• Keine Ergebnisse gefunden

If 1/2| has base points then the extension of the mapT −−→IP3 to the blow-up of T in the base locus of 1/2|is a small resolutionTeof a conic bundleT overIP1×IP1⊂IP3. T is contained in a (real) five-parameter family of conic bundles.

While the “conic bundle case” is well understood (cf. [LeB] and [Ku2]) for the “double solid case”

it is not clear for which quartics and for which small resolutions Ze of the double solid Z twistor spaces occur. One of the problems in this connection is to single out the real twistor lines. As a first step towards the solution of this problem one may try to describe the parameter space of “lines”

inZeorZ where “lines” are meant to be smooth rational curves with normal bundle O(1)⊕ O(1).

Further, one might ask which of these lines are real with respect to the real structure σ. We will give partial answers to these questions for the general case, i.e. for the case of a quartic with thirteen ordinary nodes. For this purpose we give list some basic facts that we will use in the sequel.

We give a characterisation of the quartic surfaces which may occur as branch loci. By Proposi-tion 4.1, these are real with exactly one real point. We will restrict ourselves to the case of quartics with exactly thirteen ordinary nodes. Quartics of this type have been studied by Kreußler in [Kr2]

where the following proposition may be found.

Proposition 4.2 Every real quartic surfaceB with exactly one real point and 13 ordinary double points can (in suitable coordinates) be defined by an equation of the form

F = x23f2+ 2x3L0L1L2+f22−f2 L20+L21+L22

+L20L21+L20L22+L21L22 (4.1)

= 1

4 Q2−E1E2E3E4

whereE1=x3−L0−L1−L2,E2=x3+L0+L1−L2,E3=x3+L0−L1+L2,E4=x3−L0+L1+L2, f2 =x20+x21+x22, Q = 2f2+x23−L20−L21−L22 andLj =P2

i=0aijxi with aij IR and such thatf2−L2j (j= 0,1,2)defines three smooth conics with 12 different intersection points. The real point ofB is the point(0:0:0:1). It is one of the double points.

Let T be a twistor space and let the morphism ϕ:T −→ IP3 induced by 1/2| be branched along B. Denote by Hi the trope plane Ei = 0 (i = 1, . . . ,4). Then ϕ1(Hi) splits into two components Si+ and Si. The surfaces Si± are of degree one which means that their intersection number with any twistor line is one. Moreover, theS±i are real with respect to the real structureσ ofT. (These facts can be found in [KK] Section 4.) From this we immediately get:

Corollary 4.3 If the quartic surfaceB (as above) is the branch locus of T −→ϕ IP3 for a twistor space T then the linear forms Ei (i = 1, . . . ,4) and hence the linear forms Lj (j = 1,2,3) are

real.

By Theorem 4.1 the morphismT −→IP3factors into a small resolutionT −→Z and the covering map Z −→ IP3 whereZ is the double cover of IP3 branched over B. The real structure σ onT induces a real structure onZ such that the morphismsT −→ZandZ−→IP3are real. (This real structure onZ will also be denoted byσ.) This induced real structure onZ has exactly one fixed point (namely the preimage of the real point of the quarticB).

4.2. LINES AND TWISTOR FIBRES 45

double points and such that the linear formsLj (j = 1,2,3) are real (notation as in Corollary 4.3).

We do not assume thatZe is a twistor space. We will concentrate our investigations on the study of the candidates for the twistor fibres. As already mentioned, these have to be smooth rational curves with normal bundleO(1)⊕O(1). Throughout this chapter, we will call those curves “lines”.

We need the following facts onZe(which can be found in [Kr2] Section 5.2) generalising Lemma 1.9 Lemma 4.4 The canonical line bundle of Ze is the pull-back of OIP3(2). In particular, its dual has a square root ωe1/2

Z =ϕOIP3(1)and the map Ze−→ϕ IP3 is induced by the three-dimensional linear system

ωe1/2

Z

.

For small resolutions Ze of double solids we will use the same definition of a “conic” in Ze as for double solids – cf. page 9.

