On subfields of the function field of a general surface in P 3
aa r X i v : . [ m a t h . AG ] M a r ON SUBFIELDS OF THE FUNCTION FIELD OF AGENERAL SURFACE IN P YONGNAM LEE AND GIAN PIETRO PIROLA
Abstract.
In this paper we study birational immersions from a verygeneral smooth plane curve to a non-rational surface with p g = q = 0 totreat dominant rational maps from a very general surface X of degree ≥ P to smooth projective surfaces Y . Based on the classification theoryof algebraic surfaces, Hodge theory, and deformation theory, we provethat there is no dominant rational map from X to Y unless Y is rationalor Y is birational to X . Riemann-Hurwitz Theorem (Chapter XXI in [3]) says that if φ : C → C ′ is a non-constant morphism from a very general curve C of genus g > C ′ then either φ is birational, or else C ′ is rational. It isinteresting to investigate the same statement for higher dimensional varietiesof general type under some assumption of generality in a suitable modulispace. Let X be a smooth complex projective variety of general type. Thedominant rational maps of finite degree X Y to smooth varieties ofgeneral type, up to birational equivalence of Y form a finite set. The prooffollows from the approach of Maehara [15], combined with the results ofHacon and McKernan [12], of Takayama [16], and of Tsuji [17].Motivated by this finiteness theorem for dominant rational maps on avariety of general type and by the results obtained in [11] we study dominantrational maps from a very general complex surface X of degree d ≥ P tosmooth projective surfaces Y . The main result of this paper is the following. Theorem 0.1.
Let X ⊂ P be a very general smooth complex surface ofdegree d > . Then there is no dominant rational map f from X to any nonrational surface Y unless f is a birational map. We recall that a very general element of U has the property P if P holds inthe complement of a union of countably many proper subvarieties of U. Weget immediately the following completely algebraic version of our theorem.
Theorem 0.2.
Let K be the function field of a very general complex surfacein P of degree d > . Let
C ( K ′ ( K be a proper subfield of K. Then K ′ is isomorphic either to C ( x ) , if the transcendence degree of K ′ is , or to C ( x, y ) if K ′ has transcendence degree . If one chooses a special surface X in P then it might have a domi-nant rational map to a surface of general type Y . Classical Godeaux sur-faces Y (minimal surfaces of general type with p g ( Y ) := h ( Y, K Y ) = 0, q ( Y ) := h ( Y, O Y ) = 0, K = 1, and π ( Y ) = Z ) are obtained by the Z -quotient of Z -invariant quintics [10]. Mathematics Subject Classification.
Primary 14E05, Secondary 14H10, 14J29.
As far as we know this gives the first examples of fields of transcendencedegree 2 of non-ruled surfaces that do not contain any proper non-rationalfield. This could have applications to field theory and to absolute Galoistheory.To prove our main Theorem, we study birational immersions from a verygeneral smooth plane curve to a non-rational surface with p g = q = 0 . Theorem 0.3. If C is a very general smooth plane curve of degree d ≥ then there is no birational immersion κ from C into any non-rational surface S with p g = q = 0 . Moreover, we obtain
Theorem 0.4. ( = Theorem 3.3) Let C be a very general smooth plane curveof degree d ≥ . Then there is no birational immersion κ from C into anyelliptic surface S with p g = q = 0 of Kodaira dimension if Pic( S ) is torsionfree. The method of proof combines the classification theory of algebraic sur-faces, Hodge theory and deformation theory. By Hodge theory (as Section3.5 in [11]) one has to consider only dominant rational maps f : X Y where Y is simply connected and without 2 − holomorphic global forms, thatis p g ( Y ) = 0 . From the classification theory one obtains that the Kodairadimension of Y , denoted by kod( Y ), must be ≥ . The case of surfaces ofgeneral type, that is kod( Y ) = 2 , was also considered in [11], where theproblem was solved only for d ≤ . The new idea here is to consider thefull family of smooth plane curves which are hyperplane sections of sur-faces X ⊂ P . We show that a very general plane curve, possibly up tosome small degree cases, cannot be birationally immersed in Y. The caseof elliptic surfaces, that is when kod( Y ) = 1 , is similar, but slightly moredifficult. A careful study of curves on Y as well as an estimate of the moduliof non-rational elliptic surfaces is necessary.The method presented here can be used to obtain similar results for a verygeneral point of families of surfaces of general type that contain families ofcurves of high dimension. Such are for instance symmetric products of curvesof genus g ≥ X ) = 1 and kod( X ) = 0 . For instancethe case where X is a very general quartic surface in P needs a differentapproach.In this paper we work over the field of complex numbers. Acknowledgements.
