesis. Z ezperienced also 4 gnat help from my colleagues an the Arbettsgemeanschafl "Kompleze Manntgfaltigketten" in Erlangen.

References [AvMI] ADLER, M.. and VAN MOERBEKE, P.: Geodesic Flow on SO(4) and Intersection of Quadrics,

Proc. Natl. Acad. Sci. USA 81 (1984), 4613-4616

[AvM~] ADLER, M., and VAN MOERBEKE, P.: The In te rn t ion of Four Quadrics in P', Abelian Surfaces and Their Moduli, Math. Ann. 279 (1987), 25-85

[Bl] BARTH. W.: Affine Parts of Abelian Surfacea M Complete Intersections of Four Quadrics, Math. Ann. 278 (1987), 117-131

[B2] BARTH, W.: Abelian Surfacea with (1,2)-Polarization, Advanced Studiee in Pure Mathe- matics 10 (1987), 41-84

[BPV] BARTH W.. PETERS C., and VAN DE VEN. A.: Compact Complex Surfeces, Springer-Verlag, Berlin, 1984

[Ba] BAUER. TA.: Projective Imagea of Kummer Surfaces, Math. Ann. 299 (1994), 155 - 170

Math. Nachr. 185 (1996)

DOLGACREV. I.: Weighted Projective Varieties, in kctun? Notes in Mathematics 966, Springer - Verlag, 1982 HARRIS, J.: Algebraic Geometry, Graduate rrxte in Mathematics, Springer-Verlag, 1992 LANGE, H.. and BIRXENHAKE. CH.: Complex Abelian Varietise, Grundlehren der math. Wm. 802, Springer - Verlag, New York - Heidelberg - Berlin, 1992 MIRANDA. R.. and PERSSON, U.: On Extrernal Rational Elliptic Surfacea, Math. 2. 19s

REID. M.: The Complete Intersection of Two or More Quadrics, Thesis, Cambridge, 1972 SZEMBERG, T.: A 43 Configuration of Lines and Conics in P6, Ann. Pol. Math. 60 (1994),

SZEMBERG. T.: Intertuxtione of Quadrics in P6, Kummer Surfacea and Their Moduli, Thesis, Erlangen, 1994

(1986), 537 - 558

145 - 158

Inat@ut Moiemoiyki Uniwersytet Jogielloriski Rcymonia 4 PL -so - 059 Cmww Poland c -moil: rrcmberg aim. uj. edu. pl