aa r X i v : . [ m a t h . AG ] F e b Z -actions on Horikawa surfaces. Vicente Lorenzo
Abstract
Minimal algebraic surfaces of general type X such that K X = 2 χ ( O X ) − K X = 2 χ ( O X ) − Z -actions on Horikawa surfaces. The main result is that all theconnected components of Gieseker’s moduli space of canonical models ofsurfaces of general type with invariants satisfying these relations containsurfaces with Z -actions. Let X be a minimal surface of general type over the complex numbers, whichwill be the ground field throughout the paper. If we denote by K X the selfintersection of its canonical class and by χ ( O X ) its holomorphic Euler characteristic,it is well known (cf. [2]) that the following inequalities are satisfied χ ( O X ) ≥ , K X ≥ , χ ( O X ) − ≤ K X ≤ χ ( O X ) . (1)A pair of integers ( χ, K ) is said to be an admissible pair if it satisfies theinequalities (1). The geographical question, i.e. whether there exists a minimalalgebraic surface of general type X with ( χ ( O X ) , K X ) = ( χ, K ) for any admissiblepair ( χ, K ), has received much attention [2, Section VII.8]. Several authorshave also studied the geography of surfaces with special features like genus 2fibrations (see [12]), simply-connectedness (see [13]), 2-divisibility of the canonicalclass (see [14]), global 1-forms (see [10]), etc. In this note we are interested inthe geography of surfaces with Z -actions.Let us denote by M χ,K Gieseker’s moduli space of canonical models ofsurfaces of general type with fixed self intersection of the canonical class K and fixed holomorphic Euler characteristic χ . Then, for any admissible pair( χ, K ), the moduli space M χ,K has a finite number of connected componentswhich parametrize the deformation equivalence classes of canonical models ofsurfaces of general type (cf. [6]). The main result is the following: Theorem 1.
Let ( χ, K ) be a pair of integers such that K = 2 χ − and χ ≥ or K = 2 χ − and χ ≥ . Then it is possible to find a surface with a Z -actionin any of the connected components of M χ,K . Mathematics Subject Classification (2010):
MSC 14J29
Keywords:
Surfaces of general type · Z -covers · Moduli spaces · Genus 2 fibrations emark . It follows from (1) that if we want ( χ, K ) to be the invariants of aminimal surface of general type and K = 2 χ − K = 2 χ − χ ≥ χ ≥
3) is not a restriction.The note is structured as follows. Section 2 is devoted to present someproperties of minimal surfaces of general type X with K X = 2 χ ( O X ) − X with K X = 2 χ ( O X ) −
5. In Section 4how to construct Z -covers and to obtain information about them is explained.Sections 5 and 6 include the examples that prove Theorem 1. Finally, Section7 gives an insight into these examples. K = 2 χ − . If X is an algebraic surface then χ ( O X ) = 1 − q ( X ) + p g ( X ) where p g ( X ) = h ( K X ) is the geometric genus of X and q ( X ) = h (Ω X ) is its irregularity. If X is minimal and of general type we know by [10, Proposition 2.3.2] that q ( X ) > K X ≥ χ ( O X ). On the other hand, if K X = 2 p g ( X ) − q ( X ) = 0by [3, Theorem 10].In [7] Horikawa studied minimal surfaces of general type X such that K X =2 p g ( X ) −