This work was initiated when the first named authorvisited University of Pavia supported by INdAM (GNSAGA). He would liketo thank University of Pavia for its hospitality during his visit. The firstnamed author is partially supported by the National Research Foundationof Korea(NRF) funded by the Korea government(MSIP) (No.2013006431)and (No.2013042157). The second named author is partially supported byINdAM (GNSAGA); PRIN 2012 “Moduli, strutture geometriche e loro ap-plicazioni” and FAR 2013 (PV) “Variet`a algebriche, calcolo algebrico, grafi
N SUBFIELDS OF THE FUNCTION FIELD OF A GENERAL SURFACE IN P orientati e topologici” . Finally we thank the referees for several useful sug-gestions and remarks.1. Family of curves on surfaces
We will consider birational immersions of curves into surfaces. We beginwith the following:
Definition 1.1.
Let C be a smooth curve and let S be a smooth projectivesurface. A morphism κ : C → S will be called a birational immersion if theinduced map C → κ ( C ) is birational. Our aim is to show that a very general plane curve C of degree d ≥ S with p g ( S ) = q ( S ) = 0 and Kodaira dimension kod( S ) ≥ . For this we will comparedeformations of a plane curve C of degree d and deformations of the map κ. We now explain our method starting with some general remarks. Firstlywe recall that we can find a countable number of families that contain allthe algebraic (smooth) projective varieties. This follows, for instance, fromthe fact that the Hilbert polynomials are countable and that any Hilbertscheme has a finite number of irreducible components. By the same kind ofargument we can find a countable number of families for all the birationalimmersions κ : C → S, where C is a smooth plane curve of degree d, S is anysmooth projective surface with p g ( S ) = q ( S ) = 0 and Kodaira dimensionkod( S ) ≥ . By contradiction we assume that for a very general smooth plane curveof degree d, C, a birational immersion κ : C → S exists. A Baire’s categoryargument shows that there should exist a family of deformations of thebirational immersion κ that dominates the moduli space of the plane curvesof degree d . Therefore, there should be two smooth families π : C →
W,p : S → W and a family map K : C → S , where W is smooth variety, u ∈ W such that π − ( u ) ∼ = C, p − ( u ) ∼ = S and K u ∼ = κ. Let M ( d ) be themoduli space of smooth plane curves of degree d . Under our assumption theforgetful map: W → M ( d ) x → moduli [ π − ( x )]must contain an open set of M ( d ) . Then we can compare the dimension of W with the dimension n of the vector space of the first order deformationsof the map κ. We must have: n ≥ dim M ( d ) = ( d + 1)( d + 2)2 − . Next we give a bound n ≤ n ′ + m ′ where n ′ is the dimension of the vectorspace of the first order deformation of the map κ : C → S where the targetsurface S is fixed, and m ′ = h ( T S ) is the space of the first order deformationof the complex structure of S. Finally if we can prove that m ′ + n ′ < ( d + 1)( d + 2)2 − YONGNAM LEE AND GIAN PIETRO PIROLA
In this section we will show this for surfaces S of general type and forEnriques surfaces. In the next section we will adapt the argument to thecase of surfaces of Kodaira dimension kod( S ) = 1.We recall a basic result on deformations of curves on a fixed surface (seefor instance [1]). Also we refer to [6] for a basic result on moduli of algebraicsurfaces. Let C be a smooth projective curve of genus g ( C ) ≥ S be a smooth projective surface. Let κ : C → S a birational immersion. Let T C and T S be the holomorphic tangent bundles. The differential dκ : T C → κ ∗ T S of a birational immersion κ is a sheaf inclusion and hence induces anexact sequence on C : 0 → T C → κ ∗ T S → N → . The first order deformations of κ are classified by the global sections H ( C, N )of the normal sheaf N. Let U be a Kuranishi space of deformations of κ .When κ is not rigid, that is dim( U ) >
0, we have a good bound on thedimension of U : Replace κ : C → S by a very general point of U if it isnecessary. Let N tors ⊂ N be the torsion sheaf of N and set N ′ = N/N tors . Since κ is a very general birational immersion, a simple , but basic result (see[1] Corollary (6.11), and [3] Chapter XXI ) gives that dim U ≤ h ( C, N ′ ) . When deg κ ∗ ( K S ) = m ≥ N ′ ≤ g − − m where g = g ( C ).Therefore Clifford theorem gives h ( N ′ ) ≤ g − m. We have:
Proposition 1.2.
