aa r X i v : . [ m a t h . AG ] M a y VECTOR FIELDS ON CANONICALLY POLARIZED SURFACES.
NIKOLAOS TZIOLAS
To my little daughter Eleonora.
Abstract.
This paper investigates the geometry of canonically polarized sur-faces defined over a field of positive characteristic which have a nontrivialglobal vector field, and the implications that the existence of such surfaces hasin the moduli problem of canonically polarized surfaces.In particular, an explicit integer valued function f ( x ) is obtained with thefollowing properties. If X is a canonically polarized surface defined over analgebraically closed field of characteristic p > p > f ( K X ) and X has a nontrivial global vector field, then X is unirational and the algebraicfundamental group is trivial. As a consequence of this result, large classesof canonically polarized surfaces are identified whose moduli stack is Deligne-Mumford, a property that does not hold in general in positive characteristic. Introduction
The objective of this paper is to investigate the geometry of canonically polarizedsurfaces with nontrivial global vector fields and to use the results of this investiga-tion in order to study the moduli stack of canonically polarized surfaces in positivecharacteristic. An investigation with these objectives was initiated in [Tz17a] wherethe case of smooth canonically polarized surfaces X with K X ≤ X defined over an algebraically closed field is calledcanonically polarized if and only if K X is ample and X has canonical singularities, orequivalently the singularities of X are rational double points. Canonically polarizedsurfaces are precisely the canonical models of smooth minimal surfaces of generaltype and they play a fundamental role in the classification problem of surfaces ofgeneral type. In fact, early on in the theory of moduli of surfaces of general type, itwas realized that the moduli functor of surfaces of general type is not well behavedand that the correct objects to parametrize are not the surfaces of general type butinstead their canonical models [Ko10], i.e., the canonically polarized surfaces.The property that a canonically polarized surface X has a nontrivial global vec-tor field is equivalent to the property that its automorphism scheme Aut( X ) is notsmooth. The reason is that the space of global vector fields of X is canonically iso-morphic to Hom(Ω X , O X ), the tangent space at the identity of Aut( X ). Moreover,it is well known that if X is canonically polarized then Aut( X ) is a zero dimensionalscheme of finite type over the base field. Therefore the existence of nontrivial global Mathematics Subject Classification.
Primary 14J50, 14DJ29, 14J10; Secondary 14D23,14D22.
Key words and phrases.
Algebraic geometry, canonically polarized surfaces, automorphisms,vector fields, moduli stack, characteristic p.Part of this paper was written during the author’s stay at the Max Planck Institute for Math-ematics in Bonn, from 01.02.2019 to 31.07.2019. vector fields on X is equivalent to the non smoothness of Aut( X ) and consequentlythe existence of non trivial infinitesimal automorphisms of X . Considering thatAut( X ) is a group scheme and every group scheme in characteristic zero is smooth,non smoothness of Aut( X ) can happen only in positive characteristic. Therefore acanonically polarized surface can have non trivial global vector fields only when itis defined over a field of positive characteristic.Examples of smooth canonically polarized surfaces surfaces with nontrivial globalvector fields exist but are hard to find since by [Tz17a, Lemma 4.1] such sur-faces are not liftable to characteristic zero. Such examples have been found byH. Kurke [Ku81], W. Lang [La83] and N. I. Shepherd-Barron [SB96]. Singularexamples are much easier to find and in fact there exists many examples of canoni-cally polarized surfaces with nontrivial global vector fields that are even liftable tocharacteristic zero. Such an example is given in Example 3.1.The existence of nontrivial global vector fields on canonically polarized surfaces isintimately related to fundamental properties of the local and global moduli functors,in particular the moduli stack.From the local moduli point of view, suppose that X is a canonically polarizedsurface defined over a field of characteristic p . If p = 0 then the local deforma-tion functor Def ( X ) is pro-representable since in this case, as explained earlier,Hom(Ω X , O X ) = 0 and hence X has no infinitesimal deformations [Se06, Corollary2.6.4]. The pro-representability of Def ( X ) implies the existence of a universal fam-ily for the local moduli functor, an ideal solution to the moduli problem. However,if p > X may have nontrivial infinitesimal automorphisms due to the existenceof nontrivial global vector fields and hence Def ( X ) is not pro-representable butonly has a hull.From the global moduli point of view, it is well known [KSB88] [Ko97] that themoduli stack of canonically polarized surfaces is a separated Artin stack of finitetype over the base field with zero dimensional stabilizers. In characteristic zero thestack is in fact a Deligne-Mumford stack. This implies that there exists a family X → S such that for any variety X in the moduli problem, there exists finitelymany s ∈ S such that X s ∼ = X , up to ´etale base change any other family is obtainedfrom it by base change and that for any closed point s ∈ S , the completion ˆ O S,s pro-represents the local deformation functor
Def ( X s ). However, none of these holdin general in characteristic p >
0. The reason for this failure is the existence ofcanonically polarized surfaces with non smooth automorphism scheme, or equiva-lently with nontrivial global vector fields [DM69, Theorem 4.1]. In some sense thenthe existence of nontrivial global vector fields on canonically polarized surfaces isthe obstruction for the moduli stack to be Deligne-Mumford.This investigation has two main objectives.The first objective is to find numerical conditions, which imply that the modulistack of canonically polarized surfaces is Deligne-Mumford and the local deforma-tion functor pro-representable. According to [Tz17a, Theorem 3.1] such conditionsexist. However their existence is due to purely theoretical reasons and no explicitconditions were obtained so far.The second objective is to describe the geometry of canonically polarized surfaceswhich have nontrivial global vector fields and consequently their moduli stack isnot Deligne-Mumford. The hope is to obtain a good insight in the geometry of such
ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 3 surfaces that will allow the modification of the moduli problem in order to get abetter moduli theory for these surfaces.From the existing examples of canonically polarized surfaces with nontrivialglobal vector fields and the case of smooth canonically polarized surfaces with K ≤
2, one gets the feeling that surfaces with nontrivial global vector fields tendto be uniruled and simply connected [Tz17a]. However non uniruled examples existin characteristic 2 [SB96], but it is unknown if non uniruled examples exist in highercharacteristics.The main results of this paper are the following.
Theorem 1.1.
Let X be a canonically polarized surface over an algebraically closedfield of characteristic p > . Suppose that X has a nontrivial global vector field, orequivalently Aut( X ) is not reduced and that p > max { K X ) + 12( K X ) + 3 , K X + 3 } . Then X is unirational and π ( X ) = { } . The contrapositive of the previous theorem provides numerical condition between K X , and p which implies the reducedness of the automorphism Aut( X ).If the automorphism scheme Aut( X ) of X is not smooth then Aut( X ) containsa subgroup scheme isomorphic to either α p or µ p . This is equivalent to say thatif X has a nontrivial global vector field then X has a nontrivial global vector field D such that D p = 0 or D p = D [Tz17b], [RS76]. If µ p is a subgroup scheme ofAut( X ), then finer restrictions can be imposed between K X and p which imply theunirationality of X . Theorem 1.2.
Let X be a canonically polarized surface over an algebraically closedfield of characteristic p > . Suppose that µ p ⊂ Aut( X ) , or equivalently that X hasa nontrivial vector field of multiplicative type and that one of the following happens:(1) K X = 1 and p > .(2) K X ≥ and p > K X + 3 . Then X is unirational and π ( X ) = { } . The previous results have immediate applications to the structure of the localand global moduli problems of canonically polarized surfaces.
Theorem 1.3.
Let X be a canonically polarized surface defined over an alge-braically closed field of characteristic p > . Suppose that π ( X ) = { } and that p > max { K X ) + 12( K X ) + 3 , K X + 3 } . Then
Def ( X ) is pro-representable. Theorem 1.4.
Let k be a field of characteristic p > and a ∈ Z such that p > max { a + 12 a + 3 , a + 3 } . Let M ntfg a be the moduli stack of canonically polarized surfaces X with K X = a ,and nontrivial fundamental group. Then M ntfg a is Deligne-Mumford. Theorem 1.3 is an immediate consequence of Theorem 1.1 and [Se06, Corollary2.6.4] while Theorem 1.4 is a consequence of Theorem 1.1 and [DM69, Theorem4.1] since the assumptions in both theorems imply that the automorphism schemeis reduced and that there exist no infinitesimal automorphisms.
NIKOLAOS TZIOLAS
Taking into consideration the breadth of the possible values of the fundamentalgroup of canonically polarized surfaces (it can be finite or infinite) [BCP11], onesees that the previous results apply to a very large class of canonically polarizedsurfaces.There are a few comments that I would like to make regarding the statement ofTheorems 1.1, 1.2.The reason that the cases K X = 1 and K X ≥ | mK X | . If K X = 1, then | K X | is base point free while if K X ≥ | K X | is base point free [Ek88]. Otherwise theproofs are identical. One could work with | K X | in both cases and get a unifiedstatement but in this case the bounds obtained would be weaker.The bounds on K obtained in Theorem 1.1 are not optimal if applied in specificcases. In particular, take the case when K X = 1. Then Theorem 1.1 says that X is unirational and simply connected if p > X is smooth, it hasbeen proved in [Tz17a], that X is unirational and simply connected for all p exceptpossibly for p = 3 , ,
7. I believe that the methods developed in this paper to treatsingular surfaces together with the techniques in [Tz17a] will make it possible toobtain much finer bounds than those obtained in Theorem 1.1 to the case of singularcanonically polarized surfaces with K X = 1.However, I believe that the strength of Theorem 1.1 lies in its generality and notthe optimality of the bounds obtained when applied in specific cases. The resultsapply to every canonically polarized surface and not to a specific class of them. Inindividual cases, like the cases when K X ≤ p > f ( K X ), whichimplies the smoothness of Aut( X ). Such a result will make it possible to obtaina theorem like Theorem 1.4 for canonically polarized surfaces whose fundamentalgroup is not trivial as well. However, the bounds for p are most likely going to belarger than those in Theorems 1.1, 1.2 making such a result weaker, since it wouldcover less cases, compared to Theorems 1.1, 1.2 for surfaces whose fundamentalgroup is not trivial. I believe that a method based on the methods used in thispaper should provide such a bound. However, at the moment I am unable to do so.The reason that in Theorem 1.2 I was able to obtain better bounds in the casewhen X has a vector field of multiplicative type, or equivalently when µ p is asubgroup scheme of Aut( X ), is that µ p is a diagonalizable group scheme while α p is not. As a consequence of this there are many integral curves of the vector fieldon X , something that provides a lot of information about the geometry of X .Finally I would like to say a few words about the proof of Theorems 1.1, 1.2.The main idea of the proof is to show that under certain relations between K X and p , if X has a nontrivial global vector field, then a linear system on X , usually of theform | mK X | contains a one dimensional subsystem | V | consisting of integral curvesof D . Then, to show that every irreducible component of every member of | V | isa rational curve (usually singular) which will imply that either X is birationallyruled (impossible in the case of canonically polarized surfaces) or more relationsbetween K X and p . In the implementation of this strategy, it is necessary to findconditions under which the vector field fixes the singular points of X and lifts to ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 5 the minimal resolution of X , something, unlike in characteristic zero, is not alwaystrue in positive characteristic.This paper is organized as follows.In Section 3 results about the number of singularities of a canonically polarizedsurface and conditions under which a vector field fixes the singularities of a surfaceand lifts to its minimal resolution are obtained. In particular, Theorem 3.2 providesun upper bound for the singular points of a canonically polarized surface X as afunction of K X and χ ( O X ). The result is under the assumption that the surfacehas a global vector field, an admittedly strong condition but sufficient for the pur-poses of this paper. In characteristic zero, similar bounds have been obtained byY. Miyaoka [M84]. However, in my knowledge, no such results existed yet in pos-itive characteristic. In Theorem 3.3, similar conditions are obtained which implythat a vector field fixes the singular points and lifts to the minimal resolution. Incharacteristic zero this is always true but not in general in positive characteristic.This is exhibited in Example 3.1.In Section 4 various results related to the geometry of integral curves of a vectorfield on a surface are obtained which are needed in the proofs of Theorems 1.1, 1.2.In Section 5 the general method and strategy for the proof of Theorems 1.1, 1.2are explicitly described.Sections 6, 7, 8 are devoted to the proof of the main theorems. The statements ofTheorems 1.1, 1.2 is the combination of the statements of Propositions 6.1 7.1, 8.1.2. Notation-Terminology
Let X be an integral scheme of finite type over an algebraically closed field k ofcharacteristic p > P ∈ X be a normal surface singularity and f : Y → X its minimal reso-lution. P ∈ X is called a canonical singularity if and only if K Y = f ∗ K X . Twodimensional canonical singularities are precisely the rational double points (or DuVal singularities) which are classified by explicit equations in all characteristics byM. Artin [Ar77].A normal projective surface X is called a canonically polarized surface if andonly if X has canonical singularities and K X is ample. These surfaces are exactlythe canonical models of minimal surfaces of general type.Der k ( X ) denotes the space of global k -derivations of X (or equivalently of globalvector fields). It is canonically identified with Hom X (Ω X , O X ).Let D be a nontrivial global vector field on X . D is called p -closed if and onlyif D p = λD , for some λ ∈ k . D is called of additive type if D p = 0 and ofmultiplicative type if D p = D . The fixed locus of D is the closed subscheme of X defined by the ideal sheaf ( D ( O X )). The divisorial part of the fixed locus of D iscalled the divisorial part of D . A point P ∈ X is called an isolated singularity of D if and only if the ideal of O X,P generated by D ( O X,P ) has an associated primeof height ≥ Z of X is called an integral divisor of D if and only if locallythere is a derivation D ′ of X such that D = f D ′ , f ∈ k ( X ), D ′ ( I Z ) ⊂ I Z and D ′ ( O X ) I Z [RS76].The vector field is said to stabilize a closed subscheme Y of X if and only if D ( I Y ) ⊂ I Y , where I Y is the ideal sheaf of Y in X . If Y is reduced and irreducible NIKOLAOS TZIOLAS and is not contained in the divisorial part of D then Y is also an integral curve of D .Let X be a normal surface and D a nontrivial global vector field on X of eitheradditive of multiplicative type. Then D induces an α p or µ p action on X . Let π : X → Y be the quotient of X by this action [Mu70, Theorem 1, Page 104]. Let C ⊂ X be a reduced and irreducible curve and ˜ C = π ( C ). Suppose that C is anintegral curve of D . Then π ∗ ˜ C = C . Suppose that C is not an integral curve of D .Then π ∗ ˜ C = pC [RS76].For any prime number l = p , the cohomology groups H iet ( X, Q l ) are independentof l , they are finite dimensional of Q l and are called the l -adic cohomology groupsof X . The i -Betti number b i ( X ) of X is defined to be the dimension of H iet ( X, Q l ).It is well known that b i ( X ) = 0 for any i > n , where n = dim X [Mi80, ChapterVI, Theorem 1.1]. X is called simply connected if π ( X ) = { } , where π ( X ) is the algebraicfundamental group of X .Let F be a coherent sheaf on X . By F [ n ] we denote the double dual ( F ⊗ n ) ∗∗ .3. Singular points of surfaces with vector fields.
Let X be a normal projective surface defined over an algebraically closed field k of characteristic p > X has a nontrivial global vector field D . This section has two main oblectives. Thefirst objective is to obtain an upper bound, as a function of numerical invariants of X , of the number of singular points of X . The second objective is to find conditionswhich imply that the singular points of X are fixed points of the vector field D andthat D lifts to the minimal resolution of X .If the base field has characteristic zero, then an upper bound of the numberof singular points of X was obtained by Y. Miyaoka [M84]. The proof of thatresult uses, among other characteristic zero techniques, the Bogomolov-Miyaoka-Yau inequality which fails in positive characteristic. In this section, a result in thatspirit is given under the assumption that X has a nontrivial global vector field.This is a strong restriction on X , but it suffices for the purpose of this paper.In characteristic zero, a vector field fixes the singularities and lifts to the minimalresolution [BW74]. However, this does not hold in general in positive characteristic.In fact something more interesting happens. There exist smooth minimal surfacesof general type without nontrivial global vector fields (and hence reduced auto-morphism scheme) whose canonical model has nontrivial global vector fields andtherefore non reduced automorphism scheme. This is a situation that complicatesthe structure of the moduli of surfaces of general type in positive characteristic.The next example exhibits exactly such a case. Example 3.1.
