(de Gruyter 2002
On surfaces with p
gF q F 2 and non-birational bicanonical maps
Ciro Ciliberto and Margarida Mendes Lopes
(Communicated by R. Miranda)
Abstract.The present paper is devoted to the classification of irregular surfaces of general type with pg¼q¼2 and non-birational bicanonical map. The main result is that, ifS is such a surface and ifSis minimal with no pencil of curves of genus 2, thenSis a double cover of a principally polarized abelian surface ðA;YÞ, withY irreducible. The double coverS!Ais branched along a divisorBAj2Yj, having at most double points and soKS2¼4.
2000 Mathematics Subject Classification. 14J29
1 Introduction
If a smooth surface Sof general type has a pencil of curves of genus 2, i.e. it has a morphism to a curve whose general fibre Fis a smooth irreducible curve of genus 2, then the line bundleOSðKSÞnOF is the canonical bundle onF, and therefore the bicanonical map fof Scannot be birational. Since this property is, of course, of a birational nature, the same remark applies ifShas a rational mapto a curve whose general fibre is an irreducible curve with geometric genus 2.
We call this exception to the birationality of the bicanonical map f thestandard case. Anon-standard casewill be the one of a surface of general typeSfor whichfis not birational, but there is no pencil of curves of genus 2. The classification of the non-standard cases has a long history and we refer to the expository paper [8] for in- formation on this problem. We will just mention here the fact that the non-standard cases with pgd4 are all regular.
The classification of non-standard irregular surfaces has been considered by Xiao Gang in [24] and by F. Catanese and the authors of the present paper in [6]. Xiao Gang studied the general problem of classifying the non-standard cases by taking the point of view of the projective study of the image of the bicanonical map. The out- come of his analysis is a list of numerical possibilities for the invariants of the cases which might occur. More precise results have been obtained in [6], where the first significant case pg¼3 has been considered. Indeed in [6] it is shown, among other things, that a minimal irregular surfaceSwithpg¼3 presents the non-standard case
if and only ifSis isomorphic to the symmetric product of a smooth irreducible curve of genus 3, thus pg¼q¼3 andK2¼6.
In the present paper we study this problem for surfaces with pg¼q¼2 and we prove the following result, which rules out a substantial number of possibilities pre- sented in [24]:
Theorem 1.1. Let S be a minimal surface of general type with pg¼q¼2.Then S presents the non-standard case if and only if S is a double cover of a principally polar- ized abelian surface ðA;YÞ,with Yirreducible. The double cover S!A is branched along a symmetric divisor BAj2Yj,having at most double points.One has KS2¼4.
Surfaces with pg¼q¼2 are still far from being understood. The list of known examples of surfaces of general type withpg¼q¼2 is relatively small (see [25], [26]) and there are several constraints for their existence. Here we only mention that there are various restrictions for the existence of a genus 2 fibration (see [23]) and also that M. Manetti, working on the Severi conjecture, showed in particular that if pg¼q¼2,KS is ample andKS2¼4 thenSis a double cover of its Albanese image (see [16]).
To prove our classification Theorem 1.1 we first show that the degree of the bica- nonical map is 2 for surfaces presenting the non-standard case, then we study the pos- sibilities for the quotient surface by the involution induced by the bicanonical map, and finally we show that the unique case which really occurs is the one described above. We use a diversity of techniques, which may be useful in other contexts.
The paper is organized as follows. In Section 2 we list the properties of surfacesS with pg¼q¼2 that we need. In Section 3 we characterize, by a small adaptation of a proof in [6], the surfaces Spresenting the non-standard case withKS2¼9, and in particular we verify that there is no such surface with pg¼q¼2. In Section 4 we establish some properties of the paracanonical system and then we use these results in Section 5 to conclude that for the non-standard casesSwithpg¼q¼2 the degree of the bicanonical mapis 2. Thus there is an involution iinduced by the bicanonical mapon S. We consider the quotient surface ~SS:¼S=hii and the projection map p:S!SS. In Section 6 we discuss the various possibilities for~ SS, showing that the~ only one which can really occur is that ~SSis a minimal surface of general type with
pgðSSÞ ¼~ 2,qðSSÞ ¼~ 0,K~2
S
S ¼2 and with 20 nodes. Moreover we show that the double cover pramifies exactly over the 20 nodes. Finally in Section 7, using this description, and some results on Prym varieties contained in [10], we finally prove Theorem 1.1.
Acknowledgements. The present collaboration takes place in the framework of the European contract EAGER, no. HPRN-CT-2000-00099. The second author is a member of CMAF and of the Departamento de Matema´tica da Faculdade de Cieˆn- cias da Universidade de Lisboa.
We are indebted to Rita Pardini for interesting discussions on the subject of this paper and in particular for having pointed out the use of the Kawamata–Viehweg vanishing theorem in Section 6. We thank Prof. Fabrizio Catanese for having com- municated to us some of his ideas on this subject.
We dedicate this paper to the memory of Paolo Francia, with whom we first started on this subject.
Notation and conventions.We work over the complex numbers. All varieties are as- sumed to be compact and algebraic. We do not distinguish between line bundles and divisors on a smooth variety, using the additive and the multiplicative notation in- terchangably. Linear equivalence is denoted by1and numerical equivalence by@. Anodeon a surface is an ordinary double point (i.e. a singularity of type A1). The exceptional divisor of a minimal desingularization of a node is a rational irreducible curveAwithA2¼ 2, usually called að2Þ-curve.
As already mentioned, we will say that a surfaceSof general typepresents the non- standard case, or that it is a non-standard case, ifS has no pencil of curves of geo- metric genus 2 and the bicanonical mapofSis not birational.
The remaining notation is standard in algebraic geometry.
2 Some properties of surfaces withpgFqF2
The minimal surfaces S of general type with pg¼q¼2 have various interesting properties (cf. [25], [26]). In this section we only mention those that we will need further on.
Proposition 2.1.Let S be a minimal surface of general type with pg¼q¼2.Then:
i) 4cKS2c9;
ii) if KS2¼8;9,there are no rational smooth curves on S(in particularOSðKSÞis am- ple),and,if KS2¼7andOSðKSÞis not ample,then S contains either one irreducible ð2Þ-curve,or two forming an A2configuration.Furthermore if KS2¼9,S does not contain elliptic curves.