Assume that K S is a nef divisor. Then dim U ≤ max(0 , g − deg κ ∗ ( K S )2 ) . Moreover if dim U = g − deg κ ∗ ( K S )2 , then N = N ′ and additionally either κ ∗ ( K S ) = O C or C is hyperelliptic.Proof. If dim( U ) = 0 then there is nothing to be proved. So we may assumedim( U ) > U ≤ h ( N ′ ) ≤ h ( K C ⊗ K − S ) . Since K S is nef thestatement follows from Clifford theorem. (cid:3) We give an immediate application of the above result:
Proposition 1.3. If C is a very general smooth plane curve of degree d ≥ then there is no birational immersion κ from C into any surface S of generaltype with p g = q = 0 . Proof.
We can assume that S is minimal. The proof is obtained by con-tradiction: suppose that a very general plane curve C can be birationallyimmersed in S with p g = q = 0 . Plane curves of degree d depend on( d + 1)( d + 2)2 − Ciro Ciliberto explains in this way: in an actual deformation of a subvariety its generalpoint p must move, therefore the section of the normal corresponding to the first orderdeformation cannot vanish at p . N SUBFIELDS OF THE FUNCTION FIELD OF A GENERAL SURFACE IN P dimensional moduli. By ([11], Corollary 2.5.3) surfaces of general type with p g = q = 0 depend on M ≤
19 parameters. It follows that on S we can findat least ( d + 1)( d + 2)2 − d . Sincethis number is positive if d ≥
7, and since C is not hyperelliptic we havethen by Proposition 1.2( d + 1)( d + 2)2 − < g − deg κ ∗ ( K S )2 . if d ≥
7. Therefore( d + 1)( d + 2)2 − ( d − d − < − deg κ ∗ ( K S )2 :3 d < − deg κ ∗ ( K S )2 ≤ K S is nef and big. Therefore, if d ≥ κ : C → S. (cid:3) By the same argument in Proposition 1.3, one can treat also the case when S is any Enriques surface. We note that the dimension of moduli space ofEnriques surfaces is 10. Proposition 1.4. If C is a very general smooth plane curve of degree d ≥ then there is no birational immersion κ from C into any Enriques surface S . Elliptic surfaces with p g = q = 0 of Kodaira dimension 1 In this section, we study a birational immersion from a very generalsmooth plane curve to an elliptic smooth projective surface with p g = q = 0of Kodaira dimension 1. We recall that a surface S (see [4]) is an ellipticsurface if it admits an elliptic fibration, i.e. there is a smooth projectivecurve B and a surjective map π : S → B such that the general fiber F = π − ( b ) for b ∈ B is an elliptic curve.We will assume moreover that(1) p g ( S ) = dim H ( S, K S ) = 0.(2) q ( S ) = dim H ( S, O S ) = 0.(3) S is minimal.Easy consequence of the above conditions, we have χ ( O S ) = 1 , K = 0, c ( S ) = 12, and since the irregualarity is zero B = P . Let π : S → P bethe elliptic fibration, and let N be the number of multiple fibers of π , andlet k i for i = 1 , . . . , N be their multiplicities.We first prove the following: Lemma 2.1.
