Numerical Godeaux surfaces with an involution

0.4 MB·English

by  Alberto Calabri

NUMERICAL GODEAUX SURFACES WITH AN INVOLUTION
ALBERTOCALABRI,CIROCILIBERTOANDMARGARIDAMENDESLOPES
Abstract. Minimalalgebraicsurfacesofgeneraltypewiththesmallestpos-
5 sibleinvariants have geometric genus zeroand K2=1andareusuallycalled
0 numerical Godeaux surfaces. Althoughtheyhavebeenstudiedbyseveralau-
0 thors,theircompleteclassificationisnotknown.
2 In this paper we classify numerical Godeaux surfaces with an involution,
n i.e. an automorphism of order 2. We prove that they are birationally equiv-
a alent either to double covers of Enriques surfaces, or to double planes of two
J different types: the branch curve either has degree 10 and suitable singular-
ities, originally suggested by Campedelli, or is the union of two lines and a
1
curveofdegree12withcertainsingularities. Thelattertypeofdoubleplanes
3
are degenerations of examples described by Du Val and their existence was

previouslyunknown;weshowsomeexamplesofthisnewtype,computingalso
]
G theirtorsiongroup.
A
.
h 1. Introduction.
t
a In the one-century-and-a-halfhistory of classification of algebraic varieties, sur-
m
faces having geometric genus p =0 and irregularityq =0 have been studied from
g
[ the very beginning. They were conjectured to be rational by Max Noether (about

1 1870), until Enriques, in 1894, suggested the existence of surfaces with pg =q =0
v and bi-genus P2 = 1 which now bear his name. After that, Castelnuovo, in 1896,
1 provedhiscelebratedrationality criterion,whichsaysthatasurfaceX isrationalif
5
andonlyifP (X)=q(X)=0. Sincethen,theclassificationofsurfaceswithp =0
2 g
5
has received particular attention by algebraic geometers, and not only. Indeed, in
1
the course of the years, it has been discovered that these surfaces are interesting
0
5 notonlyforclassificationpurposes,butalsofortheirintriguingrelationswithother
0 fields of mathematics, e.g. Bloch’s conjecture, classification of four-folds, etc. (see
h/ [Bl, Fr], for general information cf. [Ci2]).
t In 1931–32, Godeaux and Campedelli gave the first two examples of minimal
a
surfaces of general type with p = 0. Godeaux considered a quotient of a quintic
m g
surfaceinP3 byafreelyactingcyclicgroupoforder5ofprojectivetransformations.
:
v The smooth minimal model of this surface has K2 = 1. Campedelli constructed a
i double plane, i.e. a double cover of P2, branched along a degree 10 curve with six
X
points, not lying on a conic, all of type [3,3], that is a triple point with another
r
a infinitely near triple point. Here the smooth minimal model has K2 =2.
Campedelli also proposed the construction of a minimal surface of general type
with p = 0 and K2 = 1 as the minimal model of a double plane branched along
g
a curve of degree 10 with a 4-tuple point and five points of type [3,3], not lying
on a conic. The actual existence of such a curve was proved only 50 years later by
Kulikov, Oort and Peters in [OP]. We will say that a double plane with branch
curve having the singularities suggested by Campedelli is of Campedelli type.
MathematicsSubject Classification (2000): 14J29.
ThisresearchhasbeencarriedintheframeworkoftheEAGERprojectfinancedbytheEC,project
n.HPRN-CT-2000-00099. ThefirsttwoauthorsaremembersofG.N.S.A.G.A.-I.N.d.A.M.,which
generouslysupportedthisresearch. ThethirdauthorisamemberoftheCenterforMathematical
Analysis,GeometryandDynamicalSystems,ISTandwaspartiallysupportedbyFCT(Portugal)
throughprogramPOCTI/FEDERandProjectPOCTI/MAT/44068/2002.
1
2 A.CALABRI,C.CILIBERTOANDM.MENDESLOPES
Minimal surfaces of general type with p = 0 and K2 = 1 are nowadays called
g
numericalGodeaux surfaces. Theyhavebeenstudiedclassically,butalsobyseveral
authors in the last 30 years: it would be too long to recall here all the relevant
contributions, anyway most of them appear in our list of references.
As its construction shows, the original Godeaux’s example has non-trivial tor-
sion, more precisely its torsion group is cyclic of order 5. It is actually a result of
Miyaokain[Mi]thatthetorsiongroupofanumericalGodeauxsurfaceiscyclicofor-
deratmost5andsurfacesS withTors(S)=Z/5Zfillupanirreduciblecomponent
of the moduli space of the expected dimension 8 = χ(T ), consisting of quotients
S
of quintics in P3 by a Z/5Z-action, as in the original Godeaux’s construction.