Let k be an algebraically closed field of characteristic 2 and X ⊂ P k be the quintic given by x x ( x + x + x ) + x x + x x = 0 . I will show the following:(1) The singularities of X are rational double points of type A ( i.e., locallyisomorphic to xy + z = 0) and K X is ample.(2) X has nontrivial global vector fields and hence the automorphism schemeAut( X ) is a non reduced zero dimensional scheme. ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 7 (3) The vector fields of X do not fix all the singular points of X and thereforethey do not lift to the minimal resolution of X .(4) The minimal resolution of X is a smooth minimal surface of general typewithout vector fields and therefore with reduced automorphism schemeAut( X ′ ).I proceed to show the above properties. X is a quintic in P and hence by the standard adjunction formula, O X ( K X ) = O X (1) and hence it is very ample.The equation of X is invariant under the graded derivation D = x ∂∂x of k [ x , x , x , x ], which therefore induces a nonzero global vector field on X suchthat D = 0. Hence X has nontrivial global vector fields.The singularities of X can be checked locally. In the affine chart given by x = 1, X is given by the equation x x + x x + x x + x + x = 0in k [ x , x , x ]. The singular points of X are those with x i + x i = 0, i = 1 , x + x + x x = 0. A straightforward calculation shows that the degree two termof the polynomial defining X at every singular point is an irreducible quadric in x , x and x and hence the singularities of X are ordinary double points given locallyanalytically by xy + z = 0. Similarly one can easily check that there are no moresingularities in the other charts. Hence (1) is proved.In this chart the vector field D is given by D = ∂∂x . Hence D has no fixed pointsin the open set x = 1. In particular, none of the singular points is fixed by D .Since K X is ample and X has rational double points, Aut( X ) is zero dimensional.Then since its tangent space is Hom(Ω X , O X ) = 0, the space of global derivations,Aut( X ) is not reduced. Hence (2) is proved.Let now f : X ′ → X be the minimal resolution of X . Then X ′ is simply the blowup of the singular points of X . Since X has rational double points, K X ′ = f ∗ K X and therefore X ′ is a minimal surface of general type.Since f is the blow up of the singular points of X , a vector field on X lifts to avector field on X ′ if and only if it fixes the singular points of X . In addition, everyvector field on X ′ induces a vector field on X by the natural map f ∗ T X ′ → T X .Therefore, in order to show that X ′ has no non trivial global vector fields, it sufficesto show that there is no non trivial global vector field on X which fixes every singularpoint of X . This will be done by explicitly calculating the vector fields of X . Claim:
A vector field on X is the restriction on X of a vector field on P whichfixes X .Dualizing the exact sequence0 → O X ( − → Ω P ⊗ O X → Ω X → → Hom(Ω X , O X ) → Hom(Ω P , O X ) → Hom( O X ( − , O X ) . Moreover, there exists a natural exact sequence0 → Hom(Ω P , O P ( − → Hom(Ω P , O P ) σ → Hom(Ω P , O X ) → Ext ((Ω P , O P ( − . Now Ext ((Ω P , O P ( − H ( T P ( − P and the cohomology of P . Hence the map σ is NIKOLAOS TZIOLAS surjective and therefore every global vector field on X is induced by a vector fieldon P , and the claim is proved.Now h ( T P ) = 15 and the global vector fields on P are induced by the followinggraded vector fields of k [ x , x , x , x ]. D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x . In the calculation of the vector fields of P it was taken into considerationthat the graded derivation P i =1 x i ∂∂x i gives the zero vector field of P .In the affine chart x = 1, these derivations are given by the following derivationsof k [ x , x , x ]. D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x ∂∂x + x x ∂∂x + x x ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x x ∂∂x + x ∂∂x + x x ∂∂x , D = x ∂∂x , D = ∂∂x , D = ∂∂x , D = ∂∂x , D = x ∂∂x , D = x ∂∂x , D = x x ∂∂x + x x ∂∂x + x ∂∂x .Let now D = P i =1 λ i D i a derivation, λ i ∈ k , i = 1 , . . . ,
15. The points(0 , , , (1 , , , (0 , , , (1 , , , (0 , ,
1) are singular points of X corresponding tothe ideals ( x , x , x ) , ( x + 1 , x , x ) , ( x , x + 1 , x ) , ( x + 1 , x , x + 1) , ( x , x +1 , x + 1). A straightforward but a bit long calculation shows that the only deriva-tion fixing these ideals is D = λ ( D + D + D + D + D ) = λ (( x + x + x x ) ∂∂x + ( x + x + x x ) ∂∂x + ( x + x x + x x ) ∂∂x ) . However, this derivation does not fix the ideal ( x + 1 , x + 1 , x + a ), where a + a + 1 = 0, corresponding to the singular point (1 , , a ), neither the equation of X . Hence X does not have any nontrivial global vector fields fixing all its singularpoints and therefore its minimal resolution has no non trivial vector fields and henceit has reduced automorphism scheme.The main results of this section are the following two theorems. The first onegives an upper bound for the number of singularities of a canonically polarizedsurface X . The next one provides a condition under which a vector field fixes thesingular points and lifts to the minimal resolution. Theorem 3.2.
Let X be a canonically polarized surface defined over a field ofcharacteristic p > . Suppose that p does not divide K X and X has a nontrivialglobal vector field. Let f : X ′ → X be the minimal resolution of X . Let ν ( P ) be thenumber of f -exceptional curves over a point P ∈ X . Then(1) Suppose that K X = 1 and p = 2 . Then P P ∈ X ν ( P ) ≤ .(2) Suppose that K X ≥ and p = 3 . Then X P ∈ X ν ( P ) ≤ χ ( O X ) + 11 K X , In particular, X has at most singular points if K X = 1 and χ ( O X ) + 11 K X singular points if K X ≥ . Theorem 3.3.
With assumptions as in Theorem 3.2. Suppose also that p > and(1) p > , if K X = 1 ,(2) p > χ ( O X ) + 11 K X + 1 , if K X ≥ . ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 9
Then(1) Every singular point of X is a fixed point of D .(2) D lifts to a vector field D ′ on the minimal resolution X ′ of X .(3) Every f -exceptional curve is an integral curve of D ′ . Remark 3.4.
The proof of the theorem uses Proposition 3.14 which requires p =3 and the classification of rational double points in positive characteristic whichrequires p > χ ( O X ) > χ ( O X )+11 K X +1 >
24. However in positive characteristicit is not known at the moment of this writing if χ ( O X ) > X and so it ispossible that 12 χ ( O X ) + 11 K X + 1 may be 5 or less so p = 3 , K X ≥ Corollary 3.5.
With assumptions as in Theorem 3.2. Suppose that the singularlocus of X consists of the points A ∗ i of type A n i , i = 1 , . . . , r , D ∗ j of type D m j , j = 1 , . . . , s , E ∗ ,k of type E , k = 1 , . . . , t , E ∗ ,ν of type E , ν = 1 , . . . , w and E ∗ ,µ of type E , µ = 1 , . . . , u . Then r X i =1 n i + s X j =1 m j + 6 t + 7 w + 8 u ≤ χ ( O X ) + 11 K X . The proofs of Theorems 3.2, 3.3 will be given at the end of this section.The next proposition is a simple generalization to the case of singular surfacesof a well known result on vector fields on smooth surfaces.
Proposition 3.6.
Let X be a Gorenstein normal projective surface and D a non-trivial global vector field on X such that D p = 0 or D p = D . Let ∆ be the divisorialpart of D . Then there exists an exact sequence → O X (∆) → T X → ω − ( − ∆) → F → , where F is a zero dimensional coherent sheaf whose support is contained in theunion of the singular points of X and the isolated singularities of D .Proof. Let Z ⊂ X be the union of the singular points of X and the isolated singu-larities of D . Then Z is a finite set. Let U = X − Z . Then U is smooth and therestriction of D on U has only divisorial singularities. Therefore the quotient of U by D is smooth [RS76]. Therefore there exists an exact sequence0 → O U (∆ | U ) → T U → L U → , where L U is an invertible sheaf on U [MP97, Proposition 1.9.3]. Moreover, fromthe above sequence it follows that L U = ω − U ( − ∆ | U ). Applying i ∗ in the abovesequence, where i : U → X is the inclusion, and thaking into consideration that ω X is invertible, we get an exact sequence0 → O X (∆) → T X → ω − ( − ∆) → F → , where F is a zero dimensional coherent sheaf whose support is contained in theunion of the singular points of X and the isolated singularities of D , as claimed. (cid:3) The next proposition gives a Riemann-Roch type inequality for divisors on sur-faces with rational double points.
Proposition 3.7.
Let X be a normal projective surface over an algebraically closedfield k . Suppose that the singularities of X are rational double points. Let D be adivisor on X . Then χ ( O X ( D )) ≤ χ ( O X ) + 12 ( D − K X · D ) . Remark 3.8.
The difference between the right hand side and the left hand sidehas been calculated explicitly with respect to the analytic type of the singularitiesof X by M. Reid [Re85] in the case when the base field is C . A similar calculationmay be possible and desirable in positive characteristic. However, for the purposesof this paper, the above inequality suffices. Proof.
Let f : X ′ → X be the minimal resolution of X . Then the double dual( f ∗ O X ( D ))) [1] is invertible and hence ( f ∗ O X ( D ))) [1] = O X ′ ( D ′ ), where D ′ is adivisor on X ′ . Now by [Ar85], f ∗ O X ′ ( D ′ ) = O X ( D ) and R f ∗ O X ′ ( D ′ ) = 0. There-fore, χ ( O X ( D )) = χ ( O X ′ ( D ′ )) . (3.8.1)Then by Rieman-Roch on X ′ , χ ( O X ′ ( D ′ )) = χ ( O X ′ ) + 12 (( D ′ ) − K X ′ · D ′ ) . (3.8.2)Since X has rational double points and X ′ is the minimal resolution of X , χ ( O X ′ ) = χ ( O X ) and K X ′ = f ∗ K X . Moreover, it is clear that f ∗ D ′ = D and hence by theprojection formula, K X ′ · D ′ = f ∗ K X · D ′ = K X · C. (3.8.3)Next I will relate D and ( D ′ ) . Since X has rational double points, D is Q -Cartier.Let m ∈ Z be a positive integer such that mD is Cartier. Then mD ′ = f ∗ ( mD ) + F, where F is a divisor supported on the exceptional set of f . Then, m ( D ′ ) = m D + F < m D , since F <
0. Hence ( D ′ ) < D . Now the statement of the proposition followsfrom this and the equations (3.8.1), (3.8.2) and (3.8.3). (cid:3) The next result relates the first cohomology of the tangent sheaf of a rationaldouble point with the number of exceptional divisors over it in the minimal resolu-tion.
Proposition 3.9.
Let P ∈ X be a rational double point singularity. Let f : X ′ → X be its minimal resolution and E i , i = 1 , . . . , n the f -exceptional curves . Then h ( T X ′ ) ≥ n. Remark 3.10.
If the characteristic of the base field is zero then the inequality inthe previous proposition is in fact equality [BW74].
ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 11
Proof.
The proof follows the lines of the proof of [BW74, Pages 70, 71] with somemodifications to deal with the possible positive characteristic complications.Let Z = P ni =1 m i E i an integral effective divisor supported on the exceptionalset of f . Then for sufficiently large m i , i = 1 , . . . , n , − Z is f -ample. Therefore, H i ( T X ′ ( − mZ )) = 0, for m >> i = 1 ,
2. Taking now cohomology on the exactsequence 0 → T X ′ ( − mZ ) → T X ′ → T X ′ ⊗ O mZ → , it follows that H ( T X ′ ) = H ( T X ′ ⊗ O mZ ) . (3.10.1)Let E = P ni =1 E i be the reduced f -exceptional divisor. Then there exists an exactsequence 0 → N → O mZ → O E → , where N is supported on the exceptional set of f . Then the previous sequence givesthe exact sequence0 → T X ′ ⊗ N → T X ′ ⊗ O mZ → T X ′ ⊗ O E → . After taking cohomology in the previous sequence, and since N has 1-dimensionalsupport, it follows that h ( T X ′ ) ≥ h ( T X ′ ⊗ O E ) . (3.10.2)Next, there exists an exact sequence0 → T E → T X ′ ⊗ O E σ → ⊕ ni =1 N E i → , (3.10.3)where the map σ is the sum of the composition of the natural maps T X ′ ⊗ O E → T X ′ ⊗ O E i and T X ′ ⊗ O E i → N E i , i = 1 , . . . , n . The exactness of the sequenceabove can easily be checked locally.Now since P ∈ X is a rational double point, E i ∼ = P and N E i ∼ = O P ( − i = 1 , . . . , n . The proposition now follows from the equation (3.10.2) and by takingcohomology in (3.10.3).Finally I would like to mention that in [BW74], the equality in the statement ofthe proposition is proved by taking the exact sequence (3.10.3) with mZ in the placeof E on the left hand side of the sequence and then using a result by Tjurina that H ( T mZ ) = 0. However, this is proved only in characteristic zero and moreover, theexact (3.10.3) may not be exact with mZ in the place of E if some of the coefficientsof mZ are divisible by p . (cid:3) The next proposition gives a bound for the number of singular points of a pro-jective surface with rational double points and a nontrivial global vector field.
Proposition 3.11.
Let X be a normal projective surface over an algebraicallyclosed field of characteristic p > with rational double point singularities. Supposethat X has a nontrivial global vector field D such that D p = 0 or D p = D . Let f : X ′ → X be the minimal resolution of X . Then X P ∈ X ν ( P ) ≤ χ ( O X ) − K X + ∆ + K X · ∆ , where ∆ is the divisorial part of D and ν ( P ) is the number of f -exceptional curvesover P ∈ X . Remark 3.12.
If ∆ = 0, a case that frequently happens, then 12 χ ( O X ) − K X is abound for the singular points of X , a bound which is a function of only numericalinvariants of X . A similar bound will be given later without the assumption ∆ = 0if K X is ample and p does not divide K X . Proof.
There exists a natural exact sequence0 → f ∗ T X ′ → T X → N → , where N is a zero dimensional coherent sheaf on X supported on the singular locusof X . Hence χ ( N ) = h ( N ) ≥
0. Then from the above sequence it follows that χ ( f ∗ T X ′ ) ≤ χ ( f ∗ T X ′ ) + χ ( N ) = χ ( T X ) . (3.12.1)From the Leray spectral sequence and considering that f is birational with at mostone dimensional fibers we get the exact sequence0 → H ( f ∗ T X ′ ) → H ( T X ′ ) → H ( R f ∗ T X ′ ) → H ( f ∗ T X ′ ) → H ( T X ′ ) → . Counting dimensions we get that χ ( f ∗ T X ′ ) = χ ( T X ′ ) + h ( R f ∗ T X ′ )(3.12.2)Now from Propositions 3.6, 3.7 it follows that χ ( T X ) = χ ( O (∆)) + χ ( ω − X ( − ∆)) − χ ( F ) ≤ χ ( O (∆)) + χ ( ω − X ( − ∆)) ≤ (3.12.3) 2 χ ( O X ) + 12 (∆ − K X · ∆) + 12 (( K X + ∆) + K X · ( K X + ∆)) =2 χ ( O X ) + K X + K X · ∆ + ∆ . Now from the equations (3.12.1), (3.12.2) and (3.12.3) we get that χ ( T X ′ ) + h ( R f ∗ T X ′ ) ≤ χ ( O X ) + K X + K X · ∆ + ∆ . (3.12.4)Then by Proposition 3.9 and the previous inequality we get that χ ( T X ′ ) + X P ∈ X ν ( P ) ≤ χ ( T X ′ ) + h ( R f ∗ T X ′ ) ≤ χ ( O X ) + K X + K X · ∆ + ∆ . (3.12.5)Now by the Riemann-Roch on X ′ , Noether’s formula and the facts that K X ′ = f ∗ K X , χ ( O X ′ ) = χ ( O X ) (since X has rational double point singularities), we getthat χ ( T X ′ ) = 76 K X ′ − c ( X ′ ) = 76 K X ′ −
56 (12 χ ( O X ′ ) − K X ′ ) = − χ ( O X ′ ) + 2 K X . (3.12.6)Now the statement of the proposition follows immediately from the equations(3.12.5) and (3.12.6). (cid:3) The following lemma is an easy generalization of the Hodge index theorem tosurfaces with rational double points. It will be used throughout this paper.
Lemma 3.13.
Let X be a normal projective surface with rational double points.Let A be a nef and big line bundle on X and C a divisor on X . Then C A ≤ ( C · A ) . ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 13
Proof.
Let X ′ → X be the minimal resolution of X . Since X has rational doublepoint singularities, C is Q -Cartier. Let m > mC isCartier. Then, since f ∗ A is also nef and big on X ′ and the generalized Hodgeindex theorem for nef and big line bundles [Ba01, Corollary 2.4], it follows that m C · A = ( f ∗ ( mC )) · ( f ∗ A ) ≤ ( f ∗ ( mC ) · f ∗ A ) = m ( C · A ) . From this the lemma follows immediately. (cid:3)
The following proposition is the last ingredient needed in order to prove Theo-rems 3.2, 3.3. It will also be needed later for the proof of the main theorem of thispaper.
Proposition 3.14.