Lemma 4.5 Any “line” F ⊂Ze (i.e. any smooth rational curve F ⊂Ze is a with normal bundle O(1)⊕ O(1)) is in fact a conic.

Proof: From adjunction formula onZe we get

ωZe⊗ OF =ωFdetNF

|Ze=OF(4) and, therefore,F·ωe1/2

Z = 2.

As in Lemma 1.10 one shows that any line F is contained in an elementS of the linear system

ωe1/2

Z

and satisfiesF2= 0 andF·ωS =2 inS. In particular,ϕ:Ze−→IP3mapsF to a smooth conic (which is a tangent conic at the branch locus B) or to a double line. If F is contained in such anS∈

ωe1/2

Z

thatϕ(S) is a plane transversal to the branch locus (i.e. B∩ϕ(S) is a smooth plane quartic) thenϕ(F) is a smooth conic. (Proofs as in Section 1.3).

Remark: (See [Kr3] page 73.) If Ze is a twistor space then every irreducible real element S

ωe1/2

Z

must contain a twistor fibre. Otherwise the fibration map π:Ze −→M = CP2]3 would induce an unramified twofold coveringS−→M (F·S= 2 for any lineF). But this is impossible asM is simply connected.

From this remark and Lemma 4.5 we deduce that if Zeis a twistor space then there exist twistor fibres which are conics parametrised by XC. (Recall that the parameter spaceXC as constructed in Section 1.4 does not parametrise allconics inZ but only those that are contained in a general elementS of

ωe1/2

Z

.) Using the results of Chapter 3 we easily determine the irreducible compo-nents ofXC: The branch quartic B has thirteen nodes and at least four trope planes. (The four planesEi = 0 (i= 1, . . . ,4) are trope planes.) As a nodal quartic with thirteen nodes cannot have more than four tropes (which is clear from the table (2.5) on page 23) these are all tropes of B and Theorem 3.10, therefore, gives the number of components ofXC.

Recall that if Ze is a twistor space thenZ has a real structure with exactly one real point. For any quartic B with the properties as postulated at the start of this section we can define such a real structure on the double solid Z: By Proposition 4.2, B may be given by an equation F =F(x0, . . . , x3) = 0 of the form (4.1) and if B has all the properties that we require then the functionF takes only real values on real arguments. LetIP(1,1,1,1,2) be the weighted projective space with coordinates x0, . . . , x3 of weight 1 and y of weight 2. We define a real structure on IP(1,1,1,1,2) by (x, y) 7→(x, y). Z is contained inIP(1,1,1,1,2) as a real (i.e. invariant under

the real structure) subset – given by the equation y2+F. The real structure on IP(1,1,1,1,2) induces one onZ which has the required properties.

In the rest of this section we will be concerned with the real structure on XC which is induced from the real structure onZ. As the twistor fibres must bereallines and, in particular,realconics we want to find out which components of the spaceXC are real and which of them contain real points.

LetH ⊂IP3 be areal plane such thatB∩H is a smooth plane quartic. ThenZH is real and the real structure induces a fixed-point free involution on that Del Pezzo surface. There is, hence, an involution on Pic(ZH)=H2(ZH,Z) and as the line bundleOH(1) is real onHso is its pull-back to Z|H which equals the anticanonical line bundle. So the real structure onZ induces an involution on Pic(ZH) which respects the intersection product and which fixes the canonical class. To find out which of the components of XC are real we seek for an explicit description of this involution on Pic(ZH). As the correspondence between the (1)-curves inZH and the bitangents atB∩H is compatible with the real structures it will be helpful to know how the real structure acts on the bitangents.

Proposition 4.6 The seven irreducible components of Y0(the space of bitangents at B) are real.

Its only real points correspond to bitangents contained in the trope planes ofB.