It has to be expected that the larger is the torsion, the easier is the study and
the classification of these surfaces. This view-point has been pursued by Miles
Reid in [Re1], who proved that also surfaces with Tors(S) = Z/3Z or Z/4Z fill
up an 8-dimensional irreducible component of the moduli space, giving an explicit
description of their canonical rings.
Although a few examples of numerical Godeaux surfaces S with no torsion (cf.
[CG, DW]), or even simply connected (see [Bar2]), or with Tors(S) = Z/2Z (see
[Bar1, We1, We2]) are known, neither a classification result, nor a description of
the moduli space are known in these cases. Furthermore, all these examples turn
out to possess an involution, i.e. a birational automorphism of order 2.
This motivates the study of numerical Godeaux surfaces with an involution. A
first investigation of this subject has been done by J. Keum and Y. Lee in [KL],
undertheassumptionthatthebicanonicalsystemhasnofixedcomponents. Inthis
paper, we make no assumption of this sort and we prove the following:
Classification Theorem. A numerical Godeaux surface S with an involution σ
is birationally equivalent to one of the following:
(1) a double plane of Campedelli type;
(2) a double plane branched along a reduced curve which is the union of two
distinct linesr ,r and acurveofdegree 12with thefollowing singularities:
1 2
• the point q =r ∩r of multiplicity 4;
0 1 2
• a point q ∈r , i=1,2, of type [4,4], where the tangent line is r ;
i i i
• further three points q ,q ,q of multiplicity 4 and a point q of type
3 4 5 6
[3,3], such that there is no conic through q ,...,q ;
1 6
(3) a double cover of an Enriques surface branched along a curve of arithmetic
genus 2.
In case (3), the torsion group of S is Tors(S) = Z/4Z, whilst in case (2) Tors(S)
is either Z/2Z or Z/4Z.
Aswesaid,examplesofsurfacesoftype(1)areknownintheliterature. Surfaces
of type (3) will be called of Enriques type. Examples of such surfaces have been
produced by Keum and Naie (cf. [Ke, Na]). Double planes as in case (2) are, to
the best of our knowledge, new in the literature. They appear to be degenerations
of double planes with p = 4 and K2 = 8, introduced by Du Val in 1952 (see
g
[Du]),whenclassifyingsurfaceswithnon-birationalbicanonicalmap(cf.[Ci1]fora
modernreference). Forthis reason,wecallnumericalGodeauxsurfacesasin(2)of
Du Val type. We give explicit examples of these surfaces, both with torsion group
Z/2Z and Z/4Z, in section 9.
In order to prove our classification theorem, we proceed as follows. First of all,
in section 3, we prove some relevant properties of the fixed locus of an involution
acting on a surface of general type with geometric genus zero. In order to do so,
we follow ideas contained in joint work of the third author and Rita Pardini (e.g.
NUMERICAL GODEAUX SURFACES WITH AN INVOLUTION 3
[MP1]), namely we combine the topological and holomorphic fixed point formulas
with Kawamata-Viehweg’svanishing theorem.
Applying these results to numerical Godeaux surfaces S with an involution, in
section 4 we give a rather precise description of the fixed locus of the involution
and we prove that the bicanonical system is invariant under the involution. This
is actually the key ingredient for the proof of the classification and explains why
degenerationsofDuValdoubleplanes,whichhavenon-birationalbicanonicalmap,
come into play (cf. also recent results of Borrelli in [Bo] on the classification of
surfaces with non-birationalbicanonical map and low invariants).
Since the pencil |2K | is invariant, we can in fact consider its image on the
S
quotient surface under the involution, which is a pencil D of curves of arithmetic
genus2. Usingthis,oneseesthatthequotientsurfaceiseitherrationalorbirational
to an Enriques surface. The latter case, i.e. the Enriques type, is worked out
in section 5, where in particular we prove that numerical Godeaux surfaces of
this type have torsion group of order 4 and are birational to the double cover of
an Enriques surface branched along a curve of arithmetic genus 2, with at most
irrelevant singularities, which moves in a linear system with no fixed component.
In case the quotient of S by the involution is rational, one studies the pencil D
using adjunction. This leads to two different cases: one in which the adjoint to
this system is a base-point-free pencil of rational curves, the other in which the
adjoint is a pencil of curves of genus 1. These two cases are analysed separately in
sections6and7,respectively. TheformercaseleadstodoubleplanesofCampedelli
type: it suffices to suitably use the pencil of rational curves to map the quotient
surface to F , and then to the plane. The latter case leads to double planes of
1
Du Val type. Here the quotient surface is mapped to a weak Del Pezzo surface
X, i.e. −K is big and nef, having K2 = 1 and four disjoint (−2)-curves, whose
X X
sum is an even divisor in X. Rational surfaces with an even set of (−2)-curves
have been studied by Dolgachev, Mendes Lopes and Pardini in [DMP] and, more
recently, by us in [CCM]. Indeed we apply the main result in [CCM] to find a
suitablebirationalmorphismoftheweakDelPezzosurfaceX to P2,whichrealizes
the original numerical Godeaux surface as a Du Val double plane.