Let X be a canonically polarized surface defined over a fieldof characteristic p > . Suppose that X has a nontrivial global vector field D suchthat D p = 0 or D p = D and such that p does not divide K X . Then(1) Suppose that K X = 1 and p = 2 . Then K X · ∆ ≤ and ∆ ≤ .(2) Suppose that K X ≥ and p = 3 . Then K X · ∆ ≤ K X , (3.14.1) ∆ ≤ K X , where ∆ is the divisorial part of D .Proof. Let π : X → Y be the quotient of X by the α p or µ p action on X definedby D . Then π is a purely inseparable map of degree p and by [RS76], K X = π ∗ K Y + ( p − Y and henceeverywhere since Y is normal).By [Ek88, Theorem 1.20], the linear system | nK X | is base point free for n = 3 if K X = 1 and n = 2 if K X ≥ K X ≥ n = 2. The proof in the case when K X = 1and n = 3 is identical and is omitted. Then by [Jou83, Theorem 6.3], [Za44], thegeneral member of | K X | is of the form p ν C , where C is an irreducible and reducedcurve. Since p does not divide K X , ν = 0 and hence the general member of | K X | is a reduced and irreducible curve.Therefore there exists C ∈ | K X | such that C is reduced and irreducible andit does not pass through any singular point of X or isolated singularity of D . Let˜ C = π ( C ). Then, since C is in the smooth part of X and does not contain anyisolated singularity of D , ˜ C lies in the smooth part of Y .Suppose that C is an integral curve of D . Then [RS76], π ∗ ˜ C = C and therefore C = p ˜ C = pm , m ∈ Z since ˜ C is in the smooth part of Y . Then, since C ∈ | K X | ,it follows that p divides 9 K X and hence, since p = 3, p divides K X , which isimpossible.Hence C is not an integral curve of D and hence the map π : C → ˜ C is birational.Moreover [RS76], π ∗ ˜ C = pC . Now since ˜ C is contained in the smooth part of Y ,adjunction for ˜ C holds and hence2 p a ( ˜ C ) − K Y · ˜ C + ˜ C = π ∗ K Y · C + pC = K X · C − ( p − · C + pC =( K X · C + C ) + ( p − C − ∆ · C ) = 2 p a ( C ) − p − K X − K X · ∆) . Since the map C → ˜ C is birational, it follows that p a ( ˜ C ) ≥ p a ( C ). Then theabove equation gives that 3 K X − K X ˙∆ ≥ K X · ∆ ≤ K X , as claimed. Finally, since K X is ample, it follows from Lemma 3.13 that∆ ≤ (∆ · K X ) K X ≤ (3 K X ) K X = 9 K X , as claimed. (cid:3) We are now in a position to prove Theorems 3.2, 3.3.
Proof of Theorem 3.2 . Since X has a nontrivial global vector field, it followsfrom [RS76] that X has a nontrivial global vector field D such that D p = 0 or D p = D . Then the statement of the theorem follows immediately from Propo-sitions 3.11, 3.14. In the case when K X = 1, one must also use that fact that1 ≤ χ ( O X ) ≤ (cid:3) Proof of Theorem 3.3 . I will only do the case when K X ≥
2. The case when K X = 1 is identical and is omitted.By assumption X has canonical singularities and hence its singularities are ra-tional double points. Since p >
5, the equations classifying the rational doublepoints are the same as those in characteristic zero [Ar77]. Hence X may haveeither singularities of type A n , D m , E , E and E . Then, by Theorem 3.2, if12 χ ( O X ) + 11 K X + 1 < p , n + 1 < p and m < p . The statement of the theorem islocal at the singularities. In order to prove the theorem consider cases with respectto the singularities of X .Let P ∈ X be a singular point of X . I will do in detail only the case when P ∈ X is of type A n . The rest are similar and are left to the reader.By passing to the completion at P , we may assume that X is given by xy + z n +1 =0. Moreover, by the assumptions and Theorem 3.2, n + 1 < p . D is induced by aderivation D of k [[ x, y, z ]] such that D ( xy + z n +1 ) ∈ ( xy + z n +1 ). Now D ( xy + z n +1 ) = xDy + yDx + ( n + 1) z n Dz, with n + 1 = 0. From the above equation it follows that yDx ∈ ( x, z ) and hence,since y ( x, z ), it follows that Dx ∈ ( x, z ) ⊂ ( x, y, z ). Similarly, Dy ∈ ( y, z ).Finally, from the previous equation it follows that z n Dz ∈ ( x, y, z n +1 ). If Dz ( x, y, z ), then Dz is a unit in k [[ x, y, z ]] and hence z n ∈ ( x, y, z n +1 ), which isimpossible. Hence in this case, P is a fixed point of D .Next I will show that D lifts to the minimal resolution f : X ′ → X of X . Since X has rational double points, f is obtained by successively blowing up the singularpoints. Let f : X → X be the blow up of all singular points of X . Then, sincethe singular points of X are fixed points of D , D lifts to a vector field D on X .Moreover, X has also rational double points, of simpler type that those of X .Then, the previous argument shows that the singular points of X are fixed pointsof D . Then one can blow up again and continue this process until the minimalresolution is reached and therefore D lifts to a vector field D ′ on X ′ .It remains to show that every f -exceptional curve is an integral curve of D ′ . Inorder to prove this it suffices to prove, since f is a composition of blow ups, thefollowing. Let P ∈ Z be a rational double point on a surface Z which is a fixedpoint of a vector field D of Z and let g : ˜ Z → Z be the blow up of P . Then thereduced g -exceptional curves are integral curves of ˜ D , the vector field on Z lifting D . ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 15
Let E be a g -exceptional curve. Suppose that P ∈ X is of type A n . Then f − ( P ) = E + E , where where E , E are distinct smooth rational curves. Supposethat P ∈ X is of one of the types D n , E , E , E . Then f − ( P ) = 2 E , where E isa smooth rational curve. Then the claim that the g -exceptional curves are integralcurves of ˜ D is an immediate consequence of Lemma 3.15 which follows.This concludes the proof of Theorem 3.3. Lemma 3.15.
Let f : X → Y be a morphism between varieties defined over analgebraically closed field k of characteristic p > Such that X is normal. Supposethat D Y is a nontrival global vector field on Y and D X a nontrivial global vectorfield on X lifting D Y , i.e., there exists a commutative diagram f ∗ O X D X / / f ∗ O X O Y O O D Y / / O Y O O Let P ∈ Y be a fixed point of D Y and [ f − ( P )] = P ni =1 m i Z i , be the cycle cor-responding to the fiber f − ( P ) . Let Z i be a codimension 1 component such that p does not divide m i , Then D X ( I Z i ) ⊂ I Z i , i.e., Z i is stabilized by D X .Proof. Let Z i be a codimension 1 component of f − ( P ) such that p does not divide m i . In order to prove that D X ( I Z i ) ⊂ I Z i it suffices to prove this in an affine openset U of X which is contained in the smooth locus of X and such that U ∩ Z i = ∅ .Therefore the proof is reduced to the case when both X and Y are affine. Let then Y = Spec A , X = Spec B . Then D Y , D X are induced by derivations of A and B ,respectively. Let m p ⊂ A be the maximal ideal corresponding to P . Then f − ( P )is given by the ideal m P B of B . Moreover, since D X lifts D Y and D Y ( m P ) ⊂ m P ,it follows easily that D X ( m P B ) ⊂ m P B . Then if U is chosen small enough, m P B = I m i Z i . Moreover, since X is normal and U is in the smooth part of X , I Z i is a prime ideal of B and I Z i = ( b ), for some b ∈ B . Then D X ( b m i ) = m i b m i − D X b ∈ ( b m i ) . Since p does not divide m i it follows that b m i − D X b ∈ ( b m i ) and hence b m i − D X b = b m i c and therefore D X b ∈ ( b ). Hence D X ( I Z i ) ⊂ I Z i , as claimed. (cid:3)(cid:3) Integral curves and fixed points of vector fields on surfaces.
Let X be a normal projective surface defined over an algebraically closed field k of characteristic p >
0. Let D be a nontrivial vector field on X (or equivalently a k -derivation of O X ). This section contains various properties of integral curves of D which are needed for the proofs of the main results of this paper.The next proposition presents a method to find integral curves of D . Proposition 4.1 (Proposition 2.1 [Tz18]) . Suppose that either D p = 0 or D p = D . Then D induces an α p or µ p action on X , respectively. Let π : X → Y bethe quotient of X by this action. Let L be a rank one reflexive sheaf on Y and M = ( π ∗ L ) [1] . Then D induces a k -linear map D ∗ : H ( X, M ) → H ( X, M ) with the following properties:(1) Ker( D ∗ ) = H ( Y, L ) (considering H ( Y, L ) as a subspace of H ( X, M ) viathe map π ∗ ).(2) If D p = 0 then D ∗ is nilpotent and if D p = D then D ∗ is a diagonalizablemap whose eigenvalues are in the set { , , . . . , p − } .(3) Let s ∈ H ( X, M ) be an eigenvector of D ∗ . Then D ( I Z ( s ) ) ⊂ I Z ( s ) , where Z ( s ) is the divisor of zeros of s . In particular, if D ∗ ( s ) = λs , and λ = 0 ,then ( D ( I Z ( s ) )) | V = I Z ( s ) | V , where V = X − π − ( W ) , W ⊂ Y is the set ofpoints that L is not free. The previous proposition shows that every eigenvector of D ∗ corresponds to acurve C ⊂ X such that D ( I C ) ⊂ I C and therefore D induces a vector field on C .However it is possible that D ( O X ) ⊂ I C and hence the induced vector field on C is trivial. This implies that C is contained in the divisorial part of D . This cannothappen of course if D has only isolated singularities.Let C = n C + · · · + n k C k be a curve in X and its decomposition into its primecomponents. Suppose that D ( I C ) ⊂ I C . In general D does not induce vector fieldson C i , i.e, D ( I C i ) may not be contained in I C i . For example for any reduced andirreducible curve C , D stablizes pC but not necessarily C . The next propositionprovides some conditions in order for D to restrict to C i . Proposition 4.2.
Let C ⊂ X be a curve such that D ( I C ) ⊂ I C , where I C ⊂ O X isthe ideal sheaf of C in X . Let C = n C + · · · + n k C k be the decomposition of C inits irreducible and reduced components. If p does not divide n i , for all ≤ i ≤ k ,then D ( I C i ) ⊂ I C i , for all ≤ i ≤ k . Therefore D stabilizes the reduced part ofevery irreducible component of C and hence induces a vector field on C i , for all ≤ i ≤ k .Proof. Let i ∈ { , . . . , k } . In order to prove that D ( I C i ) ⊂ I C i it suffices to showthis on a nonempty open subset U of X such that U ∩ C i = ∅ . In fact, by taking U small enough we may assume that U ∩ C j = ∅ , for all j = i . Hence we mayassume that X = Spec A is affine and smooth and C = n i C i . Hence I C = ( t n i ),for some t ∈ A and I C i = ( t ). D is induced by a k -derivation of A . Then since D ( I C ) ⊂ I C , it follows that n i t n i − Dt ∈ ( t n i ) and hence there exists a ∈ A suchthat n i t n i − Dt = at n i . Now since p does not divide n i , n i = 0 in k and hence itfollows that Dt ∈ ( t ). Hence D ( I C i ) = I C i , as claimed. (cid:3) Corollary 4.3.
With assumptions as in Proposition 4.2. Suppose in addition that K X is an ample invertible sheaf and K X · C < p . Then D ( I C i ) ⊂ I C i , for all ≤ i ≤ k . Therefore D stabilizes the reduced part of every irreducible componentof C and hence induces a vector field on C i , for all ≤ i ≤ k .Proof. Since K X is assumed to be ample and invertible, the condition K X · C < p immediately implies that n i < p , for all 1 ≤ i ≤ k . Then the corollary followsdirectly from Proposition 4.2. (cid:3) Proposition 4.4.
Suppose that X is Q -factorial and K X is an ample invertiblesheaf. Let C ∈ | mK X | be a curve such that D ( I C ) ⊂ I C . Let C = n C + · · · + n k C k its decomposition into its reduced and irreducible components. Suppose that K X
Proof.
By Corollary 4.3, D ( I C i ) ⊂ I C i , for all 1 ≤ i ≤ k . The result is local at P .Let U = Spec A be an affine open subset of X containing P but no other point of C i ∩ C j . Since P ∈ X is a smooth point, U may be taken to be smooth. Let I and J be the ideals of C i and C j respectively. Then ( I + J ) | U = Q , with r ( Q ) = m P ,the maximal ideal corresponding to the point of intersection P of C i and C J . Nowsince D ( I ) ⊂ I and D ( J ) ⊂ J , it follows that D ( I + J ) = D ( I ) + D ( J ) ⊂ I + J .Hence D ( Q ) ⊂ Q . I will show that this implies that D ( m P ) ⊂ m P and therefore P is a fixed point of D .In order to show that D ( m P ) ⊂ m P , I will first show that C i · C j < p . Then if I = ( f ) and J = ( g ), f, g ∈ A , dim k A/ ( f, g ) < p . Hence for any a ∈ m P , thereexists ν < p such that a ν ∈ Q = I + J . Let ν < p be the smallest such ν . Then D ( a ν ) = ν a ν − Da ∈ Q = I + J . Q is a primary ideal and a ν − Q . Hence( Da ) s ∈ Q ⊂ m P , for some s ≥
0. Hence Da ∈ m P . Therefore D ( m P ) ⊂ m P , asclaimed.It remains to show that C i · C j < p . By definition, mK X ∼ P ks =1 n s C s . Let1 ≤ i, j ≤ k . Then mK X · C i = n j C i · C j + n i C i + X s = i,j n s C s · C i ≥ n j C i · C j + n i C i . (4.4.1)On the other hand, mK X = P ms =1 n s K X · C s and since K X is ample, it follows that K X · C s > ≤ s ≤ m and therefore K X · C s ≤ n s K X · C s ≤ mK X .Then from (4.4.1) it follows that C i · C j ≤ m K X − n i C i . (4.4.2)Next I will show that − C i ≤ K X · C i . Let f : X ′ → X be the minimalresolution of X . Let C ′ i = f − ∗ C i , be the birational transform of C i in X ′ . Thenby the adjunction formula for C ′ i it follows that − ( C ′ i ) = − p a ( C ′ i ) + 2 + K X ′ · C ′ i ≤ K X ′ · C ′ i . (4.4.3)Now there are adjunction formulas f ∗ C i = C ′ i + E (4.4.4) K X ′ + F = f ∗ K X Where E and F are effective f -exceptional divisors ( F is effective because f isthe minimal resolution). From these immediately follows that C i ≥ ( C ′ i ) and K X · C i ≥ K X ′ · C ′ i . From these and the equation (4.4.3) it follows that − C i ≤ K X · C i . (4.4.5)Then from the equation (4.4.2) it follows that C i · C j ≤ m K X + 2 n i + n i K X · C i . (4.4.6)But it has been shown earlier that n i K X · C i ≤ mK X and hence n i ≤ mK X and K X · C i < mK X . Hence C i · C j ≤ ( m + 3 m ) K X < p ( m + 3 m ) m + 3 m < p, (4.4.7)as claimed. This concludes the proof. (cid:3) The proof of the previous proposition shows also the following.
Corollary 4.5.
Let C , C be two different irreducible and reduced curves on X such that D ( I C i ) ⊂ I C i , for i = 1 , . Assume that C · C < p . Then every point ofintersection of C and C which is a smooth point of X is a fixed point of D . Remark 4.6.
Proposition 4.4 and Corollary 4.5 apply in particular in the casewhen the singularities of X are rational double points since they are Q -factorial.As explained in Section 3, in general, in positive characteristic a vector field ona variety Y does not fix its singular points. In section 3 conditions were obtainedwhich imply that a vector field on a surface fixes its singular points. The nextproposition shows gives a condition which implies that a vector field on a curvefixes the singular points of the curve. Proposition 4.7.