Proof: Recall the construction of the cubic threefold associated to a quartic surface with a trope as carried out in Section 2.2. The four tropes of B as well as B itself are real and, hence, this construction yields a cubicKinIP4which is real with respect to the standard real structure ofIP4. Furthermore, the projectionK\{(0:0:0:0:1)} −→IP3 from (0:0:0:0:1) is a real morphism and the preimage of the real node P B in K is a real double point P1 of the cubic. In Section 2.3 we considered the cone in IP4 over a sextic space curve S. This cone was the intersection of the tangent cone at one of the nodes (e.g. P1) of K with K. S was the intersection of this cone with a plane not containing the node. This sextic cone with vertex P1 consists of the lines in K throughP1. By Proposition 2.6 (and the proofs of the lemmas preceding this proposition) the map K\{(0:0:0:0:1)} −→ IP3 maps the lines in the cone over S to bitangents at B through the real point ofB.

The bitangents at B throughP are found as follows: The projection ofB from the nodeP onto the plane x3 = 0 is a double cover of that plane branched along a sextic curve S0. From the equation (4.1) definingBit is clear thatS0 is given by the sextic form (f2−L21)(f2−L22)(f2−L23).

By assumption the three factors (f2−L2i) are real. Now, the lines through P and a point of S0 are just the double tangents of B throughP. B has no real point except B – so S0 cannot have real points. Hence, the map K\{(0:0:0:0:1)} −→ IP3 maps the cone over S with vertex P1 onto the cone over S0 with vertex P. The latter is real and so must be the cone over S with vertex P1. Therefore, by cutting the cone with a real hyperplaneH ⊂IP4 the sextic space curveS⊂IP4 may be chosen as a real curve. Then the projection ofK from P1 ontoH is a real map and by Proposition 2.8 it maps lines inK (not throughP1) to bisecants ofS. This way we can recognise therealcomponents of Fano(K) and – using Proposition 2.6 – the real components of the spaceY0 of bitangents, as well: AsSand all its irreducible components are real all components of Fano(K) and, hence, all components ofY0are real.

To find the real points ofY0 note again thatS⊂IP4 has no real points, hence, the only bisecants ofS which arereallines must connect two conjugate points ofS. As all components ofS are real two conjugate points ofSnecessarily belong to one and the same component ofS. Thus, the only real bisecants of S are those connecting two points of the same component. So there are three components in Fano(K) that contain real points. On the other hand, the three planes inK (the

4.2. LINES AND TWISTOR FIBRES 47

preimages of the trope planes of B1)) are real and contain real lines and, therefore, the lines in these planes correspond to the three components of Fano(K) with real points.

Remark: The statements on the real points of Y0 totally conform with the remark on page 62 of [A] – namely that a real non-singular plane quartic with no real points has precisely four real bitangents.

Now, we can describe the involution on Pic(Z|H) explicitly. There are eight (1)-curves in Z|H that correspond to the four bitangents H∩ {Ei= 0} (which by Proposition 4.6 are the only real bitangents at B∩H). From the proof of Theorem 3.10 follows that these (1)-curves may be chosen asE7, K7;E5, K5;G67, C67;G56, C56. Then the following Lemma holds.

Lemma 4.7 Let the quartic B and the real structure on Z be chosen as above, let H be a real plane that intersectsB in a smooth curve, and letE7, K7;E5, K5;G67, C67;G56, C56 be the eight (1)-curves over the real bitangents ofB∩H. Then the real structure on Z induces an involution on Pic(Z|H)which is given by

Ei 7−→ Ci6 i= 1, . . . ,4

E5 7−→ K5

E6 7−→ G57

E7 7−→ K7

(1; 0,0,0,0,0,0,0) 7−→ (6; 2,2,2,2,3,1,3).

Proof: We follow the corresponding part in the proof of Lemma 5.7 in [Kr2]. The involution on Pic(Z|H) must swap the four pairs of (1)-curves over the real bitangents ofB∩H since otherwise Z|H would have real points. Therefore, the involution interchanges the four pairs of (1)-curves:

E7↔K7;E5↔K5;G67↔C67; G56↔C56. Next, remark that each (1)-curve must have even intersection with its image under the real structure since a single intersection point would be a real point of Z|H. Finally, by Proposition 4.6, there are no further real bitangents atB∩H and, hence, non of the remaining (1)-curves can be fixed by the involution or interchanged with the other (1)-curve over the same bitangent. Using the invariance of the intersection product under the involution, these facts are sufficient to determine the involution on Pic(Z|H): Denoting byC the image ofC under the involution, we have

Ei·E5=Ei·E5=Ei·K5= 1 i= 1, . . . ,4.