Proof. i) The first inequality follows from the inequalityK2d2pg for minimal irreg- ular surfaces (see [11]), and the second from the inequalityK2c3c2.
ii) follows from Miyaoka’s and Sakai’s inequalities (see [18] and [22]) for the num- ber of rational or elliptic curves on a non-ruled minimal surface. r We will also need to consider the Albanese image of these surfaces. First we recall the following facts which we will use repeatedly:
Lemma 2.2(see [2], p. 343; [1], p. 97).Let S be a minimal surface and let f :S!B be a genus b:¼gðBÞpencil of curves of genus gd2.Then
i) KS2d8ðg1Þðb1Þ, ii) c2ðSÞd4ðg1Þðb1Þand iii) qcgþb.
Furthermore if equality holds ini)then the curves of the pencil have constant modulus,
if equality holds inii)every fibre of f is smooth,and if equality holds iniii)S is bira- tionally equivalent to a product of B with the general fibre of f.
Using this lemma we obtain the following:
Proposition 2.3.Let S be a minimal surface of general type with pg¼q¼2for which the Albanese morphism a:S!A:¼AlbðSÞis not surjective.Then aðSÞ ¼B is a ge- nus2 curve, the Albanese pencil a:S!B has smooth, connected fibres F of genus2 with constant modulus and KS2¼8.
Proof. SinceqðSÞ ¼2, the Albanese image of Sis a genus 2 curve B. Then the re- mainder of the assertion is a consequence of Lemma 2.2 andwðOSÞ ¼1. r Corollary 2.4.Let S be a minimal surface of general type with pg¼q¼2.Ifo;o0are two 1-forms which generate H0ðS;WS1Þand o5o010,then the Albanese morphism a:S!A:¼AlbðSÞis not surjective,the Albanese pencil a:S!B has smooth,con- nected fibres F of genus2with constant modulus and KS2¼8.
Proof. The assertion follows from the theorem of Castelnuovo–De Franchis (see e.g.
[1], p. 123) and the previous proposition. r
Finally we notice that, if the surfaceSof general type withpg¼q¼2 has a genus 2 fibration, then the canonical system is not composed with the genus 2 fibration (see [23], Theorem 2.1, p. 16, and Theorem 5.1, p. 71). As a consequence we have:
Proposition 2.5.Let S be a minimal surface of general type with pg¼q¼2and write jKSj ¼ jMj þZ,wherejMjis the moving part ofjKSjand Z the fixed part.Then the general curve injMjis irreducible.
Proof. Assume otherwise. ThenjMjis composed with an irrational pencilP. IfFis a generic fibre ofP,jMj ¼aF wheread2, and furthermoreF2¼0. SinceFis not a genus 2 curve, KS Fd4. Since KS2c9, we see that either KS Z¼1,KS2¼9 or KS Z¼0,KS2¼8. This cannot occur. Indeed, in the former caseSwould contain a curve y with KS y¼1, hencey would be rational or elliptic, whereas in the latter caseSwould contain að2Þ-curve. In either case we would have a contradiction to
Proposition 2.1, ii). r
3 The caseK2SF9
In [21] I. Reider proved that if Sis a minimal surface of general type with KS2d10 and the bicanonical mapis not birational, thenSpresents the standard case. In Prop- osition (1.1) of [6], it is proven that the same holds ifKS2¼9 andpgd3, unlesspg¼6, KS2¼9, andSis the Du Val–Bombieri surface described in [12] and in [3], p. 193. In fact this result can be extended:
Proposition 3.1.Let S be a minimal surface of general type with KS2¼9such that the bicanonical map is not birational.Assume that S presents the non-standard case.Then
pg¼6,q¼0and S is the Du Val–Bombieri surface.
Proof. To prove the assertion it su‰ces to use the proof of Proposition (1.1) of [6]. There, the assumption pgd3 is only necessary for the proof of Claim 4.
But Claim 4 can be proved without using the assumption on pg. In fact, since KSD@2Dis big and nef, Mumford’s vanishing theorem (see [19], p. 250) yields h1ðS;OSð2KSDÞÞ ¼0. Thus the map H0ðS;OSð2KSÞÞ !H0ðD;ODð2KSÞÞis sur- jective, which in turn implies thatDis hyperelliptic. r
4 The paracanonical system in the casepgFqF2
LetSbe a minimal irregular surface of general type. IfhAPic0ðSÞis a point, we can consider the linear systemjKSþhj. A curve injKSþhjis aparacanonical curveonS.
Assume that the Albanese image ofSis a surface. Given a general pointhAPic0ðSÞ, one has, by [14], Theorem 1,h1ðS;OSðhÞÞ ¼0 and dimjKSþhj ¼wðOSÞ 1.
For hAPic0ðSÞ, let Ch be the general curve in jKSþhj. The curves Ch describe, forhAPic0ðSÞa general point, a continuous systemKof curves onS, of dimension qþdimjKSþhj ¼qþwðOSÞ 1¼pg. This is what we will call themain paracanon- ical systemofS.
Assume now thatSis a minimal surface of general type withpg¼q¼2, for which the Albanese mapa:S!A:¼AlbðSÞis surjective. The main paracanonical system of S has dimension 2 and, forhAPic0ðSÞ a general point, the curve ChAjKSþhj is linearly isolated. We writeCh¼FþMh, whereFis the fixed part of the continu- ous system K andMh the movable part, and we denote by M the continuous, 2- dimensional system described by the curveM:¼Mh. This system is parametrized by a surfacePwhich is birational to Pic0ðSÞ.
Lemma 4.1.Let S be a minimal surface of general type with pg¼q¼2presenting the non-standard case.Let Ch¼FþMh be the general paracanonical curve.Then either:
(i) M :¼Mh is irreducible and M2d3,or
(ii) F ¼0 and M is reducible as M¼M1þM2, with M1 and M2 irreducible each varying in two1-dimensional systems of curvesM1;M2.The following possibilities can occur:
(a) M12¼M22¼0,M1 M2¼4,KS2¼8
(b) M12¼M22¼M1 M2¼2,M1@M2,KS2¼8.
Proof.Suppose thatMis irreducible. ThenM2>0, otherwiseMis a pencil, whereas we know it has dimension 2. The caseM2¼1 is excluded by Proposition (0.14, iii) of [6]. The caseM2¼2 is also excluded by Theorem (0.20) of [6]. This proves (i).
Suppose thatMis reducible. SinceMis a two-dimensional system parametrized by Pic0ðSÞ,Mmust consist of two distinct irreducible componentsM¼M1þM2.