Let D be a nontrivial vector field of either additive or multi-plicative type on a smooth surface X defined over an algebraically closed field k of characteristic p > . Let C ⊂ X be a reduced and irreducible curve such that D ( I C ) ⊂ I C , where I C is the ideal sheaf of C in X . Suppose that p a ( C ) < ( p − / .Then D fixes every singular point of C and lifts to the normalization ¯ C of C .Proof. We may assume that D ( O X ) I C and hence the restriction of D on C isnot trivial (otherwise the result is obvious).Let π : X → Y be the quotient of X by the α p or µ p action on X induced by D . Then π is a purely inseparable morphism of degree p . Let ˜ C = π ( C ) ⊂ Y .Then C = π ∗ ˜ C and π ∗ C = p ˜ C [RS76]. Let P ∈ C be a singular point of C and Q = π ( P ) ∈ Y . If P is a fixed point of D then there is nothing to prove. Supposethat P is not a fixed point of D . Then Q ∈ Y is a smooth point of Y [AA86]. Hencelocally around Q ∈ Y , X → Y is an α p or µ p torsor and hence the same holds for C → ˜ C . Consider cases with respect to whether Q ∈ ˜ C is a singular or a smoothpoint of C . Case 1. Q ∈ ˜ C is singular. Then since P ∈ X is not a fixed point of D ,in suitable local analytic coordinates at P , O X = k [[ x, y ]], D = h ( x, y ) ∂/∂x and O Y = k [[ x p , y ]] [RS76, Theorem 1]. Then I ˜ C = ( f ( x p , y )) and since it is assumedthat Q ∈ ˜ C is singular, f ( x p , y ) ∈ ( x p , y ) . Then I C = ( f ( x p , y )) ⊂ k [[ x, y ]]. Write f ( x p , y ) = P i f i ( x p ) y i . Then either m P ( f ( x p , y )) ≥ p (considered in k [[ x, y ]]) orthere exists an m ≥ f m ( x p ) is a unit in k [[ x p ]].The first case is easily seen to be impossible since C is assumed to have arithmeticgenus less than p and a curve of arithmetic genus less than p cannot have a pointof multiplicity bigger than p .Suppose then that there exists an m ≥ f m ( x p ) is a unit in k [[ x p ]].By using the Weierstrass preparation theorem in k [[ x p , y ]] it follows that f ( x p , y ) = u ( x p , y )[ f ( x p ) + f ( x p ) y + · · · + f m − ( x p ) y m − + y m ] , where f i ( x p ) ∈ ( x p ), for all 0 ≤ m − u ( x p , y ) is a unit in k [[ x p , y ]] and hencealso in k [[ x, y ]]. In fact m ≥ Q ∈ ˜ C is singular. Then I C = ( y m + h ( x p , y )), where h ( x p , y ) = f ( x p ) + f ( x p ) y + · · · + f m − ( x p ) y m − ∈ ( x, y ) p +1 ⊂ k [[ x, y ]]and m ≥
2. Suppose that m ≥ p . Then m P ( C ) ≥ p and hence p a ( C ) ≥ p , whichis impossible since by assumption p a ( C ) ≤ ( p − /
2. Suppose that m < p . Thenwrite p = sm + r , 0 < r < m . After blowing up P ∈ C and its infinitely near ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 19 singular points s times we see by using the adjunction formula that2p a ( C ) ≥ sm ( m − . (4.7.1)Suppose that m ≥ ( p + 1) /
2. Then m − ≥ ( p − / a ( C ) ≥ (cid:16) sm (cid:17) (cid:18) p − (cid:19) ≥ p − , since m ≥ m < ( p +1) /
2. Then also r < m < ( p +1) /
2. Then p − r > ( p − / a ( C ) ≥ sm ( m −
1) = ( p − r ) m − ≥ (cid:18) p − (cid:19) (cid:18) m − (cid:19) . (4.7.2)Suppose that m ≥
3. Then from the above inequality it follows that p a ( C ) ≥ ( p − /
2. Suppose that m = 2. Then s = ( p − / r = 1. Then from theequation 4.7.2 it follows again that p a ( C ) ≥ ( p − / Case 2. Q ∈ ˜ C is smooth. Then C → ˜ C is a µ p or α p torsor. Hence O C = O ˜ C [ t ]( t p − s )where s ∈ O ˜ C . Let x be local analytic coordinate of ˜ C at Q . Then locally ana-lytically at Q ∈ ˜ C , O ˜ C = k [[ x ]] and s = f ( x ) ∈ k [[ x ]]. Moreover, since P ∈ C issingular, f ( x ) ∈ ( x ). Therefore O C = O ˜ C [ t ]( t p − s ) = k [[ x, t ]]( t p − f ( x )) . Then one can write f ( x ) = x m u ( x ), where u ( x ) is a unit in k [[ x ]]. If m < p then m p u ( x ) exists and therefore locally analytically at P , O C ∼ = k [[ x, y ]]( t p − x m ) . If p ≤ m then since k has characteristic p , the m p u ( x ) does not always exist.But in this case m P ( O C,P ) ≥ p which is impossible since p a ( C ) < p . Hence I C = ( t p − x m ), m ≥
2. Then by using the same argument as in Case 1 it followsthat p a ( C ) ≥ ( p − /
2, which is impossible.Hence every singular point of C is a fixed point of D . Hence D lifts to a vectorfield D ′ on the blow up X ′ of X at any singular point of C . Let C ′ be the birationaltransform of C in X ′ . Then D ′ ( I C ′ ) ⊂ I C ′ and p a ( C ′ ) < p a ( C ). Hence D ′ restrictsto a vector field of C ′ . Moreover, the previous arguments imply that the singularpoints of C ′ are fixed points of D ′ . Hence this process can continue until a birationalmap f : Y → X is reached such that Y and the birational transform ¯ C = f − ∗ C aresmooth and D lifts to a vector field ¯ D on Y such that ¯ D ( I ¯ C ) ⊂ I ¯ C and hence itinduces a vector field on ¯ C lifting D . (cid:3) Corollary 4.8.
With assumptions as in Proposition 4.7. Suppose in addition that C is singular. Let D c be the vector field on C induced by D . Suppose that D c = 0 .Let ¯ C → C be the normalization of C . Then ¯ C ∼ = P k . Moreover(1) Suppose that D p = 0 . Then D has exactly one fixed point on C . (2) Suppose that D p = D . Then D has at most two distinct fixed points on C .In particular, C is rational.Proof. By Proposition 4.7, D fixes the singular points of C and the restriction D c of D on C lifts to a vector field ¯ D on the normalization π : ¯ C → C of C .Considering that smooth curves of arithmetic genus greater or equal than 2 do nothave nontrivial global vector fields, it follows that p a ( ¯ C ) ≤ C is an elliptic curve. In this case T ¯ C = O ¯ C and hence the uniqueglobal vector field of ¯ C has no fixed points. Let P ∈ C be a singular point of C .Then by Proposition 4.7, P is a fixed point of D . Let also π − ( P ) = P ni =1 m i Q i ,be the divisor in ¯ C corresponding to π − ( P ). Then since p a ( C ) < ( p − /
2, itfollows that m i < p , for all i = 1 , . . . , m . Then by Lemma 3.15 it follows that every Q i , i = 1 , . . . , n , is a fixed point of ¯ D . This a contradiction since ¯ D has no fixedpoints.Hence ¯ C = P . In this case T ¯ C = ω − P = O P (2). Hence P has three linearlyindependent global vector fields D i , i = 1 , ,
3. These vector fields are induced fromthe homogeneous vector fields D = x ∂∂x , D = x ∂∂y and D = y ∂∂x of k [ x, y ]. Notethat D p = D and D pi = 0, i = 2 ,
3. Hence there are a i ∈ k , i = 1 , ,
3, such that¯ D = a D + a D + a D . Claim: ¯ D p = ¯ D if and only if a = a = 0 and a ∈ F ∗ p , and ¯ D p = 0 if and onlyif a + 4 a a = 0.In order to show this restrict ¯ D to the standard affine cover of P .Let U ⊂ P be the open affine subset given by y = 0. Let u = x/y . Then aneasy calculation shows that D = u ddu , D = − u ddu and D = ddu . Therefore¯ D = ( − a u + a u + a ) ddu in U . I will now show that this is additive if and only if − a u + a u + a = 0 haseither a double root or no roots and multiplicative if and only if a = 0 and a ∈ F p .Suppose that the previous equation has a double root, and hence a + 4 a a = 0.Then after a linear automorphism of k [ u ], ¯ D = au ddu , a ∈ k . This can easilyverified to be additive. Suppose on the other hand that − a u + a u + a = 0has either two distinct roots or only one simple root (hence a = 0). Suppose that a = 0 and hence it has two distinct roots. Then after a linear automorphism of k [ u ], ¯ D = a ( u + u ) ddu . Then an easy calculation shows that D p ( u p − ) = a p ( p − p ( u p + u p − ) = − a p ( u p + u p − ) = 0 . Hence in this case ¯ D is neither additive or multiplicative. Hence a = 0 and¯ D = ( a u + a ) ddu . Then ¯ D p = a p − ¯ D . Hence ¯ D p = ¯ D if and only if a p − = 1 andtherefore if and only if a ∈ F p .Let V be the affine open subset of P given by x = 0. Let v = y/x . Then in V , D = − v ddv , D = ddv and D = − v ddv . Therefore¯ D = ( − a v − a v + a ) ddv . Suppose that ¯ D is additive. Then similar arguments as before show that a +4 a a = 0. Suppose that ¯ D is of multiplicative type. Then as before we get that a = 0. This concludes the proof of the claim.Suppose now that ¯ D is of multiplicative type. Then it has been shown that¯ D = ax ∂∂x , a ∈ F ∗ p . The fixed points of this are [0 ,
1] and [1 , ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 21 exactly two distinct fixed points. These points may be over different points of C orover the same. Hence D has at most 2 fixed points on C as claimed.Suppose that ¯ D is of additive type. Then from the previous arguments it followsthat ¯ D has a single fixed point.Hence if D p = D , then D has at most two distinct points and if D p = 0 then ithas just one. (cid:3) Proposition 4.9.
Let X be a canonically polarized surface over an algebraicallyclosed field of characteristic p > . Let D be a nonzero vector field on X such thateither D p = 0 or D p = D . Assume moreover that D fixes the singular points of X and that it lifts to the minimal resolution of X . Suppose that p > ( m +3 m ) K X +3 .Then the linear system | mK X | does not contain a positive dimensional subsystemwhose members are stabilized by D .Proof. Suppose that there exists a positive dimensional linear subsystem of | mK X | ,for some m >
0, whose members are stabilized by D . Then take | V | ⊂ | mK X | aone-dimensional linear subsystem whose members are stabilized by D . Claim:
Let C ∈ | V | be any member of | V | and let C = P si =1 n i C i be itsdecomposition into its reduced and irreducible components. Then, if C i is not acomponent of the divisorial part of D , C i is a rational curve, for all i = 1 , . . . , s .Indeed. From the assumptions of the proposition it follows that K X · C < p .Then, since K X is ample, it follows by Corollary 4.3 that every C i is stabilized by D , i.e., D ( I C i ) ⊂ I C i , i = 1 , . . . , n . Hence D induces vector fields on every C i , forall i .Suppose that C i is a component of C which is not contained in the divisorialpart of D . Then the restriction of D on C i is not zero. Let π i : ¯ C i → C i be thenormalization of C i . I will show next that D lifts to ¯ C i .Let f : X ′ → X be the minimal resolution of X . Let C ′ i be the birationaltransform of C i in X ′ . Then C ′ i is stabilized by D ′ and therefore D ′ induces anonzero vector field on C ′ i . In order to show that D lifts to ¯ C i it suffices to showthat D ′ lifts to the normalization of C ′ i , which is ¯ C i . This will be done by usingProposition 4.7.Since X has canonical singularities, K X ′ = f ∗ K X . Then, since K X is ample, K X ′ · C ′ i = f ∗ K X · C ′ i = K X · C i ≤ K X · C = mK X < p − m + 3(4.9.1)by the assumptions of the proposition. Moreover, since K X ′ is nef and big, by theHodge Index Theorem and the previous inequality, it follows that( C ′ i ) ≤ ( K X ′ · C ′ i ) K X ′ ≤ m ( K X ) K X = m K X ≤ mm + 3 ( p − . (4.9.2)Now from the equations (4.9.1), (4.9.2) it follows that(4.9.3) p a ( C ′ i ) = 1 + 12 (( C ′ i ) + K ′ X · C ′ i ) < · m + 1 m + 3 ( p − < p − , Therefore, by Proposition 4.7, D ′ lifts to the normalization of C ′ i and hence D liftsto a vector field ¯ D on the normalization ¯ C i of C i . Considering that a smooth curveof genus greater or equal to 2 does not have any nontrivial global vector fields, itfollows that ¯ C i is either P or an elliptic curve. I will show that it is actually P .Next I will show that there exist fixed points of D ′ on C ′ i .Consider cases with respect to whether C ′ = f ∗ C is reducible or not. Suppose that C ′ is irreducible (and hence C does not pass through any singularpoint of X ). Then C ′ = n i C ′ i . In particular, m K X = n i ( C ′ i ) = n i C i . (4.9.4)Suppose that D ′ has no fixed points on C ′ i . Let π : X ′ → Y ′ be the quotient of X ′ by the α p or µ p action induced on X ′ by D ′ . Let ˆ C i = π ( C ′ i ). Then, by [AA86],ˆ C i is in the smooth part of Y ′ and by [RS76], π ∗ ˆ C i = C ′ i . Hence( C ′ i ) = ( π ∗ ˆ C i ) = p ˆ C i = λp, for some λ ∈ Z . Then from (4.9.4) it follows that m K X = λn i p . Since K X > λ > K X > p , which is a contradiction from the assumptions.Hence in this case there are fixed points of D ′ on C ′ i .Suppose that C ′ has at least two components. Since K X is ample, C and hence C ′ is connected. Hence C ′ i intersects another component B of C ′ . If B is containedin the divisorial part of D ′ then the intersection points C ′ i ∩ B are fixed points of D ′ . Hence in this case there are fixed points of D ′ on C ′ i . Suppose that B is not inthe divisorial part of D ′ . There are now two possibilities. B is not f -exceptionalor B is f -exceptional.Suppose that B is not f -exceptional. Then B = C ′ j , the birational transform in X ′ of a component C j of C , j = i . But now from the equation (4.4.7) in the proofof Proposition 4.4 it follows that C ′ i · C ′ j ≤ C i · C j < p. Then from Corollary 4.5 it follows that the points of intersection C ′ i ∩ C ′ j are fixedpoints of D ′ . Again then there are fixed points of D ′ on C ′ i .Suppose finally that B is f -exceptional. I will show that B · C ′ i < p and henceagain from Corollary 4.5 the points of intersection C ′ i ∩ B are fixed points of D ′ .From the adjunction formula for C ′ i it follows that( C ′ i ) ≥ − − K X ′ · C ′ i = − − K X · C i ≥ − − n i K X · C = − − mn i K X . Then C ′ = f ∗ C = s X r =1 n s C ′ s + bB + E, where b > E is an effective f -exceptional divisor. Then m K X ≥ C · C i = f ∗ C · C ′ i ≥ n i ( C ′ i ) + bB · C ′ i ≥ − n i − mK X + bB · C ′ i Therefore bB · C ′ i ≤ m K X + 2 n i + mK X (4.9.5)Now since C ∈ | mK X | and K X is ample it follows that n i ≤ mK X . Then theprevious equation becomes bB · C ′ i ≤ m K X + 2 mK X + mK X = ( m + 3 m ) K X < p, by the assumptions. Hence bB · C ′ i < p , and in particular B · C ′ i < p . Therefore,from Corollary 4.5 the points of intersection C ′ i ∩ B are fixed points of D ′ .Therefore there are fixed points of D ′ on C ′ i . Let P ∈ C ′ i be a fixed point of D ′ . Let π − ( P ) = P mi =1 n i Q i . Then since p a ( C ′ i ) < p , it follows that n i < p , i = 1 , . . . , m . Then by Lemma 3.15, every Q i is a fixed point of ¯ D i . Hence ¯ D i has ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 23 fixed points. Therefore ¯ C i ∼ = P since vector fields on an elliptic curve do not havefixed points. This concludes the proof of the claim.Let | V ′ | be the linear system which is obtained from | V | by removing the basecomponents. Hence | V ′ | has only isolated base points. Let φ : X P be therational map defined by | V ′ | . Consider now the following commutative diagram W h / / ψ ❇❇❇❇❇❇❇❇ g (cid:15) (cid:15) B σ (cid:15) (cid:15) X φ / / ❴❴❴ P Where g is the resolution of base points of | V ′ | , ψ the corresponding morphism,and h , σ is the Stein factorization of ψ . Then h is a fibration and its generic fiberis an integral normal (and hence regular) curve [Ba01, Page 91]. Moreover, by theconstruction of h , the general fiber is the birational transform in W of an irreduciblecomponent of a general member of | V ′ | . Therefore it is a rational curve.Suppose that the general fiber of h is smooth. Therefore the general fiber of h isisomorphic to P . Then the generic fiber is also a smooth curve of genus zero over K ( B ), where K ( B ) is the function field of B . Hence it is isomorphic to a smoothconic in P K ( B ) . Then by Tsen’s Theorem this conic has a K ( B )-point and thereforethe generic fiber is actually isomorphic to P K ( B ) . Therefore, X , and hence X ′ , isbirational to B × P , i.e., is birationally ruled. But then this implies that X ′ hasKodaira dimension −
1, which is a contradiction.Hence every fiber of h is singular and therefore the generic fiber is singular too.Then by Tate’s Theorem [Ta52], [Sch09], ( p − / < p a ( W g ), where W g is thegeneral fiber of h . But since the general fiber of h is the birational transform ofa component C i of a general member C of | V ′ | , it follows from the equation 4.9.3that p a ( W g ) ≤ p a ( C ) ≤ m + m ) K X < ( p − / , a contradiction. Hence | mK X | contains at most finitely many integral curves of D . (cid:3) Corollary 4.10.
Let X be a canonically polarized surface over an algebraicallyclosed field of characteristic p > . Let D be a nontrivial global vector field on X such that D p = 0 or D p = D . Suppose that(1) p > max { , m + 3 m + 3 } , if K X = 1 ,(2) p > max { χ ( O X ) + 11 K X + 1 , ( m + 3 m ) K X + 3 } , if K X ≥ .Then the linear system | mK X | does not contain a positive dimensional subsystemwhose members are stabilized by D .Moreover, suppose that D has only isolated singularities. Let π : X → Y be thequotient of X by the α p or µ p action induced by D . Then h ( O Y ( mK Y )) ≤ .Proof. From Theorem 3.3 it follows that D lifts to the minimal resolution of X .Then from Proposition 4.9 it follows that | mK X | does not contain a positive di-mensional subsystem whose members are stabilized by D .Suppose now that D has only isolated singularities. Then K X = π ∗ K Y . If h ( O Y ( mK Y )) ≥
2, then | π ∗ ( mK Y ) | gives a positive dimensional subsystem of | mK X | which consists of integral curves of D . But by Proposition 4.9 this is im-possible. (cid:3) The next two results will also be needed in the proofs of the main results of thispaper.