Analogously, one obtainsEi·E7= 1,Ei·G56= 1. Therefore,Ei= (a;∗,∗,∗,∗,1, a2,1) witha= 2 or a= 3 (i= 1, . . . ,4), i.e. Ei∈ {Cj6, Kj|j= 1. . . ,4}. FromEi6=Ki andEi·Ei0 (mod 2) one getsEi=Ci6fori= 1, . . . ,4. In the same manner one showsE6=G57and (1; 0,0,0,0,0,0,0) can be calculated from

G56= (1; 0,0,0,0,0,0,0)−E5−E6=C56

Corollary 4.8 The involution onPic(Z|H)acts on the 19 orbits found in Theorem 3.10 as follows:

The orbit with 24 elements and all its elements are fixed by the involution.

1Recall that there is one trope plane ofBthat does not correspond to a plane inK– namely the trope plane that was chosen to constructK. (cf. page 15)

Each of the other orbits is exchanged with the orbit corresponding to the same dominating component ofX(the space of tangent conics atB).

Those of the 126 elements (3.2) which are contained in orbits with eight elements are mapped to elements of the same orbit, but corresponding to a different one-parameter family of tangent

conics (atB∩H).

Corollary 4.9 All dominating components ofX are real.

The one-parameter families (of tangent conics in H) belonging to those dominating compo-nents of X with eight families in a general fibre overIPˇ3 (cf. Theorem 3.10 and its proof ) are not real. Thus, there are 24 pairs of conjugate one-parameter families. These families, hence, have no real points.

The remaining 15 one-parameter families are real.

In Section 7 of [Kr3] a family of curves in Ze is constructed. In the same section of that paper it is shown that ifZe is a twistor space then this family has to contain twistor lines. We will briefly outline the construction and afterwards we will show that this family always containsrealelements (i.e. also ifZeis not supposed to be a twistor space).

Let ZeH ωe1/2

Z

be a general element (in particular, let the image H of ZeH be a plane in IP3 which intersects B transversally). As a “line” is in particular a conic (Lemma 4.5) each twistor lineF which is contained inZeH must (by Lemma 1.10) be an element of the divisor classes given by (3.3). LetD±i =Si±∩ZeH (see Section 4.1 for the definition of theSi±). Then F·D±i = 1 for i = 1, . . .4 as the Si± are of degree one. But the D±i ZeH are just the eight (1)-curves over the four double tangentsH∩ {Ei= 0} – i.e. the four double tangents inH ⊂IP3cut out by the four trope planes. We can choose these eight (1)-curves as in the proof of Theorem 3.10 or as in Lemma 4.7. The conditionF ·D±i = 1 determines 24 of the 126 classes (3.3) which are just the classes of the monodromy orbit with 24 elements (see list on page 37).

These 24 classes are in fact determined by elementary calculations in Pic(ZeH). One only uses the intersection properties of the concerned classes. Thus, these calculations also apply to the case whereH ⊂IP3is a plane containing the real (double) point ofB such thatH intersects the quartic in a curve with one ordinary node: ZeH then is smooth (sinceZeis a resolution of the singularities in the double cover Z) and therefore ZeH is the resolution of the double cover branched along a quartic curve with one node. Such an ZeH is the blow-up ofIP2 in seven points which are not in general position any more (cf. proof of Lemma 5.7 in [Kr2]).

There is a (unique) (2)-curve C in such aZeH which by the mapZe−→ϕ IP3 is contracted to the real point ofB. If H is a real plane thenC is real and ifZe is a twistor space then every twistor fibre which is contained in ZeH must intersect C in two different points (C is real without real points). As the real point ofB is outside the tropes C must not intersect any of the divisorsDi± and, thus,C·ωZe

H =−C·(D1++D1) = 0. Consequently,Cmust represent one of the classes (3.5) on page 36 (and, moreover, must be real with respect to the real structure onZeH).