Suppose Mi2¼0 for one ofi¼1;2. ThenMi varies in a pencilMi of curves of genus at least 3 and soKS Mid4. If insteadMi2>0, then, by Proposition (0.18) of
[6],Mi2d2 and one has againKS Mid4, by the 2-connectedness of the paracanon- ical curves. In both cases
KS2 ¼KS FþKS M1þKS M2d8;
and, so, by Proposition 3.1, one hasKS2¼8 andKS M1¼KS M2 ¼4,KS F ¼0.
SinceSdoes not containð2Þ-curves, one hasF ¼0 and we have the two numerical
possibilities listed in (ii). r
Lemma 4.2.Let S be a minimal surface of general type with pg¼q¼2presenting the non-standard case and let Ch¼FþMh be as in casei)of Lemma4.1.Then:
i) if F00,then F Mh ¼2and F is1-connected;
ii) if F00andhis general,the image of the restriction map H0ðS;OSð2KSÞÞ !H0ðMh;OMhð2KSÞÞ has codimension at most1in H0ðMh;OMhð2KSÞÞ.
Proof. i) Let M:¼Mh. If F M¼2, the 2-connectedness of the canonical divisors and Lemma (A.4) of [9] implies thatF is 1-connected. To show thatF M¼2 first we claim that F Mc4. Indeed, Proposition 3.1 yields KS2c8 and Lemma 4.1, i) yieldsM2d3. Therefore
8dKS2dKS M¼M2þF Md3þF M:
So F M being even implies F Mc4. Now we show that F M¼4 cannot oc- cur. Suppose otherwise. Then from 8dKS2 ¼M2þ8þF2 and KS M¼M2 þ M Fd7 we have the possibilities:
a) KS2¼7,KS F ¼0,F2¼ 4, b) KS2¼8,KS F ¼1,F2¼ 3 or c) KS2¼8,KS F ¼0,F2¼ 4.
The first possibility implies thatFcontains two disjoint ð2Þ-curves, whilst the sec- ond and third imply that Fcontains a smooth rational curve. This is impossible by Proposition 2.1, ii). ThereforeF M¼2 and soFis 1-connected.
ii) Note that, since H1ðS;OSð2KÞÞ ¼0, the codimension of the image of the re- striction map
H0ðS;OSð2KÞÞ !H0ðMh;OMhð2KÞÞ
is exactlyh1ðS;OSðKSþFhÞÞ, which by duality is equal toh1ðS;OSðhFÞÞ. Con- sider the exact sequence
0!OSðhFÞ !OSðhÞ !OFðhÞ !0
which yields the long exact sequence
0!H0ðS;OSðhFÞÞ !H0ðS;OSðhÞÞ !H0ðF;OFðhÞÞ
!H1ðS;OSðhFÞÞ !H1ðS;OSðhÞÞ !
Since h0ðS;OSðhÞÞ ¼0 andh1ðS;OSðhÞÞ ¼0, forh general (by [14], Theorem 1), we see thath1ðS;OSðhFÞÞ ¼h0ðF;OFðhÞÞ. NowOFðhÞhas degree 0 on every compo- nent ofF. By the first part of the lemma,Fis 1-connected and so by Corollary (A.2) of [9],h0ðF;OFðhÞÞc1 (with equality holding if and only ifOFðhÞFOF). r
5 The degree of the bicanonical map In the present section we prove the following result:
Proposition 5.1.Let S be a minimal surface of general type with pg¼q¼2.Assume that S presents the non-standard case.Then the degreesof the bicanonical map is2.
Remark 5.2.For completeness let us point out that ifShas a genus 2 fibration then the degree sof the bicanonical mapis either 2 or 4, (see [23]) ands¼4 does occur (cf. Remark 7.2).
First of all we treat the caseKS2¼8, adapting a proof which appears in [17].
Proposition 5.3.Let S be a minimal surface of general type with pg¼q¼2and KS2¼8 presenting the non-standard case.Then the degreesof the bicanonical map is2.
Proof. Let f be the bicanonical mapof S. Notice that ð2KSÞ2¼4KS2¼32 and h0ððS;OSð2KSÞÞ ¼KS2þ1¼9. Then the degree of S¼:fðSÞ is 32s d7, hence s is either 2 or 4.
Supposes¼4. In this caseSis a surface of degree 8 inP8. The list of such surfaces is known (see [20], Theorem 8). Sincej2KSjis a complete linear system,Scan be one of the following:
a) the Veronese embedding inP8 of a quadric inP3;
b) a Del Pezzo surface, i.e. the image of P2 by the rational mapassociated to the linear systemj3LnIxjP2j, whereLis a line andxis a point ofP2;
c) a cone over an elliptic curve of degree 8 inP7.
We are going to prove the result by showing that none of these cases can occur. First we consider case c). Take the pull backFof a line in the cone. Then 2KS F ¼4, hence KS F ¼2. The index theorem then yields F2¼0, and therefore we would have a genus 2 pencil onS.
In case a) 2KS12H, whereHis the pull back of the hyperplane section of S. Then h¼HKS is a nontrivial 2-torsion element in PicS, since pgðSÞ ¼2 whereas h0ðS;OSðKSþhÞÞ ¼4. The e´tale double coverp:Y !Sgiven by 2h10 has invari-
antswðYÞ ¼2,KY2 ¼16. In addition pgðYÞ ¼pgðSÞ þh0ðS;OSðKSþhÞÞ ¼6 so that qðYÞ ¼5. Then, sinceqðSÞ ¼2, the subspaceV ofH0ðY;W1YÞcontaining the anti- invariant 1-forms by the involution i determined by p:Y !S has dimension 3.
Since the image of52V inH0ðY;WY2Þis contained in the subspace of invariant 2- forms which is 2-dimensional, we conclude that there are two independent 1-forms o;o0AVsuch thato5o010 and so by the theorem of Castelnuovo–De Franchis there exists a fibrationg:Y!Bwithb:¼gðBÞd2 (cf. also [5], Corollary (4.8)).
Let f be the genus of a general fibre F of g. Suppose f ¼2, bd3. Then the curve F0¼iðFÞ cannot dominateB via g. Hence F0 is again a curve of the pencil g:Y !B. It cannot be the case that F0¼F, otherwise pðFÞ would be a moving curve of genus 0 or 1 onS, a contradiction. In conclusionF0F0andpðFÞ ¼pðF0Þ would be a curve of genus 2 onSvarying in a pencil, a contradiction.
Now, by Lemma 2.2, i), we haveKY2 ¼16d8ðf 1Þðb1Þand, by Lemma 2.2, ii), 5¼qðYÞcf þb. This forces f ¼3,b¼2 or viceversa and soYis birational to BF (see again Lemma 2.2, ii)). HenceYhas a pencil of curves of genus 2, whose image on S, by what we observed above, is again a genus 2 pencil, against our hy- pothesis. Thus also case a) does not occur.