Thenwegivemoreinformationaboutthepreviouslyunknowncaseofnumerical
Godeaux surfaces of Du Val type. In particular, in section 8, we examine the
interplay between reducibility of the branch curve and torsion. Our main result in
this directionisTheorem8.5,which,undersomeassumptionsonthe branchcurve,
gives an useful criterion to decide whether the torsion is Z/4Z or Z/2Z, based on
the existence or not of plane curves of degree 8 with suitable singularities.
We remark that our results concern the birational classification of pairs (S,σ),
where S is a numericalGodeaux surface andσ aninvolutionofS. We do nottreat
here, in general, the interesting problem of determining how many involutions can
occur on a given numerical Godeaux surface and of which type according to our
classification theorem. However, our results do give some partial information. For
instance, if the ramification curve R on S has an irreducible component of genus
2, then we are in case (3) (see Proposition 7.10). In any case, using the 2-torsion,
we give a criterion, i.e. Corollary 8.9, based on the irreducibility of a certain plane
cubic,whichallowsustodistinguishbetweentheDuValandtheCampedellitypes.
As we said, in section 9, we prove the existence of numerical Godeaux double
planes of Du Val type and we are able to compute the torsion of these examples.
We do this using Maple in two different ways. One way is to find branch curves,
withirreducibledegree12component, whichareinvariantunder aprojectiveauto-
morphism of order 2 of the plane, which is an idea originally due to Stagnaro (cf.
[St]). The resulting examples turn out to have torsion group Z/4Z.
4 A.CALABRI,C.CILIBERTOANDM.MENDESLOPES
Another way is to try and find the degree 12 component of the branch curve
suitably reducible in a line and an irreducible component of degree 11. We find
examples of this type with torsion Z/2Z and with torsion Z/4Z and we prove that
the former example does not have a different involution which makes it a double
planeofCampedellitype,thereforeitiscertainlynewintheliterature. Althoughwe
donottreatheremoduliproblems,weprovethatourexamplesbothvaryinfamilies
whose images in the moduli space have dimension 5, and the general member of
each family is a Du Val double plane with an irreducible degree 12 component of
the branch curve (cf. Corollaries 9.6 and 9.7).
2. Notation and conventions.
In this section we fix the notation which will be used in this paper.
Let S be a complex projective surface. We set:
κ(S): the Kodaira dimension of S;
χ(F): the Euler characteristic of a sheaf F on S;
Pic(S): the Picard group of S;
Tors(S): the subgroup of Pic(S) composed of torsion elements;
Tors (S): the subgroup of Pic(S) composed of elements of torsion n;
n
ρ(S): the rank of the N´eron–Severigroup of S;
K : a canonical divisor of S;
S
p (S): the geometric genus of S, that is h0(S,O (K ));
g S S
q(S): the irregularity of S, that is h1(S,O );
S
P (S): the m-th pluri-genus of S, that is h0(S,O (mK )), m≥1;
m S S
If S is clear from the context, sometimes we will write p , q, K2, etc., instead of
g
p (S), q(S), K2, etc.
g S
LetX beacomplexprojectivevariety. Wedenotebye(X)thetopologicalEuler
characteristic of X and by p (X) the arithmetic genus of X. Recall that if D is a
a
curve on a surface S, then p (D)=D(D+K )/2+1.
a S
We denote by ≡ the linear equivalence of divisors on a surface and by ∼ the
numerical equivalence. We usually omit the sign · of the intersection product of
two divisors on a surface.
Recallthata(−1)-curve isasmoothirreduciblerationalcurveC withC2 =−1.
More generally, ones says that a smooth irreducible rational curve C with C2 =
−n<0 is a (−n)-curve.
We say that a divisor D on a surface is nef and big if D2 > 0 and DE ≥ 0 for
every irreducible curve E.
If x is a real number, we denote by [x] its integer part, i.e. the largest integer
number less than or equal to x.
A singular point of type [m,m] on a curve is a point of multiplicity m with an
infinitely near point again of multiplicity m.
3. Involutions on surfaces.
Let S be a smooth, irreducible, projective surface over the field C of complex
numbers. An involution of S is an automorphism σ of S of order 2. Remark that
if S is a minimal surface of general type, then any birational automorphism is an
isomorphism, therefore any birationalautomorphism of order 2 is an involution. If
X is any variety and ψ : S 99K X is a rational map, one says that ψ is composed
with the involution σ if ψ◦σ =ψ.