Let π : S → P be an irrational minimal elliptic surface with p g = q = 0 and let N be the number of multiple fibers of π . Suppose thatthere is a birational immersion κ from a very general smooth plane curve C of degree d ≥ to S . Then we have N ≤ d − . YONGNAM LEE AND GIAN PIETRO PIROLA
Proof.
We consider the map f = π ◦ κ : C → P . Let α be the degree of f .Let g be the genus of C . Then from the Hurwitz formula we have2 g − ≥ − α + ( N X i =1 α (1 − /k i )) + r ≥ − α + N α r where r is the number of branch points which are not in any multiple fiber.Note that the ramification index corresponding to the points in a multiplefiber is ≥ α . Clearly α ≥ d −
1, the gonality of the smooth plane curve C (see for instance Chapter I in [2]).We note that if we fix the number N of the multiple fibers then thereare countably many deformation types of irrational minimal elliptic surfaceswith p g = q = 0 [Theorem 7.6, Chapter I in [9]]. Therefore the Baire’scategory argument applies and we can compare the dimension of moduli ofplane curves with the dimension of the Hurwitz’ scheme defined by f. Sincevery general plane curves are obtained by α -covers of P , we must have infact N + r − ≥ ( d + 1)( d + 2) / − . This implies r ≥ ( d + 1)( d + 2) / − N −
6. Then combining it with the aboveequation, we conclude d ( d − ≥ ( d − N/ −
2) + ( d + 1)( d + 2) / − N − . So d − d +18 ≥ ( d − N − . Since we have d ≥ N ≤ d − . (cid:3) Now we want to estimate the dimension of the moduli space M ( S ) ofirrational minimal elliptic surfaces S with p g = q = 0 when the number N of the multiple fibers is fixed. We note that the moduli space M ( S ) of S is irreducible when we fix the type of the multiple fibers [Theorem 7.6,Chapter I in [9]]. Since irrational minimal elliptic surfaces S with p g = q = 0can be constructed by logarithmic transforms of rational elliptic surfaces[Section 1.6 in [9]] and the non-isotriviality of elliptic fibration is preservedby logarithmic transforms, a general element in M ( S ) has a non-isotrivialelliptic fibration. If an elliptic fibration is not isotrivial then it yields the j invariant of the smooth fibers to be non-constant. Therefore, it is enoughto estimate the dimension of the moduli space M ( S ) of irrational minimalelliptic surfaces S with p g = q = 0 under the assumption that the ellipticfibration is not isotrivial. Lemma 2.2.
Moreover, we assume that the elliptic fibration is not isotrivial.Let ¯ F be any fiber of the elliptic fibration. Then we have h (Ω S ( n ¯ F )) = n − for n > .Proof. We remark that it is enough to prove h (Ω S ( nF )) = n − F since the base curve is P . We take the conormal sequence of the immersion of F into S :(1) 0 → N ∗ F ≡ O F → Ω S | F → Ω F → . Since the surface is elliptic, K S restricts to 0 on F and therefore the sequence(1) is the same as the tangent sequence0 → O F → T S | F → O F → . N SUBFIELDS OF THE FUNCTION FIELD OF A GENERAL SURFACE IN P It follows that the associated coboundary map ∂ : H ( F, Ω F ) → H ( F, O F )is the Kodaira-Spencer map. Since the elliptic fibration is not isotrivial,the coboundary map ∂ is not zero. Therefore the natural map H ( O F ) → H ( F, Ω S | F ) is an isomorphism. This gives the first step of our proof.Let D = F + · · · + F n , where F i are general fibers for i = 1 , . . . , n . Weget then the natural map H ( O D ) → H (Ω S | D )is an isomorphism. Next we consider the commutative diagram0 / / Ω S (cid:15) (cid:15) / / Ω S (log D ) (cid:15) (cid:15) / / (cid:15) (cid:15) / / O D (cid:15) (cid:15) / / / / Ω S / / Ω S ( D ) / / Ω S | D / / ∂ : H ( F, O F ) → H ( S, Ω S ) inthe log sequence 0 → Ω S → Ω S (log F ) → O F → ∂ (1) = c ( F ) . Therefore the connecting homomorphism ∂ ′ : H ( O D ) = H ( O F ) ⊕· · · ⊕ H ( O F n ) → H (Ω S ) in the above log diagram maps the generator 1 ∈ H ( O F i ) to c ( F i ) = c ( F ), so ∂ ′ has rank 1. Since H ( O D ) → H (Ω S | D )is an isomorphism by the first step of our proof, and since h (Ω S ) = 0 weconclude that h (Ω S (log D )) = h (Ω S ( D )) = h (Ω S ( nF )) = n − (cid:3) Remark 2.3.