Finally we consider case b). We abuse notation and we denote by L the image on S of a line ofP2. Let 2LþL0 be the hyperplane section ofS. We have 2KS1 fð2LþL0Þ, and sofðL0Þ12ðKSfLÞ.
ChooseL0such thatfðL0Þis a smooth irreducible curve and consider the double coverYofSbranched overfðL0Þand determined byKSfðLÞ. The double cover formulas give wðYÞ ¼3, KY2 ¼24, pgðYÞ ¼pgðSÞ þh0ðS;OSð2KSfLÞÞ ¼7; so thatqðYÞ ¼5.
Notice thatjfðL0Þjis a genus 3 pencil on S. The pull back of it toYis either a rational pencil of curves of genus 5, or a genus 3 pencil. In the former caseYwould be birational to the product of P1 by a curve of genus 5 (see again Lemma 2.2), which is not possible. In the other case let b be the genus of the base curve of the pencil. As beforebd2, becausebþ3dqðYÞ ¼5. On the other hand Lemma 2.2, i) yieldsKY2 ¼24d16ðb1Þ. Hence b¼2 and as above we conclude that Yis bira- tional to a product of a genus 2 and a genus 3 curve, which is impossible because
pgðYÞ ¼7. r
Before continuing towards the proof of Proposition 5.1 we need to recall some facts about continuous systems of curves on a surface. For the basic definitions, we refer the reader to [6], §0. Given an irreducible, continuous systemCof curves of di- mensionron a surfaceS, theindexn:¼nCofCis the number of curves ofCpassing through rgeneral points ofS. Of coursend1. A system Cis called aninvolutionif its index isn¼1. Typical examples of involutions are:
(i) the linear systems;
(ii) pencils, or, more generally systemscomposed with pencils. This means that there is a pencil f :S!Band an involution of divisors onBsuch that the curves ofCare pull-backs via f of divisors of an involution onB.
The classical theorem of Castelnuovo–Humbert tells us that these are essentially the only involutions.
Theorem 5.4(Castelnuovo–Humbert, see [7], §5).Let S be a smooth,irreducible,pro- jective surface and letCbe an r-dimensional involution on S which has no fixed divisor and whose general divisor C is reduced.Then eitherCis a linear system or it is com- posed with a pencil.
We will use this theorem to prove the following basic result:
Proposition 5.5.Let S be a minimal surface of general type with pg¼q¼2.Assume that S presents the non-standard case.Let Ch¼FþMh be the general paracanonical curve and suppose that M :¼Mh is irreducible.Then the restriction of the bicanonical mapfto M is a birational map of M onto its image.
Proof. First we consider the case F ¼0. Then the arithmetic genus g of M is g¼KS2þ1. Since, by [14], Theorem 1, h1ðS;OSðKSþhÞÞ ¼0 for a general point hAPic0ðSÞ,j2KSjcuts out onM¼Mha non-special, base point free completeg2g2g2. We will argue by contradiction and we will suppose from now on that this series is composed with an involution t:¼tM of degreedd2. Then we must have 2g2d dðg2Þ which yields dc2þg22 ¼2þK22
S1. Since, by Proposition 2.1, one has KS2d4, we see thatd¼2. This means thatfðMÞis a linearly normal curve of degree g1 inPg2, whose arithmetic genus is 1. Notice that two distinct points x;x0 are conjugated intM if and only iffðxÞ ¼fðx0Þ.
Claim1:Let M;M0be general curves inM,then MVM0does not contain four distinct points x;y;x0;y0such thatfðxÞ ¼fðx0ÞandfðyÞ ¼fðy0Þ.
Otherwise we would have h0ðM;OMðM0ÞÞdh0ðM;OMðxþx0þyþy0ÞÞ ¼2. On the other hand, sinceh1ðS;OSðhÞÞ ¼0 forhAPic0ðSÞa general point,jM0jcuts out a complete linear series onM. SinceM0 is linearly isolated, we find a contradiction.
Letxbe a point onS. We denote byMxthe system of curves inMpassing through x.
Claim 2:Let x and x0 be general points on M conjugated int, i.e. such thatfðxÞ ¼ fðx0Þ.Every irreducible component of Mx is a 1-dimensional system of curves. Con- sider the union of all of these components containing M. Every curve in such a union contains x0.
LetM00be the general curve in a componentM0of the union in question and letxM00
be the point conjugated toxin the involutiontM00onM00. SincefðxM00Þ ¼fðxÞandf is generically finite,xM00 belongs to a finite set whenM00varies inM0, and therefore it stays fixed whenM00 varies inM0. Since xM ¼x0 we havexM00 ¼xM ¼x0, proving the claim.
It is appropriate to denote by MM;x;x0 the union of all components of Mx con- tainingM. SinceMis parametrized by a surfacePbirational to Pic0ðSÞ, the system MM;x;x0corresponds to a reduced curveDM;x;x0 onP. This curve might be reducible, but all of its irreducible components pass, by definition, through the point m of Pic0ðSÞcorresponding toM.
Claim3:When M and x;x0vary,DM;x;x0 varies in a2-dimensional systemDof curves on P with no base point.There is only one curve ofDcontaining two general points of P,i.e.Dhas index1,hence it is an involution.
LetMbe a general curve inM, thus corresponding to a general pointmof Pic0ðSÞ.
Of coursembelongs to a 1-dimensional system of curvesDM;x;x0, whenx;x0AMare conjugated byt. This proves thatDis 2-dimensional. A base point ofDwould cor- respond to a curveM ofM which belongs toDM;x;x0 for the general curveM and every pair of pointsx;x0conjugated intonM. But thenMwould have every pair of pointsx;x0onMconjugated intin common withM, a contradiction. The final as- sertion follows by Claim 1.
Claim4:Dis not a linear system.
SupposeDis a linear system. Consider the morphismfD:P!P2determined byD, which has degree at least 2. This means that, given a general curveM, corresponding tomAP, there is a curveM00Mcorresponding to m0APwithm00m, such that for every curve DADcontainingm, it also containsm0. Therefore for every pair of pointsx;x0 conjugated intonMthe curveDM;x;x0, which containsm, also contains m0, and this implies that M0 hasx andx0 in common with M. Asx;x0 vary onM staying conjugated int, we see thatMandM0have infinitely many points in com- mon, a contradiction.