Given an involution σ on S, its fixed locus is the union of a smooth, possibly
reducible, curve R and of k isolated points p ,...,p .
1 k
NUMERICAL GODEAUX SURFACES WITH AN INVOLUTION 5
Letπ: S →Σ:=S/σ bethequotientmap. andsetB :=π(R). ThesurfaceΣis
normal and π(p ),...,π(p ) are ordinary double points, which are the only singu-
1 k
larities of Σ. In particular, the singularities of Σ are canonical and the adjunction
formula gives K ≡π∗K +R.
S Σ
Letǫ: V →S be the blowing-upofS atp ,...,p andletE be the exceptional
1 k i
curve over p , i=1,...,k. Then σ induces an involution σ˜ of V whose fixed locus
i
is the union of R := ǫ∗(R) and of E ,...,E . Denote by π˜: V → W := V/σ˜ the
0 1 k
projection onto the quotient and set B := π˜(R ), C := π˜(E ), i = 1,...,k. The
0 0 i i
surface W is smooth and the C are disjoint (−2)-curves. Denote by η: W → Σ
i
the map induced by ǫ. The map η is the minimal resolution of the singularities of
Σ and there is a commutative diagram:
ǫ
V S (1)
π˜ π
η
W Σ
Themapπ˜ isaflatdoublecoverbranchedonB˜ =B + k C ,hencethereexists
0 i=1 i
a divisor L on W such that 2L≡B˜ and P
π˜∗OV =OW ⊕OW(−L). (2)
Also K ≡π˜∗(K +L). With this notation:
V W
Proposition 3.1. Let S be a minimal surface of general type and let σ be an
involution of S. Then:
(i) 2K +B is nef and big;
W 0
(ii) (2K +B )2 =2K2;
W 0 S
(iii) Hi(W,O (2K +L))=0, i=1,2.
W W
Proof. By the adjunction formula and commutativity of diagram (1), we have
k
π˜∗(2K +B )≡2K −2 E ≡ǫ∗(2K ). (3)
W 0 V i S
Xi=1
Then 2K +B is nef and big because so is 2K , proving part (i). Statement (ii)
W 0 S
also follows by formula (3).
Finally for part (iii), we have the equivalence of Q−divisors:
k
1 1
K +L≡ (2K +B )+ C .
W W 0 i
2 2
Xi=1
The divisor (2K +B )/2=η∗(2K +B)/2 is nef and big, because so is 2K +
W 0 Σ W
B , whereas 1 k C is effective, with zero integral part, and its support has
0 2 i=1 i
normalcrossingPs. Thushi(W,OW(2KW+L))=0fori>0byKawamata–Viehweg
vanishing theorem (see, e.g., Corollary 5.12, c), of [EV]). (cid:3)
For surfaces of general type with p = 0 having an involution, one can be more
g
specific. As shownin[DMP], the holomorphicandtopologicalfixedpointformulas
(see p. 566 in [AS] and formula (30.9) in [Gr]) yield:
Lemma 3.2. Let S be a surface with p = q =0 and σ an involution of S. Then
g
the number of isolated fixed points of σ is k =K R+4. (cid:3)
S
Also the following properties, which will be very useful in the sequel, hold.
Proposition 3.3. Let S be a minimal surface of general type with p = 0 and let
g
σ be an involution of S. Then:
6 A.CALABRI,C.CILIBERTOANDM.MENDESLOPES
(i) k ≥4;
(ii) K L+L2 =−2;
W
(iii) h0(W,O (2K +L))=K2 +K L;
W W W W
(iv) K2 +K L≥0;
W W
(v) k =K2 +4−2h0(W,O (2K +L)).
S W W
Proof. Since S is minimal of general type, K is nef and so statement (i) follows
S
from Lemma 3.2.
Since S is of general type with p (S)=0, also q(S)=0. Therefore:
g
p (Σ)=p (W)=0, q(Σ)=q(W)=0. (4)
g g
Bystandarddoublecoverformulas,wehaveχ(O )=2χ(O )+(L2+K L)/2,
V W W
thus statement (ii) follows from p (W)=q(W)=0.
g
By Proposition 3.1, (iii), and the Riemann-Roch Theorem, one has:
h0(W,O (2K +L))=χ(O (2K +L))=(2K +L)(K +L)/2+1=
W W W W W W
=K (K +L)+L(K +L)/2+1.
W W W
Thus (iii) follows by statement (ii), and (iv) is a trivial consequence of (iii).