Proposition 4.11.
Let f : Y → X be a composition of n blow ups starting from asmooth point P ∈ X of a surface X . Let C ⊂ X be an integral curve in X passingthrough P and let m = m Q ( C ) be the multiplicity of C at P ∈ C . Then mK Y − f ∗ C + C ′ = mf ∗ K X + n X k =1 ( km − a − a − . . . − a k ) E k , where E i , ≤ i ≤ n are the f -exceptional curves, C ′ is the birational transform of C in Y and ≤ a i ≤ m , are nonnegative integers. The proof of the proposition is by a simple induction on the number of blow ups n and is omitted. Proposition 4.12.
Let P ∈ S be a Duval singularity and let C ⊂ S be a smoothcurve such that P ∈ S . Let f : S ′ → S be the minimal resolution of P ∈ S , and E i , i = 1 , . . . , n be the f -exceptional curves. Let C ′ be the birational transform of C in S ′ and a i > , ≤ i ≤ n be positive rational numbers such that f ∗ C = C ′ + n X i =1 a i E i . Then(1) Suppose that P ∈ S is of type A n . Then ( n + 1) C is Cartier in S and ( n + 1) a i are positive integers ≤ n , i = 1 , . . . , n .(2) Suppose that P ∈ S is of type D n . Then C is Cartier in S and a i areintegers ≤ n , i = 1 , . . . , n .(3) Suppose that P ∈ S is of type E . Then C is Cartier in S and a i areintegers ≤ , i = 1 , . . . , .(4) Suppose that P ∈ S is of type E . Then C is Cartier in S and a i areintegers ≤ , i = 1 , . . . , . Notice that P ∈ S cannot be of type E because this singularity is factorial andhence there is no smooth curve passing through it.The proof of this proposition is by a straightforward computation of the coeffi-cients a i in f ∗ C depending on the type of the singularity and the position of C ′ inthe dual graph of the exceptional locus of the singularity and it is omitted. Similarcomputations can be found in [Tz03, Proposition 4.5].5. Methodology of the proof of Theorems 1.1, 1.2.
Let X be a canonically polarized surface defined over an algebraically closed fieldof characteristic p > D . The strategy for theproof of Theorems 1.1, 1.2 is to do one of the following:(1) Find an integral curve C of D on X with the following properties: Itsarithmetic genus p a ( C ) is a function of K X , p a ( ¯ C ) ≥
1, where ¯ C is thenormalization of C , and such that C contains some of the fixed pointsof D . Then by using the results of Section 4, if p a ( C ) is small enoughcompared to the characteristic p , D induces a vector field on C which liftsto ¯ C . But this would be impossible since smooth curves of genus greateror equal than two have no nontrivial global vector fields and global vector ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 25 fields on smooth elliptic curves do not have fixed points. This argumentwill allow us to conclude that if p > f ( K X ), for some function f ( K X ) of K X then X does not have any nontrivial global vector fields.(2) Find a positive dimensional family of integral curves { C t } of D whose arith-metic genus is a function of K X and χ ( O X ). Then from Corollary 4.10 theremust be a relation of the form p < f ( K X , χ ( O X )). Hence if such a relationdoes not hold, X does not have any nontrivial global vector fields.In order to achieve this, the following method will be used. It is based on amethod initially used in [RS76] and then in [Tz17a] but with different objectives.Since X has a nontrivial global vector field, then by [Tz17a, Proposition 4.1] X has a nontrivial global vector field D of either additive or multiplicative type whichinduces a nontrivial α p or µ p action. Let π : X → Y be the quotient. Then π ispurely inseparable of degree p , Y is normal and K Y is Q -Cartier. Consider now thefollowing diagram(5.0.1) Y ′ h (cid:15) (cid:15) g ❆❆❆❆❆❆❆❆ X π (cid:15) (cid:15) Z Y where g : Y ′ → Y is the minimal resolution of Y and h : Y ′ → Z its minimal model. Lemma 5.1.
Every g -exceptional curve is a rational curve (perhaps singular).Proof. let ˆ X be the normalization of Y ′ in K ( X ). Let φ : W → ˆ X be the minimalresolution of ˆ X . Then there exists a commutative diagram W φ / / ψ (cid:15) (cid:15) ˆ X ˆ π / / ˆ g (cid:15) (cid:15) Y ′ g (cid:15) (cid:15) X ′ f / / X π / / Y where ˆ π is purely inseparable of degree p , f : X ′ → X is the minimal resolutionof X and ψ is birational. Considering that X has rational double points, the f exceptional curves are smooth rational curves. Therefore, since ψ is a compositionof blow ups, it easily follows that every ˆ g -exceptional curve is a rational curve. Nowlet F be a g -exceptional curve. Then F = ˆ π ( ˆ F ), where ˆ F is a ˆ g -exceptional curve.Hence, F is a rational curve. (cid:3) Integral curves on X will be found by choosing a suitable a reflexive sheaf L on Y such that either h ( L ) ≥
2, in which case the pullbacks in X of the divisors of Y corresponding to the sections of L will be integral curves of D , or h (( π ∗ L ) [1] ) ≥ D on H (( π ∗ L ) [1] ) exhibited in Proposition 4.1.The eigenvectors of this action will be curves stabilized by D and under suitableconditions their components which are not contained in the divisorial part of D willbe integral curves of D .In order to prove Theorems 1.1, 1.2 we will distinguish cases with respect to theKodaira dimension κ ( Z ) of Z . Then results from the classification of surfaces inpositive characteristic will be heavily used [BM76], [BM77], [Ek88] and the geometry o X and Z will be compared by using the diagram (5.0.1). Moreover, since π is apurely inseparable map, it induces an equivalence between the ´etale sites of X and Y . Therefore X and Y have the same algebraic fundamental group, l -adic bettinumbers and ´etale Euler characteristic. Then by using the fact that g and h arebirational it will be possible to calculate the algebraic fundamental group, l -adicBetti numbers and ´etale Euler characteristic of X from those of Z .The proof of Theorems 1.1, 1.2 is significantly easier if the vector field D has anontrivial divisorial part as the next theorem shows. Theorem 5.2. [Tz17a, Theorem 6.1]
Suppose that D has a nontrivial divisorialpart. Suppose that K X < p . Then the Kodaira dimension of Z is − and X ispurely inseparably uniruled. Finally I collect some formulas and set up some terminology and notation thatwill be needed in the proofs.Let ∆ be the divisorial part of D . There is also the following adjunction formulafor purely inseparable maps [RS76, Corollary 1] K X = π ∗ K Y + ( p − . (5.2.1)(According to [RS76], the previous formula holds in the smooth part of X andhence everywhere since X is normal).Let F i , i = 1 , . . . , n be the g -exceptional curves and E j , j = 1 , . . . , m be the h -exceptional curves. By Lemma 5.1 the g -exceptional curves F i are all rational(but perhaps singular).Taking into consideration that g : Y ′ → Y is the minimal resolution of Y , we getthe following adjunction formulas K Y ′ + n X i =1 a i F i = g ∗ K Y , (5.2.2) K Y ′ = h ∗ K Z + m X j =1 b j E j , where a i ∈ Q ≥ , and b j ∈ Z > , j = 1 , . . . m . Moreover since both Y ′ and Z aresmooth, h is the composition of m blow ups.In the next sections I will consider cases with respect to the Kodaira dimension κ ( Z ) of Z .Finally, for the rest of the paper, fix the notation of this section.6. The Kodaira dimension of Z is 1 or 2. Proposition 6.1.
Let X be a canonically polarized surface over a field of charac-teristic p > . Suppose that X has a nontrivial global vector field D with isolatedsingularities such that D p = 0 or D p = D . Suppose moreover, with notation as inSection 5, that the Kodaira dimension κ ( Z ) of Z is 1 or 2. then(1) Suppose that K X = 1 . Then p < .(2) Suppose that K X ≥ . Then p < K X + 3 . Proof.
Suppose that the statements of the proposition are not true, i.e., p ≥ K X = 1 and that p > K X + 3, if K X ≥
2. Then, I will sow that also p > χ ( O X ) + 11 K X + 1 and therefore from Theorem 3.3 D fixes the singularpoints of X and lifts to a vector field D ′ in the minimal resolution of f . ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 27
Indeed. Let f : X ′ → X be the minimal resolution of X . Since X has canonicalsingularities, χ ( O X ) = χ ( O X ′ ) and K X ′ = f ∗ K X . X ′ is a minimal surface ofgeneral type. Therefore from Noether’s inequality, 2 χ ( O X ′ ) ≤ K X ′ + 6. Hence,since K X = K X ′ , it follows that 2 χ ( O X ) ≤ K X + 6. Hence12 χ ( O X ) + 11 K X + 1 ≤ K X + 37 < K X + 3 < p, (6.1.1)by the assumption.Now by Theorem 5.2, D has no divisorial part, i.e., ∆ = 0. Therefore, K X = π ∗ K Y and hence K Y is ample.Consider cases with respect to the Kodaira dimension κ ( Z ) of Z . Case 1: Suppose that κ ( Z ) = 2 . According to [Ek88, Theorem 1.20], the linear system | K Z | is very ample. Let W ∈ | K Z | be a smooth member which does not go through the points blown upby h in the diagram 5.0.1. Then by the adjunction formula, p a ( W ) = 10 K Z + 1.Then combining the equations 5.2.2 it follows that g ∗ (4 K Y ) = 4 K Y ′ + 4 n X i =1 a i F i = h ∗ (4 K Z ) + 4 m X j =1 b j E j + 4 n X i =1 a i F i ∼ (6.1.2) W ′ + 4 m X j =1 b j E j + 4 n X i =1 a i F i , (6.1.3)where W ′ = h ∗ W = h − ∗ W is the birational transform of W in Y ′ . By pushingdown to Y we get that 4 K Y ∼ ˜ W + 4 m X i =1 b i ˜ E i , (6.1.4)where ˜ E i = g ∗ E i , 1 ≤ i ≤ m . Note that since Y ′ is the minimal resolution of Y , g does not contract any (-1) h -exceptional curves. Hence if h is not an isomorphismthen g ∗ P mi =1 E i = 0. Now since | K Z | is very ample it follows that dim | ˜ W | ≥ | K Y | ≥
1, or equivalently h ( O Y (4 K Y )) ≥
2. But by Corollary 4.10this is impossible.
Case 2: Suppose that κ ( Z ) = 1 . Since κ ( Z ) = 1, it is well known that Z admits an elliptic fibration φ : Z → B ,where B is a smooth curve. Then one can write R φ ∗ O Z = L ⊕ T, (6.1.5)where L is an invertible sheaf on B and T is a torsion sheaf. Claim: B ∼ = P and T = 0.By Lemma 5.1, the g -exceptional curves are rational. Hence if at least one ofthem is not contracted to a point by φ ◦ h , then B is dominated by a rationalcurve and hence it is isomorphic to P . Suppose that every g -exceptional curve iscontracted to a point by φ ◦ h . Then by looking at diagram 5.0.1 we see that thereexists factorizations Y ψ ❅❅❅❅❅❅❅❅ X π > > ⑦⑦⑦⑦⑦⑦⑦⑦ σ / / B such that the general fiber of ψ is an elliptic curve. Then let Y b = ψ − ( b ) be thegeneral fiber. Then K Y · Y b = 0 and therefore, K X · π ∗ Y b = π ∗ K Y · π ∗ Y b = pK Y · Y b = 0 . But this is impossible since K X is ample. Therefore there must be a g -exceptionalcurve not contracted to a point by φ ◦ h and hence B ∼ = P .Suppose now that T = 0. Let b ∈ T . Then φ − ( b ) = pmW , m > W isan idecomposable fiber [KU85]. Moreover | KZ | defines the fibration φ [KU85].Hence 14 K Z ∼ νF , where F is a general fiber of φ and hence a smooth ellipticcurve (if p = 2 , F ∼ φ − ( b ) = pmW . and hence 14 K Z ∼ pmνW . Thenby pulling up to Y ′ it follows that14 h ∗ K Z = pmνW ′ + p ( m X i =1 c i E i ) . If h blows up a point of W then c i > h ∗ K Z has a component correspondingto a ( − h -exceptional curve with coefficient divisible by p . Considering that the( − h -exceptional curves do not contract by g , we see that in any case (if h blowsup a point on W or not) that, after pushing down to Y , 14 K Y ∼ p ˜ W + B , for somedivisor ˜ W (either the birational transform of W or the image of a − h -exceptionalcurve. Therefore by pulling up to X and since K X = π ∗ K Y ,14 K X ∼ pπ ∗ ˜ W + π ∗ B. But from this it follows that 14 K X > p , a contradiction. This concludes the proofof the claim.Next consider cases with respect to p g ( Z ). Case 1.
Suppose that p g ( Z ) ≥
2. Then, since h ( O Z ( K Z )) ≥
2, it easily followsthat h ( O Y ( K Y )) ≥
2. Then by Corollary 4.10 we get a contradiction. So this caseis impossible too.
Case 2.
Suppose that p g ( Z ) ≤
1. I will show that this case is impossible too.From the Noether’s formula on Z [Ba01, Theorem 5.1]10 − h ( O Z ) + 12 p g ( Z ) = K Z + b ( Z ) + 2(2 h ( O Z ) − b ( Z )) =(6.1.6) b ( Z ) + 2(2 h ( O Z ) − b ( Z ))it easily follows [Ba01, Page 113] that if p g ( Z ) ≤
1, then the only numerical solutionsto the equation 6.1.6 are the following:(1) p g ( Z ) = 0, χ ( O Z ) = 0, b ( Z ) = 2.(2) p g ( Z ) = 0, χ ( O Z ) = 1, b ( Z ) = 0.(3) p g ( Z ) = 1, χ ( O Z ) = 2, b ( Z ) = 0.(4) p g ( Z ) = 1, χ ( O Z ) = 1, b ( Z ) = 2.(5) p g ( Z ) = 1, χ ( O Z ) = 1, b ( Z ) = 0.(6) p g ( Z ) = 1, χ ( O Z ) = 0, b ( Z ) = 2.(7) p g ( Z ) = 1, χ ( O Z ) = 0, b ( Z ) = 4.Note that by [KU85, Lemma 3.5] the last case is not possible. Consider next eachone of the cases separately. I will only consider the first two cases. The rest aresimilar and are omitted. Case 2.1.
Suppose that p g ( Z ) = χ ( O Z ) = 0 and b ( Z ) = 2.By Igusa’s formula [IG60] it follows that the fibers of φ : Z → P are eithersmooth elliptic curves or of type mE , where m is a positive integer and E an ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 29 elliptic curve (singular or smooth). Also note that φ must have multiple fibers orelse Z cannot have Kodaira dimension 1.I will next show that in fact E is a smooth elliptic curve. Indeed. Since b ( Z ) = 2it follows that dim Alb( Z ) = 1. Hence Alb( Z ) is a smooth elliptic curve. Let then ψ : Z → Alb( Z ) be the Albanese map. Then there exist the following two maps Z ψ / / φ (cid:15) (cid:15) Alb( Z ) P Suppose that mE is a multiple fiber of φ . Suppose also that E is a rational ellipticcurve. Then E cannot dominate Alb( Z ) and hence it must contract by ψ . Hence allfibers of φ contract by ψ . But then there would be a nontrivial map P → Alb( Z ),which is impossible. Hence E is a smooth elliptic curve.It is well known [Ba01, Theorem 8.11] that the linear system | νK Z | , ν ∈ { , } contains a strictly positive divisor. Then νK Z ∼ sE , where s > E is a smooth elliptic curve. Let E ′ = h − ∗ E be the birational transformof E in Y ′ . Then E ′ is a smooth elliptic curve and since the g -exceptional curvesare all rational, it follows that E ′ does not contract by g . Therefore by pulling upto Y ′ and then pushing down to Y we get that νK Y ∼ m ˜ E + B, (6.1.7)where B is an effective divisor on Y . Hence by pulling up to X we get that νK X ∼ m ˆ E + π ∗ B. (6.1.8)As in the previous cases we see that if K X < p/ν , ˆ E is irreducible and thereforeis an integral curve of D whose normalization ¯ E is a smooth elliptic curve. I willshow that D lifts to a vector field ¯ D on ¯ E and that D has fixed points on ˆ E . Thenby Lemma 3.15, ¯ D will have fixed points which is impossible since ¯ E is an ellipticcurve and hence get a contradiction again.Let now f : X ′ → X be the minimal resolution of X . Then K X ′ = f ∗ K X andtherefore νK X ′ ∼ mE ′′ + f ∗ π ∗ B + F, (6.1.9)where E ′′ is the birational transform of ˆ E in X ′ and F is an effective f -exceptionaldivisor. Now from the equation (6.1.9), since K X ′ is nef and big, we get that K X ′ · E ′′ < νK X ′ = νK X . (6.1.10)and then from the Hodge Index Theorem it follows that that( E ′′ ) < ( K X ′ · E ′′ ) K X ′ < ν K X . Therefore from the adjunction formula it follows thatp a ( E ′′ ) < ν ( ν + 1)2 K X + 1 . Hence if K X < p − · ν ( ν + 1) , then p a ( E ′′ ) < ( p − /
2. Considering that ν ∈ { , } , the above inequality holdsif K X < ( p − /
42, which holds according by the assumptions. Also, since ˆ E is anintegral curve of D , E ′′ is an integral curve of D ′ , the lifting of D to X ′ . Thereforein this case, from Proposition 4.7 it follows that the restriction of D ′ on E ′′ fixesthe singular points of E ′′ and hence lifts to its normalization ¯ E of E ′′ .Next I will show that D ′ has fixed points on E ′′ .Suppose that D has no fixed points on ˆ E . Then ˆ E is in the smooth part of X since the singular points of X are fixed points of D . Moreover, since D has no fixedpoints on ˆ E , ˜ E = π ( ˆ E ) is in the smooth part of Y . Then K X · ˆ E = π ∗ K Y · π ∗ ˜ E = p ( K Y · ˜ E ) = λp, where, since K Y is ample, λ is a positive integer. But then from the equation(6.1.7) it follows that νK X > p , which is impossible. Therefore, there are fixedpoints of D on ˆ E . Let P ∈ ˆ E be a point which is a fixed point of D . Suppose that P ∈ X is a smooth point. Then Q = f − ( P ) is a fixed point of E ′′ . Suppose that P ∈ X is singular. Let then F be an f -exceptional curve such that F · E ′′ >
0. ByTheorem 3.3, F is an integral curve of D ′ . I will show that F · E ′′ < p and henceby Corollary 4.5, the intersection points F ∩ E ′′ are fixed points of D ′ . Write f ∗ ˆ E = E ′′ + aF + F ′ , where F ′ is f -exceptional and effective. Then by Lemma 3.13, and (6.1.8), it followsthat ˆ E < ν K X and hence ν K X > ˆ E ≥ ( E ′′ ) + a ( F · E ′′ ) . Considering now that from (6.1.10),( E ′′ ) ≥ − − K X ′ · E ′′ ≥ − − νK X We get that a ( F · E ′′ ) ≤ ν + ν ) K X , (6.1.11)and therefore F · E ′′ < p if 2 + ( ν + ν ) K X < p , which holds if 42 K X + 2 < p ( ν = 4 or ν = 6). Hence the intersection points E ′′ ∩ F are fixed points of D ′ .Hence in any case there are fixed points of D ′ on E ′′ . Then from Lemma 3.15, thepreimages of these points in ¯ D are fixed points of the lifting of D ′ on ¯ D , which isa contradiction since a vector field on an elliptic curve has no fixed points. Case 2.2.