One easily checks that for any classC satisfyingC2 =1 and C·D±i there is a unique classF0

among the 24 classesF (satisfying F2= 0, F·D±i = 1) which has intersection number two with C. Just like every of the 24 classes,F0 splits into the sum of two effective classesC1andC2 with Ci2=1 andCi·ωZe

H =1. Such a classCi need not be represented by a (1)-curve but may be reducible withC beeing an irreducible component. However, if (C1+C2)·C =F0·C = 2 then C1 and C2 must be (1)-curves: As Ci·C can only take the values1, 0 and 1 soCi·C must

4.2. LINES AND TWISTOR FIBRES 49

be one and, hence, C is not a component of a curve representing Ci. F0 admits six different of those splittings into (1)-curves and sinceF02= 0 the (1)-curves of one splitting do not meet the (1)-curves of each the other five splittings. Therefore, the linear system|F0|must be base point free. In the same manner as in Proposition 1.6 one shows that its dimentsion is one and that its elements are mapped – via Ze−→ϕ IP3 – either to tangent conics or to double lines. Finally, since F0·C = 2 and since C is contracted byϕ the linear system |F0| is mapped to the set of double lines through the real point of B.

Denote by ωe1/2

Z

IR

the real projective space of real elements in the linear system ωe1/2

Z

. This subset of the linear system contains only four singular elements: the preimages in Ze of the four trope planes at the quarticB (this follows from Lemma 5.7 in [Kr2] - note that by Proposition 4.6 there are no real bitangents at B outside the tropes). Hence by Ehresmann-fibration-theorem the family of elements of

ωe1/2

Z

IR\ Si+∪Sii= 1, . . .4 = ˇIP3IR\{Ei} is locally trivial and the fundamental group of the parametrising space IPIR3\{Ei} acts via monodromy on Pic(ZeH).

From [Kr3] we take the following Lemma 4.10 Let ZeH0

ωe1/2

Z

be a real element which contains the real exceptional lineC. Let F Pic(ZeH0) be the unique element satisfyingF2= 0,F·Di± = 1(i= 1, . . . ,4) and F·C = 2.

ThenF Pic(ZeH0) is invariant under the monodromy action ofπ1( ˇIP3IR\{Ei}).

Proof:(Sketch) Let`⊂IPˇ3= ωe1/2

Z

be a real line such that each of the corresponding elements of

ωe1/2

Z

contains the real exceptional curveC but none of the other twelve exceptional curves2).

In particular, let the surfaceZeH0 be parametrised by`. Denote byZe`the corresponding family of elements of

ωe1/2

Z

and byEthe preimage of the curveCunder the mapZe`−→Ze. Each fibre ofE overH ∈`(IR)∼=S1represents a class in the second homology of the corresponding fibreZeH ofZe`. This class is invariant under the monodromy action of π1(S1). In ZeH0, this monodromy invariant element is just the class [C] Pic(ZeH0) = H2(ZeH0,Z) = H2(ZeH0,Z). As π1(S1) = π1(`) −→

π1( ˇIP3IR\{Ei}) is surjective [C] is fixed by the monodromy action of π1( ˇIP3IR\{Ei}). And so is the class F in Pic(ZeH0) which is determined by intersection relations with monodromy-invariant

classes.

Denote by ∆0 ωe1/2

Z

the subset of singular elements of the linear system. ( ωe1/2

Z

IR0consists just of the four reducible surfacesSi+∪Si.) Letp:Z −→

ωe1/2

Z

\0 be the universal family of the linear system (i.e. the fibre overs∈

ωe1/2

Z

\0 is isomorphic to the surface parametrised by s). If Zdenotes the the constant sheaf of integers on Z then F :=R2p(Z) is a locally constant sheaf of abelian groups of rank eight. F can be thought of beeing the relative Picard group of Z over

ωe1/2

Z

\0 for the stalk in s ωe1/2

Z

\0 is isomrphic to Pic(Zs) = H2(Zs,Z). If s

ωe1/2

Z

\0 is a real element ZH0 of the linear system which contains the real exceptional curveC ofZethen there is a distinguished element in the stalkFs– namely the distinguished class F0Pic(ZH0) as constructed on page 48.