4. We know by the previous comments that this is equivalent to K X = 2 χ ( O X ) −
6. The following theorems are some of the results Horikawaproved in [7].
Theorem 2 ([7, Lemma 1.1 and Lemma 1.2]) . Let X be a minimal algebraicsurface with K X = 2 χ ( O X ) − and χ ( O X ) ≥ . Then the canonical system | K X | has no base point. Moreover, the canonical map ϕ K X : X → P p g ( X ) − is a morphism of degree 2 onto a surface that is isomorphic to either P , aHirzebruch surface F e or a cone over a rational curve of degree e in P e .Remark . It is worth noticing that P can only be obtained if χ ∈ { , } and acone over a rational curve of degree e in P e can only be obtained if χ ∈ { , , } . Remark . Suppose that the canonical image of a minimal surface of generaltype X with K X = 2 χ ( O X ) − F e . We denote by ∆ the negative section of F e and by F the fiber with respect to the projection F e → P . Then we can realize X as a double cover X → F e branched along adivisor of type 6∆ + mF for some positive integer m . By the Riemann-Hurwitzformula a general fiber F pulls back to a curve of genus 2 in X . It follows thatthe composition X → F e → P is a genus 2 fibration.Suppose now that the canonical image of X is a cone Σ over a rational curveof degree e in P e . Then we can decompose the canonical map X → Σ ⊂ P e +1 of X as X → F e → Σ ⊂ P e +1 where X → F e is a double cover and F e → Σ ⊂ P e +1 contracts the negative section of F e and is an isomorphism elsewhere. Moreoverthe branch locus of X → F e is again of type 6∆ + mF so we get a genus 2fibration on X also in this case.We can summarize Remark 3 as follows.2 roposition 1 ([7, Corollary 1.7]) . Let X be a minimal surface of general typewith K X = 2 χ ( O X ) − . If the canonical image of X is not P then it admits agenus fibration X → P .Remark . Let X be a smooth minimal surface with χ ( O X ) = 4 and K X =2. Then the canonical image of X is P and we cannot apply Proposition1. Nevertheless, in Section 5 we are going to construct a surface with theseinvariants and a genus 2 fibration. Now, if X has a genus 2 fibration then K X cannot be ample. Indeed, let us denote by F a general fiber of this genus 2fibration. The exact sequence0 → O X ( K X − F ) → O X ( K X ) → O F ( K F ) → → H ( X, K X − F ) → H ( X, K X ) → H ( F, K F ) → · · · Since h ( X, K X ) = 3 and h ( F, K F ) = 2 we conclude that H ( X, K X − F ) = 0.Let us consider D ∈ | K X − F | . Then K X D = 0 and D = −
2. It follows that D contains at least one ( − Theorem 3 ([7, Theorem 3.3, Theorem 4.1 and Theorem 7.1]) . Let ( χ, K ) be a pair of integers such that K = 2 χ − and χ ≥ . If K / ∈ · Z thenminimal algebraic surfaces X such that ( χ ( O X ) , K X ) = ( χ, K ) have one andthe same deformation type. If K ∈ · Z then minimal algebraic surfaces X suchthat ( χ ( O X ) , K X ) = ( χ, K ) have two deformation classes. The image of thecanonical map of a surface in the first class is F e for some e ∈ { , , . . . , K } .The image of the canonical map of a surface in the second class is F K +2 if K > and P or a cone over a rational curve of degree in P if K = 8 . K = 2 χ − . In [8] Horikawa studied minimal surfaces of general type X such that K X =2 p g ( X ) −
3. Arguing like we did in Section 2, it follows from [10, Proposition2.3.2] and [3, Theorem 10] that this is equivalent to K X = 2 χ ( O X ) −