Claim5:Dis not composed with a pencil.
SupposeDis composed with a pencil. By the very definition of a familyMM;x;x0, we have that the general curveDM;x;x0, if reducible, has all of its components containing the pointmAPcorresponding toM. On the other hand, by the definition of a system composed with a pencil, the general curve of such a system may have a singular point only at the base points of the pencil, which are fixed. Hence the general curve of a sys- tem composed with a pencil is not singular at a moving point. Thus we see thatDM;x;x0 must be irreducile. Since we are assuming thatDis composed with a pencil, this would imply thatDitself is a pencil, which contradicts the fact thatDhas dimension 2.
In conclusion Claims 4 and 5 above contradict the Castelnuovo–Humbert theorem above, which concludes our proof in caseF¼0.
Next we consider the case F00. By Lemma 4.2, F is 1-connected, F Mh¼2 and the linear system j2KSj cuts out onM a base point free linear series g2gr , with rdg1. Suppose that this series is composed with an involutiont:¼tM of degree dd2. Then we must have 2gddðg1Þ. This yieldsd¼2. Otherwise we would have gc3, whereasM2d3, (see Lemma 4.1), which impliesgd5.
Ifr¼g, thenMis hyperelliptic andj2KSjcuts onMtheg-fold multiple of theg21. In this situation, Claim 2 above still holds. On the other hand, by arguing as in Claim 1 above, we see that, if x;x0 are two general points onMconjugated in the hyper- elliptic involution, then Mis the unique curve inM containing them. Putting these two things together, we reach a contradiction.
If r¼g1 either M is hyperelliptic and we can argue as before, or fðMÞ has arithmetic genus 1, and then we can argue as in the caseF ¼0. r
Now we are ready to give the
Proof of Proposition 5.1. Proposition 5.3 is the statement forKS2¼8 so we can as- sume that KS2c7. Then, by Lemma 4.1, the general curve M:¼Mh in M is irre- ducible and, by Proposition 5.5, f is birational on M. Set M0¼Mh. Since M0 is also a general curve inM,f is also birational onM0. LetxAM be a general point and let x0BM be another point of S such that fðxÞ ¼fðx0Þ. By the generality of xAM, the pointx0is also a su‰ciently general point onS, hence it does not lie onF.
SinceMþM0þ2FAj2KSjwe havex0AM0. Again by the generality ofx0and ofM0, there is no other point x00AM0 such that fðx00Þ ¼fðxÞ ¼fðx0Þ. So the degree off has to be 2.
6 The bicanonical involution
LetSbe a surface with pg¼q¼2 presenting the non-standard case. By Proposition 5.1 the bicanonical mapf:S!Shas degree 2.
In general if the bicanonical mapof a surfaceShas degree 2 we can consider the bicanonical involutioni:S!S.
The involutioniis biregular, sinceSis minimal of general type, and the fixed locus ofiis the union of a smooth curveR0and of isolated pointsP1;. . .;Pt. LetSS~ be the quotient ofSbyiand let p:S!~SSbe the projection onto the quotient. The surface S~
Shas nodes at the pointsQi:¼pðPiÞ,i¼1;. . .;t, and is smooth elsewhere. Of course the bicanonical mapofSfactors throughp.
IfR00 q, the image via pofR0is a smooth curveB00not containing the singular pointsQi,i¼1;. . .;t.
Let now f :V !S be the blow-upof S at P1;. . .;Pt and set R¼ fR0, Ei¼ f1ðPiÞ, i¼1;. . .;t. The involution i induces a biregular involution ~ii on V whose fixed locus isRþP
Ei. The quotientW¼V=h~iiiis smooth and one has a commu- tative diagram
V f! S
p
??
?y
??
?yp W g! ~SS
ð6:1Þ
where p:V !W is the projection onto the quotient andg:W !SS~ is the minimal desingularization map. Of course also the bicanonical map ofV factors throughp.
Notice thatAi:¼g1ðQiÞis an irreducibleð2Þ-curve fori¼1;. . .;t. The map pis flat, since it is finite andWis smooth. SetB0¼gB00. Thus there exists a line bundle LonWsuch that 2L1B:¼B0þP
AiandpOV¼OWlL1.OW is the invariant
andL1the antiinvariant part ofpOVunder the action of~ii. Sincepis a double cover, the invariants ofVandWare related by
KV2 ¼2ðKWþLÞ2; wðOVÞ ¼2wðOWÞ þ1
2L ðKWþLÞ;
pgðVÞ ¼pgðWÞ þh0ðW;OWðKW þLÞÞ:
ð6:2Þ
Since Vis the blow-upof Sat tpoints,wðOSÞ ¼wðOVÞandKS2¼KV2þt. In this case, because we are considering double covers through which the bicanonical map factors, we can be more precise:
Proposition 6.1.Let S be a minimal surface of general type with pgðSÞd1and bica- nonical map of degree2.Then,keeping the above notation,one has:
i) h0ðW;OWð2KW þLÞÞ ¼0,h0ðW;OWð2KW þBÞÞ ¼h0ðS;OSð2KSÞÞ;
ii) either pgðWÞ ¼0 and h0ðW;OWðKWþLÞÞ ¼pgðSÞ, or pgðWÞ ¼pgðSÞ and h0ðW;OWðKWþLÞÞ ¼0;
iii) j2KVj ¼pj2KWþB0j þP
iEi, fj2KSj ¼pj2KWþB0j and furthermore OWð2KW þB0Þis nef and big;
iv) ð2KWþB0Þ2 ¼2KS2; v) wðOWð2KW þLÞÞ ¼0;
vi) KW ðKW þLÞ ¼wðOWÞ wðOSÞ.