Corollary 4.11 The locally constant sheaf F has a section over ωe1/2

Z

IR. Its value in s

ωe1/2

Z

IR

– where s corresponds to a real element ZH0 containing the real exceptional curve –

2Exceptional curves of the small resolutionZe−→r Z

corresponds to the classF0. In other words: In every smooth real elementZeH of ωe1/2

Z

there is a distinguished classFH inPic(ZeH) which forZH0 is justF0.

Proof: The distinguished class FH in Pic(ZeH) is determined as follows. Choose a path in

ωe1/2

Z

IR\ Si+∪Sii= 1, . . .4 =IPIR3\{Ei}

which connectsZeH0 (containing the real exceptional curveC) andZeH. This path induces a group isomorphism of the corresponding Picard groups Pic(ZeH0) and Pic(ZeH) which – by Lemma 4.10 – does not depend on the chosen path. The image of the distinguished classF Pic(ZeH0) (defined byF·C= 2) under this isomorphism is the distinguished class FH. The linear system of this classFH is the family which we wanted to construct.

Proposition 4.12 Let ZeH ωe1/2

Z

be a real element which contains either the real exceptional curve or non of the exceptional curves ofZe. Then the linear system of the distinguished classFH constructed above contains real elements and these are smooth.

Proof: IfZeH contains the real exceptional curve then ϕ:Ze −→ IP3 induces a bijection between the elements of |FH| and the double lines in H =ϕ(ZeH) which contain the real pointP1 of the quarticB. There are of course real double lines throughP1in H. An element of|FH|can only be singular if its image is a line which is tangent atB in a point different from P1. But there are no real bitangents throughP1 so that in this case the proposition is proven.

If ZeH does not contain an exceptional curve then H is a plane which intersects B in a smooth quartic curve. There is a pathγ: [0,1]−→

ωe1/2

Z

IR

such that γ(0) =ZeH, ZeH1 :=γ(1) contains the real exceptional curve C, and Zet :=γ(t) does not contain any of the exceptional curves for 0 < t < 1. γ induces a path in ˇIP3IR which also will be denoted by γ. There is an isomorphism Pic(ZeH) ˜−→Pic(ZeH1) induced by γ which, by assumption, takes FH to the class FH1 satisfying FH1 ·C = 2. But there is also a distinguished class Ft in Pic(Zet) for every Zet=γ(t)∈

ωe1/2

Z

(0 < t <1). Denote by F−→ [0,1] the fibration whose fibre over t [0,1] is the linear system

|Ft| ∼= IPC1. By Corollary 1.7 and Lemma 1.8, ϕ : Ze −→ IP3 maps each element of the linear systems |Ft| to tangent conics at B. The induced map from F into the parameter space X of tangent conics atB is injective and compatible with the real structures. The imageFX ofFinX is aIPC1-bundle overγ⊂IPˇ3. The fibre ofFX overγ(t)∈IPˇ3 is a one-parameter family of tangent conics atB∩γ(t) and is real with respect to the induced real structure onX (cf. Corollary 4.9).

In the consequence we have got a IPC1-bundle FX over [0,1] endowed with a fibre preserving real structure (inherited fromX). ButIPC1has only two different real structures: The antipodal map of the sphere and the natural real structure ofIPC1as complexification ofIPIR1. In particular, the real structure onIPC1can not change in a continuous family. Therefore, as the fibre ofFX over 1[0,1]

hasreal points, every fibre ofFX overγ must have real points. Thus|FH|has real points.

To show that all real elements of|FH|are smooth note that the only singular elements of|FH|are reducible and split into two (1)-curves inZeH which have intersection product 1. But conjugate (1)-curves cannot have odd intersection since then their intersection would contain real points of

e

ZH of which there are non.