Finally, as for statement (v), it suffices to remember that k =K2 −K2, K2 =
S V V
2(K +L)2 and use statements (ii), (iii). (cid:3)
W
Corollary 3.4. Let S be a minimal surface of general type with p = 0 and let σ
g
be an involution of S. Then, with W as above,
k k
(i) |2K + B | =6 ∅, |2K + B + C | = |2K + B | + C and
W 0 W 0 i=1 i W 0 i=1 i
h0(W,OW(2KW +B0)=KS2 +1−Ph0(W,OW(2KW +L)); P
(ii) if D ∈|2K +B |, then D is nef and big, 1-connected and thus h0(D,O )=
W 0 D
1, h1(D,O )=p (D);
D a
(iii) p (D)>0.
a
Proof. The Hurwitz formula gives 2K ≡ π˜∗(2K +B + k C ). By the pro-
V W 0 i=1 i
jection formula (2), one has P
k
H0(V,O (2K ))=H0(W,O (2K +L))⊕H0(W,O (2K +B + C )). (5)
V V W W W W 0 i
Xi=1
By Proposition 3.3, (v) and (i), K2 −2h0(W,O (2K +L))=k−4≥0.
S W W
Sinceh0(V,O (2K ))=h0(S,O (2K ))=K2+1,weconcludethatthesecond
V V S S S
summand in (5) is not 0. For every i = 1,...,k, one has that C (2K +B +
i W 0
C )=−2, and so |2K +B + C |=|2K +B |+ C . This proves (i).
i i W 0 i i W 0 i i
PIn Proposition 3.1, (i), we alreadPy proved that D is nefPand big. Therefore D
is 1-connected, see e.g. Lemma 2.6 in [Me], hence h0(D,O ) = 1 and the final
D
assertion of (ii) follows by the Riemann–Roch Theorem.
Finally for assertion (iii), it suffices to note that, by Propositions 3.1 and 3.3,
K D ≥0 and D2 >0. (cid:3)
W
Corollary 3.5. Let S be a minimal surface of general type with p = 0 and σ an
g
involution of S. Then, with W as above, K2 ≥K2.
W V
Proof. By Noether’s formula, because χ(O ) = χ(O ) and e(W) ≤ e(V), since
W V
the map π˜ determines an injection of H2(W,C) into H2(V,C). (cid:3)
Corollary 3.6. Let S be a minimal surface of general type with p = 0, let ϕ :
g
S → PKS2 be the bicanonical map of S and let σ be an involution of S. Then the
following conditions are equivalent:
(i) ϕ is composed with σ;
(ii) h0(W,O (2K +L))=0;
W W
NUMERICAL GODEAUX SURFACES WITH AN INVOLUTION 7
(iii) K (K +L)=0;
W W
(iv) the number of isolated fixed points of σ is k =K2 +4.
S
Proof. The bicanonical ϕ is composed with σ if and only if one of the summands
in formula (5) above vanishes. By Corollary 3.4, (i), the second summand is never
0 and so ϕ is composed with σ if and only if h0(W,O (2K +L)) = 0, which
W W
in turn, by Proposition 3.3, (iii) and (v), is equivalent to K (K +L) = 0 and
W W
k =K2 +4. (cid:3)
S
Corollary 3.7. Let S be a minimal surface of general type with p = 0 and σ an
g
involution of S. If the bicanonical map ϕ is composed with σ, then:
(i) h0(W,O (2K +B ))=P (S)=1+K2;
W W 0 2 S
(ii) for D ∈|2K +B |, h0(D,O (D))=K2;
W 0 D S
(iii) K D =0, D2 =2K2 and p (D)=K2 +1;
W S a S
(iv) −4≤K2 ≤0, and K2 =0 if and only if K ∼0;
W W W
(v) either W is rational or the minimal model of W is an Enriques surface.
Proof. Part(i)is immediate fromCorollaries3.4,(i), and3.6, (ii). Part(ii) follows
from part (i) by considering the long exact sequence obtained from
0→O →O (D)→O (D)→0
W W D
and using that h1(W,O ) = 0. Now note that DK = 2(K + L)K , so
W W W W
K D =0followsfromCorollary3.6,(iii). NotealsothatD2 =2K2 byProposition
W S
3.1, (ii), therefore the adjunction formula implies that p (D) = K2 +1. Since
a S
k =K2 +4, one has that K2 =−4 and so the first inequality in (iv) follows from
S V
Corollary 3.5. The second inequality and the remainder of statement (iv) follow
from the Index Theorem because D is nef and big by Proposition 3.1, (i).
Finally, again because D is nef and big, K D = 0 implies that κ(W) ≤ 0. So
W
statement(v)followsbytheclassificationofsurfaces,sincep (W)=q(W)=0. (cid:3)
g
Thefollowinglemmawillbeusedlater. We keepthenotationintroducedabove.
Namely D is a general member of the non-empty linear system |2K +B |.