5. Thefollowing theorems are some of the results Horikawa proved in [8].
Theorem 4 ([8, Section 1]) . Let X be a minimal algebraic surface with K X =2 χ ( O X ) − and χ ( O X ) ≥ . Then the canonical system | K X | has a unique basepoint b and the canonical map ϕ K X : X P p g ( X ) − is a rational map whoseimage is isomorphic to either a Hirzebruch surface F e or a cone over a rationalcurve of degree e in P e . Moreover, let us denote by ε : e X → X the blow-up of X at b . Then e X is the minimal resolution of singularities of the normal doublecover X ′ of one of the following surfaces.a) A blow-up of the Hirzebruch surface F e at two points of a fiber that maybe infinitely near. ) A blow-up of the Hirzebruch surface F at a point on the negative section.c) The Hirzebruch surface F .Remark . It is worth noticing that assuming χ ≥ χ, K ) = (6 , b ) of Theorem 4 can only happen when ( χ, K ) = (6 ,
7) and case c ) of Theorem 4 can only happen when ( χ, K ) = (7 , Proposition 2 ([8, Corollary 1.4]) . Let X be a minimal surface of general typewith K X = 2 χ ( O X ) − . If X ′ is a double cover of the blow-up of F e at twopoints of a fiber, then X admits a genus fibration X → P . Theorem 5 ([8, Introduction]) . Let ( χ, K ) be a pair of integers such that K = 2 χ − and χ ≥ . If K + 1 / ∈ · Z and ( χ, K ) = (7 , then minimalalgebraic surfaces X such that ( χ ( O X ) , K X ) = ( χ, K ) have one and the samedeformation type. If K + 1 ∈ · Z or ( χ, K ) = (7 , then minimal algebraicsurfaces X such that ( χ ( O X ) , K X ) = ( χ, K ) have two deformation classes.Remark . Let ( χ, K ) be a pair of integers such that K = 2 χ − χ ≥ K +1 ∈ · Z or ( χ, K ) = (7 , M χ,K , that we denote by M Iχ,K and M IIχ,K , will be described below.In the case K = 2 χ − K = 2 χ − X is not enough todetermine its deformation class. When ( χ, K ) = (6 ,
7) or ( χ, K ) = (7 ,
9) wemay need to know X ′ too. Theorem 6 ([8, Theorem 8.1]) . Let ( χ, K ) be a pair of integers such that K = 2 χ − , χ ≥ , K + 1 ∈ · Z and K = 7 . The image of the canonicalmap of a smooth minimal surface in M Iχ,K is F e for some e ∈ { , , . . . , ( K +1) − } . The image of the canonical map of a smooth minimal surface in M IIχ,K is F ( K +1)+1 . Theorem 7 ([8, Theorem 6.1]) . Let ( χ, K ) = (6 , . Smooth minimal surfaces X in M Iχ,K have canonical image F and X ′ is a double cover of a blow-up of F at two points of a fiber. Smooth minimal surfaces X in M IIχ,K either havecanonical image a cone over a rational curve of degree in P or they havecanonical image F and X ′ is a double cover of a blow-up of F at a point onthe negative section. Theorem 8 ([8, Theorem 7.1]) . Let ( χ, K ) = (7 , . Smooth minimal surfaces X in M Iχ,K have canonical image F e for some e ∈ { , } and X ′ is a doublecover of a blow-up of F e at two points of a fiber. Smooth minimal surfaces X in M IIχ,K have canonical image F and X ′ is a double cover of F . Z -covers. A Z -cover of a variety Y is a finite map f : X → Y together with a faithfulaction of Z on X such that f exhibits Y as X/ Z . The structure theorem forsmooth Z -covers was first given by Catanese [4]. According to [4, Section 2] or[11, Theorem 2.1], to define a smooth Z -cover of a smooth variety Y it sufficesto consider both:- Smooth divisors D , D , D such that the branch locus B = D + D + D is a normal crossing divisor.- Line bundles L , L , L satisfying 2 L ≡ D + D , L ≡ D + D andsuch that L ≡ L + L − D .The set { L i , D j } i,j is called the building data of the cover. Remark . We can also consider Z -covers of a variety Y with singularities. If Y is normal similar building data is required to define the cover, but there aresome differences (see [1]):- The divisors D j are not necessarily Cartier divisors but Weil divisors.