Suppose that p g ( Z ) = 0, χ ( O Z ) = 1, b ( Z ) = 0. I will show that thiscase is also impossible. Claim: dim | K Z | ≥ F t i = m i P i , t i ∈ P , i = 1 , . . . , r be the multiple fibers of φ . Since T = 0,they are all tame. Then by the canonical bundle formula [Ba01, Theorem 7.15 andPage 118] we get thatdim | nK Z | = n ( − χ ( O Z )) + r X i =1 (cid:20) n ( m i − m i (cid:21) = − n + r X i =1 (cid:20) n ( m i − m i (cid:21) , (6.1.12)where for any m ∈ N , [ m ] denotes its integer part. Also, in the notation [Ba01,Remark 8.3] if, λ ( φ ) = − r X i =1 m i − m i , ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 31
Then κ ( Z ) = 1 if and only if λ ( φ ) >
0. Hence φ has at least two multiple fibers.Suppose that φ has at least three multiple fibers, i.e., r ≥ m i ≥
2. Thenfor every 1 ≤ i ≤ r , (cid:20) − m i ) (cid:21) ≥ (cid:20) (cid:21) = 3 . Then from the equation 6.1.12 it follows that dim | K Z | ≥ − · φ has exactly two multiple fibers with multiplicities m and m .Then in order to have λ ( φ ) >
0, at least one of them must be greater or equal than3. Say m ≥ m ≥
2. Then from the equation 6.1.12 it follows thatdim | K Z | = − (cid:20) − m (cid:21) + (cid:20) − m (cid:21) ≥ − (cid:20) · (cid:21) + (cid:20) · (cid:21) = 1 . Hence 6 K Z ∼ mE , where m > E is a smooth ellipticcurve. By repeating now the argument used in Case 2.1 we see that this is impossibleif 42 K X + 3 < p . This concludes the study of the case when κ ( Z ) = 1. (cid:3) The Kodaira dimension of Z is 0. Fix the notation as in Section 4. The main result of this section is the following.
Proposition 7.1.
Let X be a canonically polarized surface defined over an alge-braically closed field of characteristic p > . Suppose that X admits a nontrivialglobal vector field D such that D p = 0 or D p = D . Suppose that Z has Kodairadimension zero. Then p < max { K X ) + 12( K X ) + 3 , K X + 3 } . Moreover, suppose that D p = D . Then(1) Suppose that K X = 1 . Then p < .(2) Suppose that K X ≥ . Then p < K X + 3 . Proof.
I will only do the case when K X ≥
2. The case when K X = 1 is identicaland is omitted. Then only difference between the two cases is that in the Case 3.1below, where the case when D p = D is studied, if K X = 1 then | K X | is base pointfree while if K X ≥ | K X | is base point free [Ek88]. So in the case K X = 1,onehas to work with the linear system | K X | instead.From now on assume K X ≥
2. Suppose that the assumptions of the propositiondo not hold, in their respective cases. Then in particular, K X < p . Hence byTheorem 5.2, D has only isolated singularities, i.e., ∆ = 0. Therefore from theequation (5.2.1) it follows that K X = π ∗ K Y . Hence, since K X is ample, K Y isample as well. Moreover, Y is singular since if this was not true, then K X = pK Y ≥ p .Let f : X ′ → X be the minimal resolution of X . Then, as before, since X hascanonical singularities, K X = f ∗ K Y and therefore X is a minimal surface of generaltype. Moreover, from the equation (6.1.1) it follows that12 χ ( O X ) + 11 K X + 1 ≤ K X + 37 < K X + 3 < p. Hence by Theorem 3.3 every singular point of X is a fixed point of D , D lifts to avector field D ′ on X ′ and that every f -exceptional curve is stabilized by D ′ . According to the classification of surfaces [BM76], [BM77], Z is one of the fol-lowing: An abelian surface, a K3 surface, an Enriques surface or a hyperellipticsurface. Case 1: Suppose that Z is an abelian surface. Then every g -exceptionalcurve is also h -exceptional since by Lemma 5.1 every g -exceptional curve is rationaland there do not exist nontrivial maps from a rational curve to an abelian surface.Hence there exists a factorization Y ′ g / / φ (cid:15) (cid:15) Y θ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ Z (7.1.1)Let B j , j = 1 , . . . , r be the θ -exceptional curves. Then one can write K Y = θ ∗ K Z + r X j =1 γ j B j . But then, since { B j , ≤ j ≤ r } is a contractible set of curves, it easily follows that K Y = r X j =1 γ j B j ≤ , which is impossible since K Y is ample. Therefore Z cannot be an abelian surface. Case 2: Suppose that Z is a hyperelliptic surface. I will show that thiscase is also impossible. It is well known that if Z is hyperelliptic, then b ( Z ) =2 [BM77] and hence dim Alb( Z ) = 1. Then the morphism φ : Z → Alb( Z ) is anelliptic fibration [BM77]. Since every g -exceptional curve is rational, they must becontracted to points in Alb( Z ). Hence there exists a factorization Y ˜ ψ ❋❋❋❋❋❋❋❋❋ X π ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ ψ / / Alb( Z )The general fiber Y b of ˜ ψ is an elliptic curve. Hence K Y · Y b = 0. hence K X · π ∗ Y b = π ∗ K Y · π ∗ Y b = pK Y · Y b = 0 , which is impossible since K X is ample. Hence Z can be either a K3 surface or anEnriques surface. Case 3: Suppose that Z is a K surface. Consider now two cases withrespect to whether D is of multiplicative or additive type. Case 3.1.
Suppose that D is of multiplicative type, i.e., D p = D .By [Ek88, Theorem 1.20], | K X | is base point free. Also, since K X = π ∗ K Y , byProposition 4.1, there exists a k -linear map D ∗ : H ( O X (3 K X )) → H ( O X (3 K X )) . (7.1.2)Moreover, since D p = D , D ∗ is diagonalizable (with eigenvalues in the set { , , . . . , p − } ) and their eigenvectors correspond to integral curves of D . Let H ( O X (3 K X )) = ⊕ ki =1 V ( λ i ) , (7.1.3)the decomposition of H ( O X (3 K X )) in eigenspaces of D ∗ , where λ i ∈ F p , 1 ≤ i ≤ k . ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 33
Suppose that dim | K X | = m . Let Z i , i = 1 , . . . , m be a basis of | K X | corre-sponding to eigenvectors of D ∗ . Since K X is ample it follows from [Ha77, Corollary7.9] that Z i is connected for all i . Now since Z i are eigenvectors of D ∗ , Z i arestabilized by D and hence D induces nontrivial vector fields on each Z i . Moreover,if K X < p/
3, something which is true if the assumptions of the proposition hold,then from Corollary 4.3, D restricts to every reduced and irreducible component of Z i , for all 1 ≤ i ≤ m .Since Z is a K ω Z ∼ = O Z . Hence from the equations (5.2.2) it followsthat K Y = s X j =1 b j ˜ E j , (7.1.4)where ˜ E j is the birational transform in Y of the h -exceptional curves not contractedby g (note that such curves exist because if this was not the case then Y ′ = Z andhence since K Z = 0 it would follow that K Y = 0 which is impossible since K Y isample). In particular p g ( Y ) = 0 and hence p g ( X ) = 0. Let C = π ∗ ˜ E , where ˜ E isany irreducible component of K Y in the equation (7.1.4). Then, since K X < p , C is reduced and hence is an integral curve of D . Claim: D has at most two fixed points on C .Indeed. From the equation (7.1.4) it follows that K X · C = π ∗ K Y · π ∗ ˜ E = pK Y · ˜ E ≤ pK Y = K X . (7.1.5)Moreover, from Lemma 3.13, C ≤ K X . Let C ′ = f − ∗ C be the birational transformof C in X ′ . Then K X ′ · C ′ = f ∗ K X · C ′ = K X · C ≤ K X . Moreover, ( C ′ ) ≤ C ≤ K X = K X ′ . Therefore p a ( C ′ ) ≤ K X + 1. Then, theassumptions of the proposition imply that K X + 1 < ( p − /
2. Hence it followsfrom Corollary 4.8 that D ′ fixes the singular point of C ′ and lifts to its normalization¯ C . Suppose that C ′ is singular. Then by Corollary 4.8, ¯ C ∼ = P and D ′ has at mosttwo fixed points on C ′ . Suppose that C ′ is smooth. Then it must be either asmooth rational curve or an elliptic curve. In the first case D ′ has exactly two fixedpoints on C ′ . Suppose that C ′ is an elliptic curve. Then the map C ′ → C factorsthrough the normalization ˜ C → C . Therefore there exists a purely inseparable mapof degree p map C ′ → ˜ C of smooth curves. Moreover, since C is the pushforwardin Y of an h -exceptional curve, C is rational and hence ˜ C = P . Therefore thereexists a purely inseparable map of degree p , C ′ → P . But this implies that thereexists a map P → ( C ′ ) ( p ) , where C ′ → ( C ′ ) ( p ) is the k -linear Frobenius. But this isimpossible since ( C ′ ) ( p ) is also an elliptic curve. Therefore, C ′ cannot be an ellipticcurve and hence in any case D ′ has at most two fixed points on C ′ .Next I will show that this implies that D has at most two fixed points on C . Let P ∈ C be a fixed point of D .Suppose that P is a smooth point of X . Then Q = f − ( P ) is a fixed point of D ′ on C ′ .Suppose that P is a singular point of X . Let then E be an f -exceptional curvewhich intersects C ′ . By Theorem 3.3, E is stabilized by D ′ . Then by repeating wordby word the arguments that lead to the equation (6.1.11) we find that E · C ′ ≤ K X + 2 and hence the assumptions of the proposition imply that E · C ′ < p . Therefore, by Corollary 4.5, every point of intersection of E and C ′ is a fixed pointof D ′ on C ′ . Therefore D has at most as many fixed points on C as D ′ has on C ′ and hence at most 2. This concludes the proof of the claim.Therefore C is a rational curve and D has at most two fixed points on D . Let P , P be the fixed points of D on C , with the possibility that P = P .Let 1 ≤ i ≤ m be such that C is not an irreducible component of Z i . Since K X is ample, it follows that C · Z i >
0. For the same reason, Z i · Z j > Z i ∩ Z j = ∅ , for all 1 ≤ i, j ≤ m .Let now again Z i be a member of the basis of | K X | . Let A be an irreducibleand reduced component of Z i different from C such that C · A >
0. I will showthat every point of intersection of C and A is a fixed point of D . Indeed, from thedefinition of C and Z i , it follows that C · A ≤ C · Z i = 3 K X · C ≤ K X < p by the equation (7.1.5) and the assumptions of the proposition. Hence by Corol-lary 4.5, every point of intersection of A and C which is a smooth point of X is afixed point of D . The points of intersection of A and C which are singular pointsof X are fixed points of D always. Hence every point of intersection of C and A isa fixed point of D . In particular, every point of intersection of C and Z i is a fixedpoint of D (in the case C is not a component of Z i ).Suppose that P = P . Let 1 ≤ i ≤ m . Then either C is a component of Z i or( Z i ∩ C ) red = { P } , for all 1 ≤ i ≤ m . But this implies that P is a base pointof | K X | , which is impossible. Hence P = P . For the same reason, it is notpossible that either ( Z i ∩ C ) red = { P } , for all i or ( Z i ∩ C ) red = { P } , for all i .Therefore there exist indices 1 ≤ i = j ≤ m , such that ( Z i ∩ C ) red = { P } and( Z j ∩ C ) red = { P } . But then, since Z i ∩ Z j = ∅ and the curves Z i and Z j areconnected, the curve W = Z i + Z j + C contains loops. Let ˜ Z i = π ( Z i ), ˜ Z j = π ( Z j ).Then ˜ W = ˜ Z i + ˜ Z j + ˜ E is a curve whose reduced curve ˜ W red contains loops. Hencedim H ( O ˜ W red ) ≥ H ( O ˜ W ) ≥ Z is a K H ( O Z ) = 0 Hence H ( O Y ′ ) = 0and therefore from the Leray spectral sequence it follows that H ( O Y ) = 0. Thenfrom the exact sequence 0 → O Y ( − ˜ W ) → O Y → O ˜ W → · · · H ( O Y ) → H ( O ˜ W ) → H ( O Y ( − ˜ W )) → H ( O Y ) → H ( O ˜ W ) = 0 . Considering now that h ( O ˜ W ) ≥ H ( O Y ( − ˜ W )) = H ( O Y ( ˜ W + K Y )), H ( O Y ) = H ( O Y ( K Y )) and that p g ( Y ) = 0, it follows that h ( O Y ( ˜ W + K Y )) ≥ . (7.1.6)Now since π ∗ ( ˜ W + K Y ) = W + K X ∼ K X it follows that | K X | contains a posi-tive dimensional subsystem whose members are stabilized by D . Then by Proposi-tion 4.9, and the assumptions of the proposition, this is impossible. Hence Z cannotbe a K3 surface. Case 3.2.
Suppose that D is of additive type, i.e., D p = 0.The main idea in order to treat this case this is the following. I will show thatthere exists a ”small” positive number ν such that dim | νK Y | ≥ p large enough by Corollary 4.10. ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 35
The main steps of the proof are the following.Let F = P nj =1 F j be the reduced g -exceptional divisor. Then write F = F ′ + F ′′ ,where F ′ = P rj =1 F j , where F j , j = 1 , . . . , r are the g -exceptional curves which arenot h -exceptional, and F ′′ = P nj = r +1 F j are the g -exceptional curves that are also h -exceptional. Notice that F ′ = 0 because if that was the case then there would bea birational morphism ψ : Y → Z . Then by the adjunction formula, K Y = ψ ∗ K Z + ˜ F = ˜ F , since K Z = 0, where ˜ F is a ψ -exceptional divisor. Then K Y = ˜ F ≤
0, which isimpossible since K Y is ample.Then I will show that at least one of the following is true.(1) h ( O Y (2 K Y )) ≥ | K Y | ≥ B = P rj =1 n j F j , and a positive number ν such thateither ν ≤ K X or ν = 44, and such that dim | νK Y ′ + B | ≥
1. Moreover,the linear system | ˆ B | , where ˆ B = h ∗ B in Z is either base point free or itsmoving part is base point free. This implies that dim | νK Y | ≥ K X + 3 < p and 4 K X + 3 < p .Therefore, by Corollary 4.10, H ( O Y ( K Y )) = H ( O Y (2 K Y )) = k (7.1.7)and hence p g ( Y ) = 1.Next I will show that Y has rational singularities. Indeed. The Leray spectralsequence for g gives0 → H ( O Y ) → H ( O Y ′ ) → H ( R g ∗ O Y ′ ) → H ( O Y ) → H ( O Y ′ ) → H ( R g ∗ O Y ′ ) . Now since g is birational it follows that H ( R g ∗ O Y ′ ) = 0. Moreover, by Serreduality, H ( O Y ) ∼ = H ( O Y ( K Y )) = k and H ( O Y ′ ) ∼ = H ( O Y ′ ( K Y ′ )) = k and H ( O Y ′ ) = 0, since Z is a K H ( O Y ) = 0 and R g ∗ O Y ′ = 0. Therefore Y has rational singularities asclaimed. In particular, every g -exceptional curve is a smooth rational curve.Let ˆ F i = h ∗ F i , i = 1 , . . . , r , be the birational transforms of the F i in Z . Considernext cases with respect to whether the curves ˆ F i are either all smooth or there existsa singular one among them. Case 1.