W 0
Lemma 3.8. Let S be a minimal surface of general type with p = 0 and an
g
involution σ and let W be as above. If E ⊂W is a curve such that ED = 0, then
E2 < 0 and the intersection form on the components of E is negative definite. In
particular if E is a curve such that E2 =−1 and ED =0, then E( k C )≤0.
i=1 i
P
Proof. The first part of the lemma is obvious by the Index Theorem, because D is
nef and big by Proposition 3.1, (i). For the second part, note that, since E(B +
0
k C ) is even and ED = 0, also E( k C ) is even. By the first part applied
i=1 i i=1 i
Pto the curve E+C , i=1,...,k, one haPs EC ≤1. Suppose that E( k C )>0.
i i i=1 i
Thentherewouldbeatleasttwocurves,sayC1 andC2,suchthatECP1 =EC2 =1,
thereforeA=2E+C +C wouldsatisfyA2 =0andAD =0,acontradiction. (cid:3)
1 2
Proposition 3.9. Let S be a minimal surface of general type with p = 0 and
g
an involution σ and let W be as above. Then there exists a birational morphism
f :W →W′ and an effective divisor D′ on W′ with the following properties:
(i) there are k (−2)-curves C′ on W′ such that f∗(C′)=C , i=1,...,k;
i i i
(ii) D′ is nef and such that f∗(D′)=D, D′2 =D2 and p (D′)=p (D);
a a
(iii) KW′ +D′ is nef.
Proof. If K +D is nef, there is nothing to prove. Otherwise, since K +D is
W W
effective (see Corollary3.4), there is an irreducible curve E such that E2 <0 and
1 1
E (K +D)<0. Then K E <0, thus E is a (−1)-curve and E D =0. Hence
1 W W 1 1 1
k
B E =2, and furthermore E ∩ C =∅ by Lemma 3.8.
0 1 1 i=1 i
S
8 A.CALABRI,C.CILIBERTOANDM.MENDESLOPES
Letf :W →W bethecontractionofE toapointp . OnW wehavek(−2)-
1 1 1 1 1
curvesC(1) suchthatf∗(C(1))=C ,i=1...k,andanefdivisorD onitsuchthat
i 1 i i 1
D2 = D2 and f∗(D ) = D. We set B(1) = f (B ). Note that B(1) has a double
1 1 1 0 1 0 0
point at p and it does not meet ∪k C(1). Note also that D ≡2K +B(1).
1 i=1 i 1 W1 0
If K +D is nef, we have finished. Otherwise, reasoning as above, there is a
W1 1
(−1)-curve E on W such that E (K +D ) < 0, D E = 0 and E B(1) = 2.
2 1 2 W1 1 1 2 2 0
The curve E′ = f∗(E ) satisfies E′D = 0, E′2 = −1 and K E′ = −1 and, since
1 2 W
k (1)
E is disjoint from the curve C , none of the curves C is a component of
1 i=1 i i
E′. Hence, by Lemma 3.8, E′P∩ k C(1) = ∅, implying that also E is disjoint
i=1 i 2
from the curves C(1). So we can cPontract E and proceed.
i 2
Finally, the existence of the morphism f : W → W′ is shown by iterating the
above procedure. (cid:3)
Remark 3.10. We notice that, by the proof of Proposition 3.9, the divisor D′+
C′ +···+C′ is divisible by 2 in Pic(W′).
1 k
Later, we will need the following lemma, due to Beauville (Lemme 2 in [Be]):
Lemma3.11. Letπ˜ :V →W beaflatdoublecoverbetweentwosmoothsurfacesV
andW,branchedoverthesmoothcurveB˜ =B +···+B ,whereB ,...,B arethe
1 n 1 n
irreduciblecomponentsofB˜. SupposethatPic(W)hasno2-torsionelement. Define
a group homomorphism ψ : Z⊕n → Pic(W)⊗Z by ψ(ε ,...,ε ) = n ε B .
2 2 1 n i=1 i i
Then Tors2(V)∼=ker(ψ)/h(1,...,1)i. P (cid:3)
4. Numerical Godeaux surfaces with an involution.
In the remainder of this paper, we focus on the study of a numerical Godeaux
surface S, i.e. a minimal surface of general type with p = 0 and K2 = 1, having
g S
an involution σ. We will freely use the notation introduced so far.