- The sheaves L i are not necessarily invertible sheaves but reflexive divisorialsheaves.In addition, the branch locus may have non-divisorial components. Proposition 3 ([4, Section 2] or [11, Proposition 4.2]) . Let Y be a smoothsurface and f : X → Y a smooth Z -cover with building data { L i , D j } i,j . Then: K X ≡ f ∗ (2 K Y + D + D + D ) ,K X = (2 K Y + D + D + D ) ,p g ( X ) = p g ( Y ) + X i =1 h ( K Y + L i ) ,χ ( O X ) = 4 χ ( O Y ) + 12 X i =1 L i ( L i + K Y ) . Remark . Let us consider one of the line bundles defining a Z -cover f : X → Y ,for instance L . Then we can decompose f as f ◦ f where f : X → Y is adouble cover of Y branched along D + D and f : X → X is a double cover of X branched along f ∗ D ∪ f ∗ ( D ∩ D ). In particular K X = f ∗ ( K X + f ∗ L ).Let us denote N = h ( K X + f ∗ L ) and i : X → P N − the morphism definedby the complete linear system | K X + f ∗ L | . Then ( i ◦ f ) ∗ ( O P N − (1)) = K X .Hence i ◦ f is a morphism induced by some sections of K X . It follows that if h ( K X ) = h ( K X + f ∗ L ) then i ◦ f is the canonical map of X and i ( X ) isits canonical image. 5 Constructions when K = 2 χ − . In this section we are going to prove Theorem 1 in the case K = 2 χ − χ, K ) such that K = 2 χ − χ ≥ χ is even then K / ∈ · Z and M χ,K has a unique connected component byTheorem 3. Then we just need to find a surface in M χ,K with a Z -action. If χ is odd it may happen that K ∈ · Z and in this case M χ,K has two connectedcomponents again by Theorem 3. What we are going to do is to find a surfacein M χ,K and by studying its canonical image we are going to check that it doesnot belong to the second deformation class when K ∈ · Z . Finally, we aregoing to find surfaces in the second deformation class for any K ∈ · Z .Let us assume first that χ is even. We denote by F the Hirzebruch surfacewith negative section ∆ of self-intersection ( −
2) and fiber F . The smooth Z -cover π : S → F of F whose branch locus B = D + D + D is a normalcrossing divisor consisting of smooth and irreducible divisors D ∈ | ∆ | D ∈ | ∆ + 2 F | D ∈ | + ( χ + 4) F | satisfies χ ( O S ) = χ and K S = K . The canonical divisor of S is big and nefbecause 2 K S is the pullback via π of the big and nef divisor ∆ +( χ − F . Hence S is minimal and its canonical model, that has a Z -action, belongs to M χ,K .We notice that when χ > K S is ample and S itself is a canonicalmodel. When χ = 4 it follows from Remark 4 that K S is not ample because S has a genus 2 fibration. The canonical model of S , obtained by contracting the( − π ∗ ∆ ) red , is a Z -cover of the quadric cone.Let us assume now that χ is odd, which happens always if K ∈ · Z . Wedenote by ∆ and F the two classes of fibers of P × P . The smooth Z -cover π : S → P × P of P × P whose branch locus B = D + D + D is a normalcrossing divisor consisting of smooth and irreducible divisors D ∈ | ∆ | D ∈ | ∆ | D ∈ | + ( χ + 1) F | satisfies χ ( O S ) = χ and K S = K . The canonical divisor of S is ample because2 K S is the pullback via π of the ample divisor ∆ + ( χ − F . Hence S ∈ M χ,K and it has a Z -action. According to Remark 8 we can decompose π as π = π ◦ π where π : P × P → P × P is a double cover of P × P branched along D + D and π : S → P × P is a double cover of P × P branched along π ∗ D ∈| + ( χ + 1) F | . Then K S is the pullback via π of the very ample divisor K P × P + π ∗ D ∈ | ∆ + ( χ − ) F | . Since h ( K S ) = h ( P × P , ∆ + ( χ − ) F )the canonical image of S is P × P . When K ∈ · Z the surface S belongs tothe first deformation class by Theorem 3.6inally, we are going to consider a smooth Z -cover π : S → F e of theHirzebruch surface F e with negative section ∆ of self-intersection ( − e ) ≤ − F . In this case, the branch locus B = D + D + D is a normalcrossing divisor consisting of D ∈ | F | D = 0 D ∈ | F | if e / ∈ · Z D ∈ | F | if e ∈ · Z D ∈ | + 5 eF | D ∈ | + 5 eF | where D is the union of the negative section ∆ and a smooth and irreducibledivisor D ′ ∈ | + 5 eF | . Then χ ( O S ) = 4 e − K S = 8 e −