Suppose that there exists an 1 ≤ i ≤ r such that ˆ F i is singular. Inthis case I will show that dim | ( K X ) K Y | ≥ g -exceptional curves we can assume that i = 1. Thenby the adjunction formulaˆ F = 2 p a ( ˆ F ) − − K Z · ˆ F = 2 p a ( ˆ F ) − ≥ . Hence the linear system | ˆ F | in Z is base point free [Hu16, Propositions 3.5, 3.10]. Claim 7.2.
Let Q ∈ ˆ F be a singular point of ˆ F and m = m Q ( ˆ F ) be the multi-plicity of the singularity. Then m Q ( ˆ F ) ≤ K X . (7.2.1) In order to prove the claim, observe the following. Over a neighborhood of anysingular point of ˆ F , F can meet at most two distinct h -exceptional curves E i and E j , and moreover it must intersect each one of them with multiplicity 1. Indeed.Suppose that F meets three distinct h -exceptional curves E i , E j and E s (overthe same point of Z ). Since h is a composition of blow ups, it follows that E i ∩ E j ∩ E s = ∅ . Hence the intersection of F and E i ∪ E j ∪ E s consists of at least twodistinct points, say P and Q . Up to a change of indices we can assume hat P ∈ E i and Q ∈ E j . Then the union Ex( h ) ∪ F , where Ex( h ) is the exceptional set of h ,contains a cycle. Therefore from the equations (5.2.2) it follows that K Y = n X j =1 b j ˜ E j , (7.2.2)where ˜ E j = g ∗ E j , j = 1 , . . . , n . Moreover if E j = −
1, then ˜ E j = 0. But then, if F meets at least two distinct h -exceptional curves, ∪ nj =1 ˜ E contains either a singularcurve or a cycle. In any case, if ˜ C = P nj =1 b j ˜ E j then H ( O ˜ C ) = 0. But then fromthe equation in cohomology H ( O Y ) → H ( O ˜ C ) → H ( O Y ( − ˜ C )) → H ( O Y ) → , and since H ( O Y ) = 0, H ( O Y ) = k , it follows that dim H ( O Y ( − ˜ C )) ≥ h ( O Y ( K Y + ˜ C )) = h ( O Y (2 K Y )) ≥ , a contradiction to the equations (7.1.7). Hence F meets at most two distinct h -exceptional curves. Suppose that F meets an h -exceptional curve E i and E i · F ≥
2. Then there are two possibilities. Either E i is also g -exceptional or it is not.Suppose that E i is g -exceptional. But this is impossible because Y has rationalsingularities and in such a case two g -exceptional curves cannot intersect withmultiplicity bigger than one. Suppose that E i is not g -exceptional. Then ˜ E i = g ∗ E i is singular and therefore h ( O ˜ E i ) ≥
1. But then h ( O ˜ C ) ≥ h ( O Y (2 K Y )) ≥
2, which is again a contradiction to theequations (7.1.7). Hence it has been shown that over a neighborhood of any singularpoint of ˜ F , F meets at most two h -exceptional curves with multiplicity at mostone.Next I will show that m Q ( ˆ F ) ≤ K Y ′ · F . (7.2.3)The map h is a composition of blow ups of points of Z . Since ˆ F is singular, h mustblow up the singular points of ˆ F . Let h : Y → Z be the blow up of Q ∈ Z . Thenthere exists a factorization Y ′ h ❆❆❆❆❆❆❆❆ h / / ZY h ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ Then also h ∗ ˆ F = ( h ) − ∗ ˆ F + m Q ( ˆ F ) E , where E is the h -exceptional curveand ( h ) − ∗ ˆ F is the birational transform of ˆ F in Y . From this it follows that ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 37 E · ( h ) − ∗ ˆ F = m Q ( ˆ F ). Also K Y = h ∗ K Z + E = E . Therefore K Y · ( h ) − ∗ ˆ F = m Q ( ˆ F ). Moreover, K Y ′ = h ∗ K Y + E ′ , where E ′ is an effective h -exceptional divisor. But then K Y ′ · F = h ∗ K Y · F + E ′ · F ≥ K Y · ( h ) ∗ F = K Y · ( h ) − ∗ ˆ F = m Q ( ˆ F ) . This proves the claim.As it has been shown earlier, F meets at most two h -exceptional curves E j and E s , with the possibility j = s , each one of them with intersection multiplicity one.Suppose that E i = E j and that F intersects E j and E s at the same point Q .Hence E j ∩ E s ∩ F = ∅ . Then since Y has rational singularities it is not possiblethat E j and E s are both g -exceptional.Suppose that E s is g -exceptional but E j is not g -exceptional. Then g ∗ E j wouldbe singular. But then from the equation (7.2.2) and the arguments following it, weget again that dim H ( O Y (2 K Y )) ≥
2, a contradiction to the equation (7.1.7).Hence neither of E j and E s is g -exceptional. Now write K Y ′ = b j E j + b s E s + X r = j,s b r E r . Then from the equation (7.2.3) and the facts that E j · F = E s · F = 1, F · E r = 0,for r = j, s , it follows that m Q ( ˆ F ) ≤ K Y ′ · F = b j + b s . Then from the equation (7.2.2) and the fact that E j and E s are not g -exceptionalit follows that K Y = b j ˜ E j + b s ˜ E s + ˜ W , where W is an effective divisor. Then since K X = π ∗ K Y we get that K X = b j π ∗ ˜ E j + b s π ∗ ˜ E s + π ∗ ˜ W .
Now considering that K X is ample we get that m Q ( ˆ F i ) ≤ K Y ′ · F i = b j + b s ≤ b j π ∗ ˜ E j · K X + b s π ∗ ˜ E s · K X ≤ K X , as claimed.Suppose finally that E j = E s , i.e., F meets exactly one h -exceptional curve.Then K Y ′ · F = b j . If E j is not g -exceptional then the previous argument proves theclaim. Suppose that E j is also g -exceptional. Then there exists a − h -exceptionalcurve E λ such that b λ ≥ b j . The previous argument now shows that b λ ≤ K X andhence m Q ( ˆ F i ) ≤ b j ≤ b λ ≤ K X . This concludes the proof of Claim 7.2.
Claim 7.3.
Let B be any member of the linear system | ( K X ) K Y ′ + F | . Then B ∼ W ′ + m X i =1 γ i E i , (7.3.1)where γ i ≥ i and W ′ is the birational transform in Y ′ of a smooth curve W in Z such that | W | is base point free and p a ( W ) ≥ By [Hu16, Proposition 3.5 and 3.10], the linear system | ˆ F | is base point freeand contains a smooth curve. Let W ∈ | ˆ F | . be a general member. Then W isreduced and irreducible and moreover it does not pass through h (Ex( h )). Let W ′ be the birational transform of W in Y . Then W ′ ∼ = W . Now from Proposition 4.11it follows that µK Y ′ − h ∗ ˆ F + F = m X i =1 γ i E i , (7.3.2)where γ i ≥
0, for all 1 ≤ i ≤ m , and µ is the maximum of the multiplicities of thesingular points of ˆ F . But from Claim 7.2 it follows that µ ≤ K X . Hence( K X ) K Y ′ − h ∗ ˆ F + F = m X i =1 γ ′ i E i , (7.3.3)for some γ ′ i ≥
0, for all 1 ≤ i ≤ m . Let now W ∈ | ˆ F | be a general member. Then W ′ = h ∗ ˆ F = F i + ( h ∗ ˆ F − F ). Then from the equation (7.3.2) it follows that( K X ) K Y ′ + F i = ( K X ) K Y ′ + W ′ − h ∗ ˆ F i + F i = W ′ + m X i =1 γ ′ i E i , for some γ ′ i ≥
0, 1 ≤ i ≤ m . This concludes the proof of Claim 7.3.Now pushing down to Y by g ∗ , and considering that F i is g -exceptional, we seethat ( K X ) K Y ∼ ˜ W + m X j =1 γ ′ j ˜ E j . (7.3.4)Moreover notice that from the construction of W , dim | ˜ W | ≥ | ( K X ) K Y | ≥
1. But according to the assumptions of the proposition,( K X ) + 3( K X ) + 3 < p, and therefore from Corollary 4.10 we get a contradiction. Hence there is no g -exceptional curve F i such that ˆ F i = h ∗ F i is singular. Case 2.
Suppose that ˆ F i is smooth for any i = 1 , . . . , r . In this case I will showthat K X > ( p − / F i is smooth it follows that ˆ F i ∼ = P and that ˆ F i = −
2, for all i = 1 , . . . , r .Consider now cases with respect to whether or not every connected subset of theset { ˆ F , . . . , ˆ F r } is contractible. Case 2.1.
Suppose that every connected subset of { ˆ F , . . . , ˆ F r } is contractible.Let φ : Z → W be the contraction. Since ˆ F i = −
2, for all i = 1 , . . . , r , W hasDuval singularities. Therefore K Z = φ ∗ K W . Hence, since K Z = 0, K W = 0. Thenthere exists a factorization Y ′ g / / φh ❇❇❇❇❇❇❇❇ Y ψ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ W Hence K Y = ψ ∗ K W + ˜ E = ˜ E , where ˜ E is a divisor supported on the ψ -exceptionalset. But then K Y = ˜ E ≤
0, which is impossible since K Y is ample. ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 39
Case 2.2.
There exists at least one connected subset of { ˆ F , . . . , ˆ F r } which isnot contractible. Claim 7.4.
There exists integers 0 ≤ γ j ≤ j = 1 , . . . , r such that the linearsystem | K Y ′ + P rj =1 γ j F j | has dimension at least one. Moreover, let B ∈ | K Y ′ + P rj =1 γ j F j | be any member. Then if K X < p/ B ∼ W ′ + m X i =1 γ i E i , where γ i ≥ i and W ′ is the birational transform in Y ′ of a reduced andirreducible curve W in Z such that | W | is base point free and p a ( W ) ≥ Claim 7.5.
There exist numbers 0 ≤ γ i ≤ i = 1 , . . . , r such that if Γ = P ri =1 γ i ˆ F i , then Γ · ˆ F i ≥
0, for all 1 ≤ i ≤ r , and Γ ≥ { ˆ F , . . . , ˆ F s } , s < r , be the maximal connectedsubset of { ˆ F , . . . , ˆ F r } which is contractible. Since the rank of Pic( Z ) is at most22 [Hu16] it follows that s ≤ φ : Z → Z ′ be the contraction of { ˆ F , . . . , ˆ F s } . Then Z ′ has Du Val singu-larities. Since ∪ ri =1 ˆ F is connected, there exists a curve ˆ F j ∈ { ˆ F s +1 , . . . , ˆ F r } , suchthat ˆ F j ∩ ( ∪ ˆ F si =1 ) = ∅ and of course ˆ F j does not contract by φ . Let F ′ j = φ ∗ ˆ F j .Observe now that one of the following happens.(1) F ′ j is singular. In this case one of the following happens.(a) ˆ F j meets two distinct φ -exceptional curves, say ˆ F λ , ˆ F µ , 1 ≤ λ < µ ≤ s .(b) ˆ F j meets one φ -exceptional curve ˆ F i , i ≤ s , such that ˆ F j · ˆ F i ≥ F j meets exactly one φ -exceptional curve ˆ F i and ˆ F i · ˆ F j = 1.(2) F ′ j is smooth.Suppose that the case 1.a happens. Then let Γ = ˆ F j + P µi = λ ˆ F i . Then this is acycle of − · ˆ F i = 0, for all i ∈ { j, λ, λ + 1 , . . . , µ } , and Γ = 0.Suppose that the case 1.b happens. Then let Γ = ˆ F j + ˆ F i . Then Γ · ˆ F j ≥ · ˆ F i ≥ ≥ W is not reduced, i.e., when W has either a D s , E , E or E singularity.Suppose that W has a D s singularity. The fundamental cycle of the singularityis ˆ F + 2 P s − i =1 ˆ F i + ˆ F s − + ˆ F s . Hence in this case ˆ F j must intersect some ˆ F i ,2 ≤ i ≤ s −
2. Let Γ = ˆ F j + ˆ F i − + 2 P s − k =1 ˆ F k + ˆ F s − + ˆ F s . Then Γ · ˆ F j = 0,Γ · ˆ F k = 0, i − ≤ k ≤ s and Γ = 0.The cases when W has E , E or E singularities are treated similarly.Suppose finally that case 2 happens, i.e., F ′ j is smooth. Then write φ ∗ F ′ j = ˆ F j + s X i =1 a i ˆ F i . Let m be the index of F ′ j in S . Then according to Proposition 4.12, m ∈ { , , , s +1 } (the exact value of m depends on the type of singularities of S ). Moreover, if S has an A s or D s singularity, then ma i ≤ s , for all i = 1 , . . . , s . If S has an E
60 NIKOLAOS TZIOLAS singularity then ma i ≤ i and if S has an E singularity then ma i ≤ i . In any case ma i are positive integers at most 22, for all i = 1 , . . . , s , and m ≤ s + 1 ≤
23. Let γ i = ma i , for all i = 1 , . . . , s and γ j = m . Let alsoΓ = mφ ∗ F ′ j = γ j ˆ F j + s X i =1 γ i ˆ F i . Then Γ · ˆ F i = 0, i = 1 , . . . , s , and Γ · ˆ F j = m ( F ′ j ) ≥ F ′ ) <
0, then the set { ˆ F j , ˆ F , . . . , ˆ F s } would be contractible which is not true). Moreover, Γ ≥
0. Thisconcludes the proof of Claim 7.5.So it has been proved that there exists a nontrivial effective divisor Γ = P ri =1 γ i ˆ F i in Z , such that 0 ≤ γ i ≤ i = 1 , . . . , r , and Γ · ˆ F i ≥ i = 1 , . . . , r andΓ ≥
0. In particular, if three of the ˆ F i meet at a common point or two havea tangency then B is reduced. Now since ˆ F i is smooth for all i , every multiple γ i ˆ F i can be considered singular with multiplicity γ i ≤
23 at every point. If two,say ˆ F i and ˆ F j meet at a point with multiplicity 1 then Γ has at this point mul-tiplicity γ i + γ j ≤
23 + 23 = 46. Therefore from Proposition 4.11 it follows that46 K Y ′ − h ∗ Γ + Γ ′ is an effective divisor, where Γ ′ = P ri =1 γ i F i .Consider now cases with respect to Γ .Suppose that Γ = 0. Then by [Hu16, Proposition 3.10], the linear system | B Γis base point free. Moreover, by [Jou83, Theorem 6.3], if p = 2 ,
3, Γ ∼ p ν W , where W is a smooth irreducible elliptic curve. In fact | Γ | is also base point free [Hu16,Proposition 3.10]. I claim that if ν >
0, then K X > p ν /
44. Indeed.46 K Y ′ + Γ ′ = 46 K Y ′ + Γ ′ − h ∗ Γ + h ∗ Γ = (46 K Y ′ − h ∗ Γ + Γ ′ ) + p ν W ′ =(7.5.1) p ν W ′ + E, where E is an effective divisor whose prime components are g -exceptional and h -exceptional curves and W ′ is the birational transform of W in Y ′ ( W can be chosento avoid the points blown up by h ). Then by pushing down to Y and then pullingup on X we find that 46 K X = p ν π ∗ ˜ W + π ∗ ˜ E, (7.5.2)where ˜ W = g ∗ W and ˜ E = g ∗ E . Also notice that since W moves in Z , W ′ is not g -exceptional and hence ˜ W = 0. Then, since K X is ample, it follows that 46 K X ≥ p ν .But this is impossible since we are assuming that the inequalities of the statementof Claim 3.2 do not hold. Hence ν = 0. Then by pushing the equation 7.5.1 downto Y we get that 46 K Y = ˜ W + g ∗ E, where ˜ W is the birational transform of W in Y . Now since dim | W | ≥ | K Y | ≥
1. But according to the assumptions of the proposition,(46 · K X + 3 < p. Then from Corollary 4.10 gives a contradiction. So it is not possible that Γ = 0.Suppose finally that Γ >
0. Then Γ is nef and big. Then by [Hu16, Corollary3.15], Γ ∼ mW + C , where W is a smooth elliptic curve and C ∼ = P . Moreover, asbefore, the linear system | W | is base point free [Hu16, Proposition 3.10]. Repeatingnow the arguments of the previous case we find that46 K Y ′ + Γ ′ = mW ′ + C ′ + E, ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 41 where W ′ and C ′ are the birational transforms of W and C in Y ′ and E is effective.Repeating now word by word the arguments of the case when Γ = 0 we get againa contradiction.Therefore, under the assumptions of the proposition, Z is not a K Case 4: Suppose that Z is Enriques. In this case, since we assume p = 2, π ( X ) = π ( Z ) = Z / Z . Then there existsan ´etale double cover ν : W → X of X . Then K W = ν ∗ K X and K W = 2 K X .Also D lifts to a nontrivial global vector field D ′ on W . Then in the correspondingdiagram (5.0.1) for W , Z is going to be a K W show that Z cannotalso be an Enriques surface. (cid:3) The Kodaira dimension of Z is − . Proposition 8.1.