We start by recalling the following:
Theorem 4.1 (Miyaoka, [Mi]). Let S be a numerical Godeaux surface. Then the
torsion group Tors(S) of S is cyclic of order n ≤ 5. Moreover the linear system
|3K | has no fixed part and has at most 2 base points. If there is no base point,
S
then |Tors(S)|≤2. If there is one base point, then 3≤ |Tors(S)|≤ 4. If there are
two base points, then |Tors(S)|=5. (cid:3)
Lemma 4.2 (Lemma 5 in [Mi]). Let S be a numerical Godeaux surface. If D is
an effective divisor with h0(S,D)≥2, then DK ≥2. (cid:3)
S
Lemma 4.3 (Reid,seep.158in[Do]). Let S be a numerical Godeaux surface. If η
is a non-trivial element of Tors(S), then there is a unique element P in |K +η|.
η S
Furthermore, if η and η′, η 6=η′, are non-trivial elements of Tors(S), then P and
η
Pη′ have no common components. (cid:3)
As an immediate consequence of Lemma 4.3, we have the following:
Corollary 4.4. Let S be a numerical Godeaux surface. If |Tors(S)|=4 or 5, then
|2K | has no fixed components. (cid:3)
S
Our results in §3 imply that S enjoys the following properties (cf. also [KL]):
Proposition 4.5. If S is a numerical Godeaux surface with an involution σ, then:
(i) the number of isolated fixed points of σ is k =5;
(ii) the bicanonical map ϕ is composed with σ;
(iii) K R=1;
S
(iv) R2 is odd and −7≤R2 ≤1.
NUMERICAL GODEAUX SURFACES WITH AN INVOLUTION 9
Furthermore R=Γ+Z +···+Z where:
1 h
(v) Γ is a smooth curve with K Γ=1, 0≤p (Γ)≤2 and Γ2 =2p (Γ)−3;
S a a
(vi) if p (Γ)=2, then Γ∼K and S has non-trivial torsion;
a S
(vii) Z ,...,Z are disjoint (−2)-curves, which are disjoint also from Γ, and
1 h
h=p (Γ)−K2 −2≥0. (6)
a W
Proof. By Proposition 3.3, (i) and (v), the number k is odd and 4≤k ≤5. Hence
k = 5, that is part (i). Then, part (ii) follows by Corollary 3.6, whilst part (iii)
follows from (i) and Lemma 3.2.
Let us prove part (iv). Since 2R2 = B2, one has 4L2 = 2R2 −10 and thus
L2 =(R2−5)/2. ByProposition3.3,(ii)andCorollary3.6,(iii),L2 =−2−K L=
W
−2+K2 , hence R2 =1+2K2 . Then by Corollary 3.7, (iv), −7≤R2 ≤1.
W W
Part(iii)implies thatR hasa uniqueirreduciblecomponentΓ suchthatK Γ=
S
1. Since R is smooth, so is Γ. We can write R = Γ+Z, where Z is effective.
Then K Z = 0, thus the irreducible components of Z are (−2)-curves, which are
S
pairwise disjoint and disjoint from Γ because R is smooth.
By the Index Theorem, Γ2 ≤ 1. Since K Γ = 1 by adjunction, one has Γ2 =
S
2p (Γ)−3, which ends the proof of statement (v).
a
IfΓ2 =1,thenΓishomologoustoK ,butnotlinearlyequivalenttoK ,because
S S
p =0. ThereforeΓ−K isanon-trivialtorsionelementoftheN´eron-Severigroup
g S
of S, which shows (vi).
Finally (vii) follows from R2 =1+2K2 =Γ2−2h and Γ2 =2p (Γ)−3. (cid:3)
W a
Remark 4.6. ThestatementsaboutRonS inProposition4.5canbe readaswell
as about the curve B on the surface W, cf. diagram (1) and Corollary 3.7. For
0
instance,π˜(ǫ−1(Z )) is a smoothrationalcurveonW with self-intersection−4and
i
is an irreducible component of B .
0
Now Beauville’s Lemma 3.11 and Theorem 4.1 imply the following:
Corollary 4.7. Let h be the number of (−2)-curves of R as in Proposition 4.5. If
W is a rational surface, then
−K2
h≤1+ W (7)
(cid:20) 2 (cid:21)
and if equality holds, then S has non-trivial 2-torsion.
Proof. Consider the map ψ of Lemma 3.11. The domain of ψ is Z⊕(h+6), because
2
the branch locus of π˜ : V → W has h+6 irreducible components. Moreover the
image of ψ is a totally isotropic subspace of Pic(W)⊗Z =H2(W,Z ).
2 2
Since W is simply connected, h2(W,Z ) = h2(W,Z). Noether’s formula im-
2
plies that b (W) = 10 − K2 , hence a totally isotropic subspace of H2(W,Z )
2 W 2
has dimension at most 5+ [−K2 /2]. By Theorem 4.1, dimker(ψ) ≤ 2, there-
W
fore h+6 ≤ 2+5+[−K2 /2], which proves (7). If equality holds in (7), then
W
dimker(ψ)=2, so Tors (V) and Tors (S) are not trivial by Lemma 3.11. (cid:3)
2 2
InordertoprovetheClassificationTheorem,statedintheintroduction,weneed
tounderstandthesurfaceW andthedivisorD ≡2K +B onW. ByProposition
W 0
4.5andCorollaries3.4and3.7,onehasthatDisnef,D2 =2,K D =0,p (D)=2,
W a
and h0(W,O (D))=2.