8. In particular K S = 2 χ ( O S ) − K S ∈ · Z . According to Remark 8 we can decompose π as π = π ◦ π where π : F e → F e is the double cover of F e branchedalong D + D and π : S → F e is the double cover of the Hirzebruch surface F e with negative section Γ of self-intersection ( − e ) and fiber G branchedalong π ∗ D ∈ | + 10 eG | . Then K S is the pullback via π of the divisor K F e + π ∗ D ∈ | Γ + (3 e − G | . Since h ( K S ) = h ( F e , Γ + (3 e − G ) thecanonical image of S is the image of F e via the map ϕ defined by the completelinear system | Γ + (3 e − G | . If e > ϕ is an embedding. If e = 2 then ϕ embeds F e as a surface of minimal degree.In any case the surface S belongs to the second deformation class by Theorem3. K = 2 χ − . In this section we are going to prove Theorem 1 in the case K = 2 χ − χ, K ) = (7 ,
9) such that K = 2 χ − χ ≥
3. If χ is odd then K + 1 / ∈ · Z and M χ,K has a unique connectedcomponent by Theorem 5. Then we just need to find a surface in M χ,K with a Z -action. If χ is even it may happen that K + 1 ∈ · Z and in this case M χ,K has two connected components again by Theorem 5. What we are going to dois to find a surface in M χ,K and by studying its canonical image we are goingto check that it does not belong to M IIχ,K when K + 1 ∈ · Z . Finally, we aregoing to find surfaces in M IIχ,K when K + 1 ∈ · Z . The case ( χ, K ) = (7 , χ is odd and bigger than 3. We denote by F theHirzebruch surface with negative section ∆ of self-intersection ( −
3) and fiber F . Then the smooth Z -cover π : S → F of F whose branch locus B = D + D + D is a normal crossing divisor consisting of smooth and irreducibledivisors D ∈ | ∆ | D ∈ | ∆ + 4 F | D ∈ | + ( χ + 5) F | χ ( O S ) = χ and K S = K . The canonical divisor of S is ample because2 K S is the pullback via π of the ample divisor ∆ + ( χ − F . Therefore, S ∈ M χ,K and it has a Z -action. Let us suppose that χ = 3. Then wecannot assume D to be smooth in the previous example. Nevertheless a smooth Z -cover π : S → P of P whose branch locus B = D + D + D is a normalcrossing divisor consisting of smooth and irreducible divisors D ∈ |O P (1) | D ∈ |O P (1) | D ∈ |O P (5) | satisfies ( χ ( O S ) , K S ) = (3 , S is ample because 2 K S is the pullback via π of the ample sheaf O P (1). Hence S ∈ M , and it has a Z -action.Let us assume now that χ is even and greater or equal than 4, which happensalways if K + 1 ∈ · Z . We denote by F the Hirzebruch surface with negativesection ∆ of self-intersection ( −
1) and fiber F . Then the smooth Z -cover π : S → F of F whose branch locus B = D + D + D is a normal crossingdivisor consisting of smooth and irreducible divisors D ∈ | ∆ | D ∈ | ∆ + 2 F | D ∈ | + ( χ + 2) F | satisfies χ ( O S ) = χ and K S = K . The canonical divisor of S is ample because2 K S is the pullback via π of the ample divisor ∆ + ( χ − F . Hence S ∈ M χ,K and it has a Z -action. Let us restrict now to the case K + 1 ∈ · Z . If wedenote by q : e F → F the blow-up of F at the point of intersection of D and D with exceptional divisor E ′ , the cover π induces a Z -cover e π : e S → e F withbranch locus e B = e D + e D + e D where e D = q ∗ D − E ′ e D = q ∗ D − E ′ e D = q ∗ D + E ′ If we denote by π : e F → e F the intermediate Z -cover of e π with branch locus e D + e D (see Remark 8), we have that e F is a blow-up b : e F → F of F at twopoints of a fiber. In addition the induced map ε : e S → S is the blow-up of S at the point over D ∩ D with exceptional divisor E . We obtain the followingcommutative diagram: S π (cid:15) (cid:15) e S ε o o π (cid:15) (cid:15) f / / F e F π (cid:15) (cid:15) b ; ; ✈✈✈✈✈✈ F e F q o o π is the Z -cover of e F branched along π ∗ e D and f := b ◦ π . I claimthat we are in case a ) of Theorem 4 and the canonical image of S is F . Indeed,let us denote by Σ (resp. G ) the