Proof. i), ii) By the projection formulas for double covers, one has
H0ðV;OVðKVÞÞ ¼H0ðW;OWðKWÞÞlH0ðW;OWðKW þLÞÞ and
H0ðV;OVð2KVÞÞ ¼H0ðW;OWð2KWþBÞÞlH0ðW;OWð2KWþLÞÞ:
In both the above decompositions, the first summand is the invariant, the second the anti-invariant part under the action of the involution~ii. The fact that the bicanonical mapof V factors through p implies the vanishing of one of the two summands in each of the decompositions. Thus assertion ii) follows immediately. Since pgðSÞd1, either the invariant or the anti-invariant part of H0ðV;OVðKVÞÞis non-zero. Hence the invariant part of H0ðV;OVð2KVÞÞ is certainly non-zero, and therefore i) also holds.
iii) Recall that B¼B0þP
Ai. Part i) implies that j2KVj ¼pj2KWþBj. Since j2KSjis base point free (see [8]), the fixed part ofj2KVjis 2P
iEi. More precisely, one
hasj2KVj ¼ fj2KSj þ2P
iEi. Thus one has fj2KSj ¼pj2KW þB0jand therefore OWð2KWþB0Þis nef and big becauseOSð2KSÞis nef and big.
iv) follows immediately from fj2KSj ¼pj2KW þB0j because fð2KSÞ2¼4KS2 andpis a double cover.
v) Since 2ðKW þLÞ1ð2KWþB0Þ þP
Ai and OWð2KW þB0Þ is nef and big by iii), we can apply the Kawamata–Viehweg vanishing theorem to the divisorKWþL (see [13], Corollary 5.12, c), pp. 48–49) obtaining:
hiðW;OWð2KW þLÞÞ ¼0 i¼1;2:
By i)h0ðW;OWð2KW þLÞÞ ¼0, thuswðOWð2KW þLÞÞ ¼0.
vi) By the Riemann–Roch theorem and by the formulas (6.2) we have
wðOWð2KW þLÞÞ ¼1
2ð2KW þLÞ ðKWþLÞ þwðOWÞ
¼KW ðKW þLÞ þ1
2L ðKW þLÞ þwðOWÞ
¼KW ðKW þLÞ þwðOSÞ wðOWÞ:
Then the assertion follows from part v). r
If Sis a minimal surface of general type with pg¼q¼2 and bicanonical mapof degree 2, we can be more specific.
Lemma 6.2.Let S be a minimal surface of general type with pg¼q¼2 for which the Albanese map is surjective.Suppose the bicanonical map of S has degree2and let W be as above.Then either
i) pgðWÞ ¼2,qðWÞ ¼2,or ii) pgðWÞ ¼0,qðWÞ ¼1or iii) pgðWÞ ¼2,qðWÞ ¼0.
Proof. By ii) of Proposition 6.1 we know that eitherpgðWÞ ¼2 or pgðWÞ ¼0. By the projection formulas for double covers, one has
2¼qðSÞ ¼h1ðV;OVðKVÞÞ ¼h1ðW;OWðKWÞÞ þh1ðW;OWðKWþLÞÞ and thereforeqðWÞc2 with equality holding if and only ifh1ðW;OWðKWþLÞÞ ¼0.
Assume thatqðWÞ ¼2. ThenH0ðV;WV1Þis generated by two 1-formso;o0which are invariant under the bicanonical involution and therefore o5o0 is an invari- ant element of H0ðV;WV2Þ. Since, by Corollary 2.4,o5o020, pgðWÞ00 and so
pgðWÞ ¼2.
Assume now thatqðWÞ ¼1. ThenH0ðV;WV1Þhas invariant and antiinvariant sub- spaces both of dimension 1. Ifoþandoare generators of such subspaces, they form a basis of H0ðV;WV1Þ. Since, as before, oþ5o20, oþ5o is a nonzero anti- invariant element ofH0ðV;WV2Þ. SopgðWÞis not 2 and therefore pgðWÞ ¼0.
Suppose now thatqðWÞ ¼0. ThenH0ðV;WV1Þis generated by two 1-formso;o0 which are antiinvariant under the bicanonical involution and therefore o5o0 is an invariant element of H0ðV;WV2Þ. As in the preceding paragraphs we conclude that
pgðWÞ ¼2. r
We keep the same assumptions as in Lemma 6.2, and we analyse the possibilities given by the lemma.
Lemma 6.3.The case qðWÞ ¼2cannot occur.
Proof. Suppose otherwise. By Proposition 6.1, vi) we haveKW ðKW þLÞ ¼0 and so KW ð2KW þB0Þ ¼0. Therefore, sincejKWjis a pencil we get a contradiction to the fact that 2KW þB0is nef and big (see Proposition 6.1, iv)). r Lemma 6.4.Keep the assumptions in Lemma6.2and assume furthermore that S has no genus2pencils.Then the case qðWÞ ¼1does not occur.
Proof. We notice first that kðWÞ<0 and thus W is a ruled surface. In fact sup- pose otherwise. Then some multiple ofKW is an e¤ective divisor. By Proposition 6.1, vi) we have KW ðKWþLÞ ¼ 1, and so KW ð2KWþB0Þ<0, which contradicts 2KW þB0being nef and big.
In this case we have, by Proposition 6.1, ii),h0ðW;OWðKW þLÞÞ ¼2 and thus we can writejKW þLj ¼ jYj þZ, wherejYjis the moving part andZis the fixed part.
Since for eachð2Þ-curveAiwe haveAi ðKW þLÞ ¼ 1, we infer thatZ00. No- tice thatpðjYjÞis exactly the moving part ofjKVjand therefore by Proposition 2.5 the general curve YinjYjis irreducible. Furthermore, since Wis not rational, and jYjis a linear system of dimension 1, the geometric genus of a general curveY AjYj is at least 1.
Claim 1: for every e¤ective, non-zero divisor N<KW þL, one has h0ðN;ONÞ þ paðNÞc2.
By the Riemann–Roch theorem, we have
h0ðW;OWðKW þNÞÞ þh2ðW;OWðKWþNÞÞ
¼1
2ðKW NþN2Þ þh1ðW;OWðKW þNÞÞ: ðÞ Now notice thath0ðW;OWðKW þNÞÞ ¼0. If not, sinceN<KW þL, we would have
h0ðW;OWð2KWþLÞÞ00, a contradiction to Proposition 6.1. SinceNis e¤ective we have alsoh2ðW;OWðKW þNÞÞ ¼0, and soðÞcan be written as
paðNÞ 1þh1ðW;OWðKW þNÞÞ ¼0:
Since, by duality, h1ðW;OWðKWþNÞÞ ¼h1ðW;OWðNÞÞ and one has h1ðW;OWðNÞÞdh0ðN;ONÞ 1 for any e¤ective divisorN, we obtainpaðNÞ 1 þ h0ðN;ONÞ 1c0, proving the claim.
Claim2:If T is a general ruling of W,then Y T¼1.
As we noticed already, the geometric genus of a general curve YAjYj is at least 1, and of courseh0ðY;OYÞ ¼1. By Claim 1 we conclude thatYis smooth and elliptic.