W
If R has a component Γ with p (Γ)=2, we have further information on W,D:
a
Corollary 4.8. Let S be a numerical Godeaux surface with an involution σ. Sup-
pose that R has an irreducible component Γ of genus 2. Then either one of the
following two cases occurs:
(i) W is a minimal Enriques surface, R=Γ and |D|=|B |;
0
10 A.CALABRI,C.CILIBERTOANDM.MENDESLOPES
(ii) W is a rational surface, −2≤K2 ≤−1 and Γ−K ∈Tors (S).
W S 2
Proof. On W, let Γ =η∗(π(Γ))≤B , thus Γ2 =2 and K Γ =0. Write:
0 0 0 W 0
D ≡2K +B ≡2K +Γ +(B −Γ ). (8)
W 0 W 0 0 0
Now Γ D = Γ2 = D2, so, by the Index Theorem, Γ ∼ D. Hence (8) implies that
0 0 0
2K +B −Γ ∼ 0. By Corollary 3.7, (v), either the minimal model of W is an
W 0 0
Enriques surface or W is rational. In the former case 2K is an effective divisor.
W
SinceB −Γ isalsoeffective,2K +B −Γ ∼0impliesthat0≡2K ≡B −Γ ,
0 0 W 0 0 W 0 0
that is case (i) of the statement.
If W is rational, numerical equivalence is the same as linear equivalence and so
Γ ≡DandB −Γ ≡−2K ,therefore2Γ≡2K . Byformula(6),B has1−K2
0 0 0 W S 0 W
irreducible components, i.e. h=−K2 , thus formula (7) implies K2 ≥−2. (cid:3)
W W
Remark 4.9. We will see later, in Proposition7.10, that case (ii) in Corollary4.8
does not actually occur.
We now study W and the pencil |D| on W by using adjunction:
Lemma 4.10. With the above notation, write |K + D| = F +|M|, where F
W
denotes the fixed part and |M| the movable part. Then:
(i) h0(W,O (M))=2;
W
(ii) the general curve of the pencil |M| is irreducible;
(iii) MD =2;
(iv) if F 6=0, then every component E of F is such that DE =0 and E2 <0.
Proof. Assertion(i)followssincep (D)=2andW isregular. Assertion(ii)follows
a
by Bertini’s Theorem, since every pencil on the regular surface W is rational.
Let us prove part (iii). Since D is nef and D(M +F) = 2, one has MD ≤ 2.
Suppose by contradiction that MD ≤1. Consider the pull–back |M˜| of |M| to V,
whichisalsothepull–backofapencil|N|onS. Sinceπ˜∗(D)=ǫ∗(2K ),onewould
S
have that NK ≤1, which is impossible by Lemma 4.2. This proves (iii).
S
Assertion (iv) follows now by the Index Theorem. (cid:3)
We have already seen that −4≤K2 ≤0. Now we want to consider the surface
W
W′ as in Proposition 3.9.
Lemma 4.11. One has −2 ≤ K2 ≤ 0. Furthermore K2 = 0 if and only if W′
W′ W′
is a minimal Enriques surface.
Proof. Since D′(KW′+D′)=2,the Index Theoremimplies that(KW′+D′)2 ≤2,
or equivalently KW2 ′ ≤ 0, because KW′D′ = 0 and D′2 = 2. On the other hand,
(KW′ +D′)2 ≥0 and therefore KW2 ′ ≥−2.
If KW2 ′ = 0, then KW′ ∼ 0, and therefore W′ is an Enriques surface. The
converse is trivial. (cid:3)
Lemma 4.12. If KW2 ′ <0, then |KW′ +D′|=|M′| has no fixed part.
Proof. Write,asusual,|KW′+D′|=F′+|M′|,whereF′ isthefixedpartand|M′|
is the movable part. By Lemma 4.10, (iii), and the construction of the morphism
f :W →W′, which contracts only curves in F, we see that M′D′ =2.
NoticethatF′KW′ =F′M′+F′2,soM′F′iseven. SinceM′F′ ≤M′F′+M′2 ≤
(KW′ + D′)2 = KW2 ′ + 2 < 2, it follows that M′F′ = 0. This forces F′ to be
0. Otherwise, since D′F′ = 0, then F′2 < 0, implying that F′(KW′ + D′) =
F′2+M′F′ <0, a contradiction because (KW′ +D′) is nef. (cid:3)