negative section (resp. the class of a fiber)of the latter Hirzebruch surface F . Taking into account the standard formulasfor Z -covers and the fact that K e S = ε ∗ K S + E , a straightforward calculationgives: ε ∗ K S = f ∗ (cid:18) Σ + χ − G (cid:19) + E. (2)Now, it follows from the equality h ( K S ) = h (Σ + χ − G ) together with (2)that E is the fixed part of the linear system | ε ∗ K S | and | f ∗ (cid:0) Σ + χ − G (cid:1) | isits moving part. We conclude that the canonical image of S coincides with theimage of the map induced by the complete linear system | f ∗ (Σ + χ − G ) | . SinceΣ + χ − G is very ample (recall the assumption K + 1 ∈ · Z that implies χ ≥
6) we conclude that the canonical image of S is F as claimed. Therefore S ∈ M Iχ,K by Theorem 6.Now we are going to consider a Z -cover π : T → F k +1 of the Hirzebruchsurface F k +1 with negative section ∆ of self-intersection − ( k + 1) ≤ − F . In this case, the branch locus B = D + D + D consists of D ∈ | F | D = 0 D ∈ | F | if k ∈ · Z D ∈ | F | if k / ∈ · Z D ∈ | + 5( k + 1) F | D ∈ | + 5( k + 1) F | The divisor D is the union of the negative section ∆ and an irreducible divisor D ′ ∈ | + 5( k + 1) F | with an ordinary triple point p . The divisor D passesthrough p intersecting D with multiplicity 3. Then χ ( O T ) = 4 k + 3 and K T = 8 k . Nevertheless T has an elliptic singularity over p such that its minimalresolution r : S → T satisfies χ ( O S ) = 4 k + 2 and K S = 8 k − K S = 2 χ ( O S ) − K S + 1 ∈ · Z . If we denote q : e F k +1 → F k +1 the blow-up of F k +1 at p with exceptional divisor E ′ , the cover π induces another Z -cover e π : S → e F k +1 . The canonical divisor of S is amplebecause 2 K S is the pull-back via e π of q ∗ (2∆ + (3 k + 1) F ) − E ′ =: D and thisdivisor is ample by the Nakai-Moishezon criterion. Indeed, let us suppose thatthere exists an irreducible curve C ∈ | q ∗ ( a ∆ + bF ) − cE ′ | such that CD < a, b, c . This implies that q ( C ) ∈ | a ∆ + bF | isan irreducible curve of F k +1 with a point of multiplicity c > ( k − a + 2 b , whichis clearly impossible. Now, π induces one more Z -cover ee π : e S → ee F k +1 on theblow-up ee F k +1 → e F k +1 of e F k +1 at the point of intersection p ′ of E ′ with thestrict transform via q of the fiber of F k +1 through p . It turns out that one ofthe intermediate surfaces of the cover ee π is a blow-up e F k +2 → F k +1 of F k +1 attwo points of a fiber, one of them belonging to the negative section and e S → S is the blow-up of S at the point over p ′ . It can be proved as above that weare in case a ) of Theorem 4 and the canonical image of S is F k +1 . Therefore S ∈ M IIχ,K by Theorem 6 and it has a Z -action.9hen k = 1, i.e. when ( χ, K ) = (6 , S is not ample because ( e π ∗ q ∗ ∆ ) red is a( − Z , obtained by contracting this ( − Z -cover of a blow-up of the quadric cone. Arguing as before, and taking intoaccount Remark 7, it can be proved that the canonical divisor of Z is ampleand its canonical image is a cone over a rational curve of degree 3 in P . Hence Z ∈ M II , by Theorem 7 and it has a Z -action.Let us study now the case ( χ, K ) = (7 , M Iχ,K it suffices to consider the example that we constructed in thegeneral case with χ odd. Indeed, the smooth Z -cover π : S → F of F whosebranch locus B = D + D + D is a normal crossing divisor consisting of smoothand irreducible divisors D ∈ | ∆ | D ∈ | ∆ + 4 F | D ∈ | + 12 F | satisfies ( χ ( O S ) , K S ) = (7 , S is ample because 2 K S is the pullback via π of the ample divisor ∆ + 6 F . The cover π induces another Z -cover e S → e F where e F → F is the blow-up of F at the point of intersectionof D and D . It turns out that one of the intermediate surfaces of this Z -coveris a blow-up e F → F of F at two points of a fiber and e S → S is the blow-upof S at the point over D ∩ D . It can be proved as above that we are in case a ) of Theorem 4 and the canonical image of S is F . Therefore S ∈ M Iχ,K byTheorem 8 and it has a Z -action.To obtain a smooth minimal surface in M IIχ,K we can consider a smooth Z -cover π : S → P of P whose branch locus B = D + D + D is a normalcrossing divisor consisting of smooth and irreducible divisors D ∈ |O P (1) | D ∈ |O P (1) | D ∈ |O P (7) | This cover satisfies ( χ ( O S ) , K S ) = (7 ,