Claim 1 implies also that each irreducible componentyofZis rational and such that y Yc1. Consider the penciljYj. By the Riemann–Roch theorem
h0ðW;OWðYÞÞ ¼Y2þh1ðW;OWðYÞÞ
and so 0cY2c2. We claim thatjYjhas no multiple fibres. IfY2>0 the claim is trivial, sinceY2c2. AssumeY2¼0 and notice thatY L>0 because otherwise we would have a pencil of curves of genus 1 onV, which is impossible. HenceY Z¼ Y2þY Z¼Y ðKW þLÞ ¼Y L>0 and thus there exists an irreducible curve y inZsuch thatY y¼1. So also in this casejYjhas no multiple fibres.
We can consider now the relatively minimal fibration h: ~WW !P1 associated to jYj, i.e. we blow upthe base points ofjYj, if any, and contract theð1Þ-curves con- tained in fibres ofjYj. SincejYj has no multiple fibres andwðOWW~Þ ¼1, we have by [1], corollary V.12.3, p. 162,KWW~1 2F, whereFis a general fibre ofh.
Let now Tbe a general ruling ofWandTT~ the corresponding ruling of WW~. Since KWW~ TT~ ¼ 2, we conclude that F TT~¼1 and therefore also Y T ¼1 proving Claim 2.
Now we can finish our proof. LetTbe the general ruling of W. Since each com- ponent of Z is rational, TZ¼0, and so we have ðKW þLÞ T¼ ðZþYÞ T ¼ Z TþY T¼1. SinceKW T ¼ 2, we haveL T¼3. This implies that, by pull- ing back to Vthe ruling of W, we obtain a pencil of curves of genus 2, against our
hypothesis. r
Finally we come to the caseqðWÞ ¼0.
Proposition 6.5.Keep the assumptions as in Lemma6.2and assume furthermore that S has no genus2 pencils. If qðWÞ ¼0 then B0¼0,W is a minimal surface of general type with pgðWÞ ¼2,KW2 ¼2 and p:S!~SSis ramified only at20 nodes ofSS.~ Fur- thermore,if C and C0 are the general curves injKWjandjKSj respectively,C and C0 are smooth,irreducible and non-hyperelliptic.
Proof. We keepthe notation as in the beginning of the section. Let a:S!A:¼ AlbðSÞbe the Albanese map. We can define a morphism~aa: ~SS!Aby associating to
each pointxASS~the sum of the Albanese images of the two points in the cycle pðxÞ.
SinceqðSSÞ ¼~ 0 this mapis constant and, upto a translation, we may assume that its image is the point 0AA. Hence if pðxÞ ¼y1þy2we haveaðy1Þ ¼ aðy2Þ. Thus we can define a morphism a: ~SS!KðAÞ, where KðAÞis the Kummer surface ofA, by associating toxASS~ the point yAKðAÞcorresponding toaðy1Þ ¼ aðy2Þ.
Given any pointx0 in the branch locus, we have pðx0Þ ¼y0þy0, soaðy0Þis a 2- torsion point inA. In particular the ramification divisorRmust be contracted by the Albanese mapand so also B00 is contracted by a. Notice thatK~SS¼aðKKðAÞÞ þD, whereDis the divisor where the di¤erential ofadrops rank, in particularDcontains all the curves contracted bya. SinceKðAÞis aK3 surface, we see that there is an ef- fective canonical divisor onSS~containing the smooth curveB00. Hence alsoKW can be written asB0þD, whereDis an e¤ective divisor.
Notice that by the classification of surfacesWis either elliptic or of general type.
LetjKWj ¼ jYj þZ, whereZis the fixed part andjYjthe movable part ofjKWj. Since pgðWÞ ¼2, the system jYj is a pencil. SinceW is regular, by Bertini’s theorem the general curve ofjYjis irreducible.
Remember that the bicanonical mapof Vhas degree 2 to its image, and factors throughpand through the mapdefined by the linear systemj2KW þB0jonW. This implies that the linear series cut out byj2KW þB0jon the general curveY AjYjde- termines a birational mapon Y. In particular it has projective dimension at least 2.
SinceWis not ruled, one hasð2KW þB0Þ Yd3, with equality being possible only if gðYÞ ¼1, which in turn is only possible ifY2¼KW Y¼0.
By the formulas (6.2) and by Proposition 6.1, we haveKW ðKWþLÞ ¼2, hence ð2KW þB0Þ KW ¼4.
Since 2KW þB0is nef we haveð2KWþB0Þ Ycð2KWþB0Þ KW ¼4. SinceYis nef, one hasB0 Yd0, hence we obtainKW Yc2. By the adjunction formulaY Z is even, hence eitherY Z¼0 orY Z¼2. On the other hand we have seen above that we can writeKW ¼B0þD, where Dis an e¤ective divisor, and so 2KW þB0¼ 3B0þ2D, hence 3c3B0 Yþ2Y Dc4, so either B0 Y ¼0 or B0 Y ¼1 and gðYÞ ¼1. This is impossible because then Y2¼Y KW ¼0, thus 0¼Y KW ¼ Y B0þY D¼1þY D and the nef divisor Y would be such that Y D¼ 1, a contradiction. Thus the only possibility is Y B0¼0, Y D¼2 and therefore KW Y ¼2. Since Y Z is even and non-negative, we have that either Y2¼0, Y Z¼2 orY2¼2;Y Z¼0.
In the first caseY2¼0,Y Z¼2, we getOYð2KW þB0ÞFOYð2KYÞ. This is im- possible because in this casejYjis a genus 2 pencil and soj2KWþB0jwould determine a non-birational maponW.
If Y2¼2;Y Z¼0, then we have 2KW þB012Yþ ð2ZþB0Þ and 2Y ð2ZþB0Þ ¼0. Since 2KWþB0 is nef and big, the only possibility is that 2ZþB0¼0. So B0¼Z¼0, hence KW1Y is nef and therefore W is minimal.
Moreover KW2 ¼Y2¼2. Furthermore 2KW þB0¼2KW and, by Proposition 6.1, iv) we have KS2¼4. In addition, by the formulas (6.2) and by Proposition 6.1, we have ðKW þLÞ2¼ 8 and so KV2 ¼ 16. Hence t¼16þKS2¼20, where t is, as before, the number of isolated fixed points of the bicanonical involution. Thus p is ramified exactly over 20 nodes.
By the above the general curveCin the linear systemjKWj ¼ jYjis irreducible and non-hyperelliptic, because the bicanonical map ofWis birational. SinceKW2 ¼2 and jKWjis a rational pencil,Cis necessarily smooth. The assertion for the general curve
C0injKSjis then obvious. r
7 The main theorem
In the previous sections we saw that if the bicanonical mapfof a surfaceSwithpg¼ q¼2 is not birational andS has no pencil of curves of genus 2, thenf has degree s¼2 and we have described in Proposition 6.5 some properties of the quotient ofS by the involution induced by the bicanonical map.