Let X be a canonically polarized surface defined over an alge-braically closed field of characteristic p > . Suppose that X admits a nontrivialglobal vector field D such that D p = 0 or D p = D . Suppose also that, with notationas in section 5, Z has Kodaira dimension − and that one of the following holds(1) K X = 1 and p > .(2) K X ≥ and p > K X + 3 .Then X is unirational and π ( X ) = { } .Proof. I will only do the case when K X ≥
2. The only difference between the twocase is that in the proof one has to use the inequalities in Proposition 3.14 thatcorrespond to each case. Otherwise the proofs are identical.Let f : X ′ → X be the minimal resolution of X . Then from the inequalities(6.1.1) it follows that12 χ ( O X ) + 11 K X + 1 ≤ K X + 37 < K X + 3 < p, from the assumptions. Therefore from Theorem 3.3, D lifts to a vector field D ′ on X ′ . Moreover, every f -exceptional curve is stabilized by D ′ .Since κ ( Z ) = − Z is a ruled surface. Hence there exists a fibration of smoothrational curves φ : Z → B , where B is a smooth curve. Claim:
Under the conditions of the proposition, B ∼ = P .Suppose that the claim has been proved. Then Z and hence Y ′ are rational. Inparticular π ( Y ′ ) = π ( Z ) = { } . Then there exists a commutative diagram Y ′ ( p ) σ / / F ( p ) ❆❆❆❆❆❆❆ ˆ X ˆ g / / ˆ π (cid:15) (cid:15) X π (cid:15) (cid:15) Y ′ g / / Y Where ˆ X is the normalization of Y ′ in X , ˆ π and σ are purely inseparable mapsof degree p , ˆ g is birational and F ( p ) is the k -linear Frobenius. Therefore, since Y ′ is rational, Y ′ ( p ) is also rational and hence X is purely inseparably unirational.Moreover, since ˆ π is purely inseparable, it follows that π ( ˆ X ) = π ( Y ′ ) = { } .Then, since ˆ g is birational and ˆ X and X are normal, it follows by [Gr60, ChapterX] that the natural map π ( ˆ X ) → π ( X ) is surjective. Therefore π ( X ) = { } . Therefore it remains to prove the claim.Suppose that a g -exceptional curve F does not map to a point in B by the map φh . Then there exists a dominant morphism F → B . But since F is a rationalcurve then B ∼ = P .Suppose that every g -exceptional curve is contracted to a point in B by φh .Then there exists a factorization Y ′ g / / h (cid:15) (cid:15) Y ψ (cid:15) (cid:15) Z φ / / B (8.1.1)The general fiber of ψ is a smooth rational curve. Also, since the g -exceptional setis contained in fibers of φh , Y has rational singularities. Let σ : X → B be thecomposition ψπ .Consider next cases with respect to whether the divisorial part ∆ of D is zeroor not. Case 1: ∆ = Then K X = π ∗ K Y and hence, since π is a finite map, K Y is ample. Let Y b be a general fiber of ψ . Then Y b ∼ = P . Therefore since Y b = 0, it follows that K Y · Y b = − K Y is ample. Therefore in this case B ∼ = P . Case 2: ∆ = In order to show that B ∼ = P I will show that there exists a rational curve (ingeneral singular) C in X which dominates B . The method to find such a rationalcurve is to show that there exists an integral curve C of D on X which dominates B . Then by Corollary 4.8, if the arithmetic genus of C is small compared to thecharacteristic p , C is rational. Finally, integral curves of D will be found by utilizingProposition 4.1.By [Ek88, Theorem 1.20], the linear system | K X | is base point free. Thenby [Jou83, Theorem 6.3], the general member of | K X | is of the form p ν C , where C is an irreducible and reduced curve. Suppose that ν >
0. Then K X > p/
3, whichis impossible from the assumptions of the proposition. Hence the general memberof | K X | is reduced and irreducible (but perhaps singular).The assumptions of the proposition imply that p does not divide K X . Therefore,from Proposition 3.14 it follows that K X · ∆ ≤ K X (8.1.2) ∆ ≤ K X . Claim:
There exists a rank 1 reflexive sheaf M on Y such that O X ( K X + ∆) = ( π ∗ M ) [1] . I proceed to prove the claim. According to the adjunction formula (5.2.1) for π , K X + ∆ = π ∗ K Y + p ∆ , (8.1.3)Let now U ⊂ X be the smooth part of X and V = π ( U ) ⊂ Y . Then V is alsoopen. Since π is purely inseparable of degree p , if L is an invertible sheaf on U ,then L p = π ∗ N , where N is an invertible sheaf on V [Tz17b, Proposition 3.8].Therefore, ( O X (∆) | U ) p = π ∗ M V , ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 43 where M V is an invertible sheaf on V . Since X and Y are normal, U and V havecodimension 2 in X and Y , respectively, and therefore it easily follows that O X ( p ∆) = ( π ∗ M ) [1] , where M = i ∗ M V , i : V → Y is the inclusion. From this and the equation (8.1.3)the claim follows. Therefore also O X (3 K X + 3∆) = ( π ∗ N ) [1] , where N = M [3] . Hence by Proposition 4.1, there exists a k -linear map D ∗ : H ( O X (3 K X + 3∆)) → H ( O X (3 K X + 3∆)) . Let C ∈ | K X +3∆ | be a curve which corresponds to an eigenvector of D ∗ . Thenby Proposition 4.1, C is stabilized by D . Moreover, from the equations (8.1.2) itfollows that K X · C = 3 K X + 3 K X · ∆ ≤ K X , (8.1.4) C = 9 K X + 9∆ + 18 K X · ∆ ≤ K X Let now C = P si =1 n i C i be the decomposition of C into its prime divisors.The assumptions of the proposition imply that K X · C = 3 K X < p . Hence byCorollary 4.3, every component C i of C is stabilized by D and hence D induces avector field on C i , for all i . The induced vector field will be non zero if and only if C i is not contained in the divisorial part ∆ of D . Claim:
Suppose that C i is not contained in the divisorial part ∆ of D . The C i a rational curve.I proceed now to prove the claim. Let C ′ i = f − ∗ C i be the birational transform of C i in the minimal resolution X ′ of X . Then C ′ i is stabilized by D ′ . Let ν : ¯ C i → C ′ i be the normalization of C ′ i , which is also the normalization of C i .Since K X is ample,it follows from the equations (8.1.4) than K X · C i ≤ K X · C ≤ K X . Then also K ′ X · C ′ i = f ∗ K X · C ′ i = K X · C i ≤ K X , and from the Hodge Index Theorem,( C ′ i ) ≤ ( K X ′ · C ′ i ) K X ′ ≤ K X . Therefore from the adjunction formula,p a ( C ′ i ) ≤ K X + 1 < ( p − / , from the assumptions of the proposition. Hence from Proposition 4.7 it follows that D ′ fixes every singular point of C ′ i , for all i = 1 , . . . , s and D ′ lifts to a vector field ¯ D in the normalization ¯ C i of C ′ i . Therefore ¯ C i is either a smooth rational curve or anelliptic curve. I will show that D ′ has fixed points on C ′ i and hence by Lemma 3.15,¯ D has also fixed points and hence ¯ C i ∼ = P . Therefore C i is rational.Next I will show that there exists a fixed point of D on C i . Suppose that this wasnot the case and that D has no fixed points on C i . Then C i is in the smooth partof X since by Theorem 3.3, D fixes every singular point of X . Then if ˜ C i = π ( C i ),˜ C i is in the smooth locus of Y . Since there are no fixed points of D on C , C · ∆ = 0.Then from the adjunction formula for π we get that K X · C i = π ∗ K Y · C i = π ∗ K Y · π ∗ ˜ C i = pK Y · ˜ C i , and therefore K X · C i ≥ p . On the otherhand it has been shown that K X · C i ≤ K X < p , by the assumptions of the proposition. Hence there exists fixed pointsof D on every C i and therefore ¯ C i ∼ = P and hence C i is rational as claimed.Now let ∆ ′ = P νi =1 n i C i , where C i , 1 ≤ i ≤ ν ≤ s are the irreducible componentsof C that are also components of ∆ (and hence the restriction of D on C i is zero).Let also Z = P sj = ν +1 n i C i , where C j are the irreducible components of C whichare not contained in ∆ and therefore the restriction of D on C j , j ≥ ν + 1, is notzero (if ν = s then Z = 0). Then C = ∆ ′ + Z .Next I will show that Z = 0 and that there is a component of it which dominates B . Hence B is rational.Suppose that this is not true and that either Z = 0 or no component of Z dominates B . Therefore either Z = 0 or Z is contained in a finite union of fibers of ψh : X → B . Let F be a general fiber of ψh . Then in both cases F · Z = 0. Thenif we write 3 K X = C −
3∆ = ∆ ′ + Z − π becomes∆ ′ + Z = 3 π ∗ K Y + 3 p ∆ . Intersecting this with a general fiber F and taking into consideration that F · Z = 0and that F · π ∗ K Y = − p we find that∆ ′ · F = − p + 3 p (∆ · F ) . (8.1.5)Now ∆ ′ · F = ν X i =1 n i ( C i · F ) ≤ m ν X i =1 ( C i · F ) ! ≤ m ∆ · F, (8.1.6)where m is the maximum among the n , . . . , n ν such that C i · F = 0. Notice thatit is not possible that C i · F = 0, for all i = 1 , . . . , ν . If this was the case, then∆ ′ · F = 0. But since also we assume that Z · F = 0, it would follow that C · F = 0and hence ( K X + ∆) · F = 0. But then K X · F = − ∆ · F ≤ , for a general fiber F . But this is impossible since K X is ample. Hence ∆ ′ · F > m > m ≤ K X . Indeed. From the definition of ∆ ′ and theequation (8.1.4) it follows that m ≤ ν X i =1 n i ≤ ν X i =1 n i ( K X · C i ) = K X · ∆ ′ ≤ K X · C ≤ K X , as claimed. Then from the equations (8.1.5), (8.1.6) it follows that(12 K X − p )∆ · F + 6 p > . (8.1.7)Notice now that from the adjunction formula for π it follows that K X · F = π ∗ K Y · F + ( p − · F = − p + ( p − · F. Then since K X · F >
0, it follows that ∆ · F ≥
3. Now the assumptions of theproposition imply that K X < p/
12. Then it is easy to see that(3 p − K X )∆ · F − p > , which is a contradiction to the equation (8.1.7). Therefore it is not possible that Z · F = 0. Hence there exists a component C i of C such the restriction of D on C ECTOR FIELDS ON CANONICALLY POLARIZED SURFACES. 45 is not zero and C i dominates B . Then since C i is rational, it follows that B ∼ = P .This concludes the proof of Proposition 8.1. (cid:3) References [AA86] A. Aramova, L. Avramov,
Singularities of quotients by vector fields in character-istic p >
0, Math. Ann. 273, 1986, 629-645.[Ar77] M. Artin,
Coverings of the rational double points in characteristic p , Complexanalysis and algebraic geometry, Iwanami Shoten, Tokyo, 11-22,1977.[Ar85] M. Artin,
Reflexive modules over rational double points , Math. Ann. 270, 79-82,1985.[Ba01] L. Badescu,
Algebraic surfaces , Springer Universitext, 2001.[BM76] E. Bombieri, D. Mumford,
Enriques classification of surfaces in char.p, II , inComplex analysis and algebraic geometry, Cambridge Univ. Press, 23-42, (1977).[BM77] E. Bombieri, D. Mumford,
Enriques classification of surfaces in char.p, III , Invent.Math. 35, 197-232 (1976).[BCP11] I. Bauer, F. Catanese, R. Pignatelli.
Surfaces of general type with geometricgenus zero: a survey , Complex and differential geometry, Springer Proc. Math., 8,Springer, Heidelberg, 2011, 1-48.[BW74] D. M. Burns, J. Wahl,
Local contributions to global deformations of surfaces ,Inventiones Math. 26, 1974, 67-88.[DM69] P. Deligne and D. Mumford,
The irreducibility of the space of curves of givengenus , Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 36, 75109.[Ek88] T. Ekedhal,
Canonical models of surfaces of general type in positive characteristic ,Inst. Hautes ˆEtudes Sci. Publ. Math. No. 67, 1988, 97-144.[Gr60] A. Grothendieck,
Revˆetements ´etales et groupe fondamental , S´eminaire deG´eom´etrie Alg´ebrique du Bois Marie, 1960-1961.[Ha77] R. Hartshorne,
Algebraic geometry , Springer, 1977.[Hu16] D. Huybrechts,
Lectures on K3 surfaces , Cambridge University Press, 2016.[IG60] J. Igusa,
Betti and Picard numbers of abstract algebraic varieties , Proc. Nat. Acad.Sci. USA 46, 1960, 724-726.[Jou83] J. -P. Jouanolou,
Th´eor´emes de Bertini at applications , Progr. Math. vol. 42,Birkh¨auser, Boston, 1983.[KU85] T. Katsura, K. Ueno,
On elliptic surfaces in characteristic p , Math. Ann. 272,1985, 291-330.[KM98] J. Koll´ar, S. Mori,
Birational geometry of algebraic varieties , Cambridge UniversityPress,[KSB88] J. Koll´ar, N. I. Shepherd-Barron,
Threefolds and deformations of surface singu-larities , Invent. Math. 91, 1988, 299-338.[Ko10] J. Koll´ar,
Moduli of varieties of general type , preprint, available at arXiv:1008.0621[math.AG].[Ko97] J. Koll´ar,
Quotient spaces modulo algebraic groups , Ann. of Math. (2) 145 (1997),no. 1, 33-79.[Ku81] H. Kurke,
Examples of false ruled surfaces , Proceedings of the symposium in al-gebraic geometry, Kinosaki, 1981, 203-223.[La83] W. E. Lang,
Examples of surfaces of general type with vector fields , Arithmeticand Geometry vol. II, 167-173, Progress in Mathematics 36, Birkh¨auser 1983.[Li09] C. Liedtke,
Non-Classical Godeaux surfaces , Math. Ann. 343, 2009, 623-637.[Mi80] J. Milne, ´Etale Cohomology , Princeton University Press, 1980.[M84] Y. Miyaoka,
The maximal number of quotient singularities on surfaces with givennumerical invariants , Math. Ann. 268, 159-171, 1984.[MP97] Y. Miyaoka, T. Peternell,
Geometry of Higher Dimensional Algebraic Varieties ,Birkh¨auser Verlag, 1977.[Ray70] M. Raynaud,
Specialization du foncteur de Picard , Publ. Math. Math. IHES 38,1970, 27-76.[Mu70] D. Mumford,
Abelian varieties , Tata Studies in Math., Oxford University Press,1970. [Re85] M. Reid,
Yound person’s guide to canonical singularities , in Algebraic Geometry(Bowdoin 85), Proc. Sympos. Pure Math. 46, Part 1, AMS 1987, 345-414.[RS76] A. N. Rudakov, I. R. Shafarevich,
Inseparable morphisms of algebraic surfaces ,Izv. Akad. Nauk SSSR 40, 1976, 1269-1307.[Sch09] S. Schr¨oer,
On genus change in algebraic curves over imperfect fields , Proc. AMS137, no 4, 2009, 1239-1243.[Se06] E. Sernesi,
Deformations of Algebraic Schemes , Springer, 2006.[SB96] N. I. Shepherd-Barron,
Some foliations on surfaces in characteristic 2 , J. AlgebraicGeometry 5, 1996, 521-535.[Ta52] J. Tate,
Genus change in inseparable extensions of function fields , Proc. AMS 3,1952, 400-406.[Tz03] N. Tziolas,
Terminal 3-fold divisorial contractions of a surface to a curve I , Com-positio Mathematica 139, 2, 03, 239-261.[Tz17a] N. Tziolas,
Automorphisms of smooth canonically polarized surfaces in positivecharacteristic , Adv. Math. 310, 2017, 235-289.[Tz18] N. Tziolas,
Corrigendum to ”Automorphisms of smooth canonically polarized sur-faces in positive characteristic” , Adv. Math. 334, 2018, 585-593.[Tz17b] N. Tziolas,
Quotients of schemes by α p or µ p actions , Manuscripta Mathematica,2017, 152, 247-279.[Wa85] B. Wajnryb, Divisor class group descent for affine Krull domains , Journal ofAlgebra 92, 1985, 150-156.[Wi17] J. Witaszek,
Effective bounds on singular surfaces in positive characteristic , Michi-gan Math. J. 66, 2017, 367-388.[Za44] O. Zariski,
The theorem of Bertini on the variable singular points of a linearsystem of varieties ,Trans. Amer. Math. Soc., 56, 1944, 130-140.
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