9) and the canonical divisor of S is amplebecause 2 K S is the pullback via π of the ample sheaf O P (3). In addition, π induces another Z -cover e S → F where F → P is the blow-up of P at thepoint p of intersection of D and D . It turns out that one of the intermediatesurfaces of this Z -cover is F and e S → S is the blow-up of S at the point over D ∩ D . Arguing as usual it can be proved that we are in case c ) of Theorem4 and the canonical image of S is F . Therefore S ∈ M IIχ,K by Theorem 8 andit has a Z -action. Remark . Let us fix an admissible pair ( χ, K ) such that K = 2 χ − χ ≥ K + 1 ∈ · Z . Our examples of surfaces X ∈ M IIχ,K are constructed as minimalresolutions of surfaces Y with a singularity obtained by contracting a smoothelliptic curve with self-intersection ( − Z -cover Y → F e with branch locus B = D + D + D such that D has anordinary triple point p , D passes through p intersecting D with multiplicity3 and D does not pass through p . We can resolve the singularity over p in acanonical way. Let b : e F e → F e be the blow-up of F e at p with exceptional divisor E . Then there is a Z -cover X → e F e with branch locus e B = f D + f D + f D where f D = b ∗ D − E f D = b ∗ D − E f D = b ∗ D + E The induced map X → Y resolves the singularity over p . In addition, using theformulas for Z -covers we can check that the holomorphic Euler characteristicand the self intersection of the canonical class of X are reduced in 1 with respectto those of Y . In particular K Y = 2 χ ( O Y ) −
6. Nevertheless, the surfaces Y soconstructed do not belong to the moduli space M χ +1 ,K +1 because these ellipticsingularities are not canonical. They belong to the KSBA-compactification M χ +1 ,K +1 of M χ +1 ,K +1 .A question which arises now is whether they are in the closure of M χ +1 ,K +1 or if, on the contrary, they belong to a different irreducible component of M χ +1 ,K +1 . To answer this question we notice that we can assume D tobe smooth and the surface Z so obtained has the same invariants as Y but isnot singular. Since Y and Z are Z -covers of the same surface and the divisorsforming the branch locus belong to the same linear systems in both cases, itfollows that Y ∈ M χ +1 ,K +1 is a natural deformation (see [11, Section 5]) of Z ∈ M χ +1 ,K +1 . This shows that Y belongs to the closure of M χ +1 ,K +1 in M χ +1 ,K +1 . fibrations. In [9] Horikawa studied genus 2 fibrations. One of the things he showed is thatif we have a genus 2 pencil f : X → P , then the following formula holds: K X = 2 χ ( O X ) − X p ∈ P H ( p ) . The contribution H ( p ) is a non-negative integer that is bigger than 0 if and onlyif the fiber over p ∈ P is not 2-connected.Most of our examples of surfaces on the line K = 2 χ − K = 2 χ − H ( p ) = 1.Moreover, this fiber is usually obtained by making the divisors D and D of thebranch locus of the Z -cover intersect and consists of two elliptic curves withself-intersection ( −
1) intersecting transversally in one point.11 cknowledgements.
The author is deeply indebted to his supervisor MargaridaMendes Lopes for all her help. The author also thanks the anonymous reviewerfor her/his thorough reading of the paper and suggestions.
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Topology35 , 4:845–862, 1996. 12icente Lorenzo Center for Mathematical Analysis, Geometry and Dynamical SystemsDepartamento de MatemáticaInstituto Superior TécnicoUniversidade de LisboaAv. Rovisco Pais1049-001 LisboaPortugal