In this section we will classify these quotients. Let us start by presenting an exam- ple, which was first pointed out by F. Catanese (cf. [8], Example (c), page 70, and Remark 3.15, page 72).
Example 7.1. Let A be an abelian surface with an irreducible symmetric principal polarization Y, and suppose that A contains no elliptic curves. Let h:S!A be the double cover branched on a smooth divisor BAj2Yj so that hOS¼OAl OAðYÞ. Since KS ¼hðYÞ, the invariants of the smooth surface S are pgðSÞ ¼2, qðSÞ ¼2, KS2¼4. Notice that the map h:S!A factors through the Albanese mapa:S!AlbðSÞ. Sinceh has degree 2 and AlbðSÞis a surface, we see that that AlbðSÞFA. In addition we observe thatShas no genusb pencil of curves of genus 2. Indeed, by Lemma 2.2, ii) and by the assumption that AFPic0ðSÞ contains no elliptic curve, one should have b¼2, and by part i) of the same lemma we would findKS2d8, a contradiction.
Remark now thatBis symmetric with respect to the involution jofAdetermined by the multiplication by1. Hence jcan be lifted to an involutionionSthat acts as the identity onH0ðS;OSðKSÞÞ. We denote byp:S!SS~:¼S=hiithe projection onto the quotient. We observe that pgðSSÞ ¼~ 2,qð~SSÞ ¼0,K~2
S
S ¼2 and the only singularities of the surface~SSare 20 nodes. Sinceh0ðSS;~ OSS~ð2KSS~ÞÞ ¼wðO~SSÞ þK~2
S
S ¼5¼h0ðS;OSð2KSÞÞ, the bicanonical mapofSfactors through p:S!SS. Since~ Shas no pencil of curves of genus 2, we have the situation described in Proposition 6.5.
For the sake of completness, we want to point out the following alternative de- scription of~SS. One embeds, as usual, the Kummer surface KumðAÞofAas a quartic surface inP3¼PðH0ðA;2YÞÞ. The surfaceSS~is a double cover of KumðAÞbranched along the smooth plane sectionHof KumðAÞcorresponding toBand on 6 nodes, cor- responding to the six points of order 2 ofAlying onY. The ramification divisorRof S~
S!KumðAÞis a canonical curve isomorphic toH, and thus it is not hyperelliptic.
Remark 7.2.The same construction can also be done with a reducible polarizationY on A. Then A is isomorphic to the productE1E2 of two elliptic curves and the surfaceSconstructed as above has two elliptic pencils of genus 2 curves. In this case the bicanonical mapofShas degree 4 (see [23], Theorem 5.6).
We are finally going to prove our classification theorem:
Theorem 7.3. Let S be a minimal surface of general type with pg¼q¼2,presenting the non-standard case.Then S is as in Example7.1.
For the proof we need a preliminary lemma and some notation. LetXandY be smooth, projective surfaces and f :X !Y be a surjective map. Let Rbe the rami- fication curve onX, i.e. the subscheme ofXwhere f drops rank. LetCbe a smooth, irreducible curve onXnot contained inR. SetG:¼ fðCÞand fðGÞ ¼CþD. No- tice thatCandDhave no common component. For every point pAC, denote byrp
[resp. by dp] the coe‰cient of p in the divisor cut out on C byR[resp. by D]. Set dp¼rpdpand p0:¼ fðpÞ. Then:
Lemma 7.4.With the above notation,if Gis smooth at p0,thendpd0.
Proof. Use local coordinatesðs;tÞcentered at p0in such a way thatGhas equation t¼0. Use local coordinatesðx;yÞcentered at pin such a way thatChas equation x¼0 and fðx;yÞ ¼0 is the equation of D. Then f has local equations s¼cðx;yÞ andt¼xfðx;yÞ. ThereforeRhas equation
fqc
qyþxqðf;cÞ qðx;yÞ¼0
whence the assertion follows immediately. r
Now we can prove our classification theorem:
Proof of Theorem7.3. The main stepin our proof is to show that the Albanese map a:S!A:¼AlbðSÞhas degreen¼2. This is what we are going to prove first.
As we saw in Section 6, the bicanonical mapofSfactors through the degree 2 finite cover p:S!SS~ branched only at the 20 nodes ofSS. By Proposition 6.1, iii),~ K~SSis a nef and big line bundle onSS. More precisely, from Proposition 6.5 it follows that~ jKSS~j is a pencil with no fixed component and with two base points which do not occur at any of the nodes ofSS. Hence~ ðp:S!SS;~ KSS~Þis a good generating pair in the sense of [10].
Let C be a general curve in jK~SSj and let C0:¼pðCÞ. Since C does not contain any of the nodes ofSS, the cover~ p:C0!Cis an e´tale double cover. Theorem (6.1) of [CPT] yields that the Prym varietyP:¼PrymðC0;CÞrelated to the double cover p:C0!C is isomorphic to the Albanese surface A. Therefore Ais principally po- larized, and we denote by Y its principal polarization. Furthermore, after having identifiedAwithP, the Abel–Prym mapa:C0!Pcoincides, upto translation, with the restriction to C0 of the Albanese map a:S!A. Notice thatC0 is not hyper- elliptic and setG:¼aðC0Þ. By the results in [15], chapter 12, the mapajC:C0!G is an isomorphism and therefore G is smooth. FurthermoreG is in the class of 2Y by Welters’ criterion (see again [15], chapter 12).
Let us setaðGÞ ¼C0þDand let us denote byRthe ramification curve ofa. By Lemma 7.4 we haveKS D¼C0 DcC0 R¼C0 KS ¼4, with equality holding if
and only ifD@KS. By the index theorem we haveD2c4. Thusn 8¼n ð2YÞ2¼ aðGÞ2¼ ðC0þDÞ2c16. This proves thatn¼2 and, in addition, thatD@KS.
Now we can finish our proof by showing that the branch curve B of a:S!A is a divisor in the class of 2Y. This immediately follows from the fact that 16¼ 2R ðC0þDÞ ¼2B G, henceB Y¼4, so thatBis numerically equivalent to 2Y.
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Received 1 October, 2001
C. Ciliberto, Dipartimento di Matematica, Universita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy
Email: cilibert@mat.uniroma2.it
M. M. Lopes, CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Email: mmlopes@lmc.fc.ul.pt