A similar result was proved for rational elliptic surfaces in( [13] , Lemma(2)). Moreover we note that Lemma 2.2 follows under the onlyassumption that the fibration is not isotrivial.
We recall the canonical divisor formula (see [4] [Chapter V, Corollary(12.3)]) for elliptic fibrations:(2) K S = − F + N X i ( k i − F i = ( N − F − X i F i . Proposition 2.4.
Let S be a minimal elliptic surface with p g = q = 0 ofKodaira dimension . Moreover, we assume that the elliptic fibration is notisotrivial. Let M be a Kuranishi space of the deformation space of S . Then M ( S ) = dim M ≤ N. Proof.
We would like to estimate h ( T S ) = dim H ( S, T S ). We recall that H ( S, T S ) = 0 . In fact the j invariant is not constant. This implies that theconnected component of the identity of the automorphism group of S mustcommute with the fibration. So if it is not trivial then all smooth fibersare isogeneous, again it contradicts the non-triviality of the j invariant.Therefore from Hirzebruch-Riemann-Roch Theorem and Serre duality (see[6]) we have h ( T S ) = h ( T S ) + 10 χ ( O S ) − K S = 10 + h (Ω S ( K S )) . We get h (Ω S ( K S )) = h (Ω S (( N − F − P F i )) ≤ h (Ω S (( N − F )) ≤ N − h ( T S ) ≤ N − M ( S ) ≤ N. (cid:3) Proposition 2.5.
A very general smooth plane curve of degree d ≥ cannotbe birationally immersed into any elliptic surface S with p g = q = 0 ofKodaira dimension . YONGNAM LEE AND GIAN PIETRO PIROLA
Proof.
By Lemma 2.1 we may assume that N ≤ d −
2, we first observe thatthe number of moduli of S is by Proposition 2.4 M ( S ) ≤ N ≤ d < ( d + 1)( d + 2)2 − d ≥ . It follows that κ ( C ) cannot be rigid in S. By the similar argument in the proof of Proposition 1.3 and using Propo-sition 2.4 ( d + 1)( d + 2)2 − − − N < g − deg κ ∗ ( K S )2 . Therefore by Lemma 2.1 we have( d + 1)( d + 2)2 − ( d − d − <
17 + d − − deg κ ∗ ( K S )2 :2 d < − deg κ ∗ ( K S )2 . Therefore, since K S is nef, if d ≥ κ : C → S. (cid:3) Combining Propositions 1.3, 1.4, and 2.5, we get Theorem 0.3.3.
Elliptic surfaces without torsion versus plane curves
This section treats a birational immersion from a very general smoothplane curve to some special families of elliptic surfaces. We keep the sameassumption made at the beginning of Section 2, that is, we assume that S is an elliptic surface with p g = q = 0 and the Kodaira dimension 1 , but nowwe will also assume that • Pic( S ) is torsion free.In other words, since q = 0 we assume the vanishing of the first integralhomology group of S : H ( S, Z ) = 0 . Let ≡ denote the linear equivalence of divisors which, in our case, is thehomological equivalence. Let F be a general fiber of π and { F i } { i =1 ,...,N } the multiple fibers, i.e. F i is effective and k i F i ≡ F where k i ∈ Z , k i > . We remark from ([7], Chapter 2) thatTors(Pic( S )) = ker( ⊕ Ni =1 Z /k i Z ψ → Z /M Z )where M = Q Ni =1 k i , ψ (( a , . . . , a N )) = P a i M i (mod M ), M i = M/k i . Itfollows k i and k j are relatively prime. We may assume 1 < k < k < · · · M > . We have F ≡ M λ, F i = M i λ. Therefore if we set K S = ρλ then ρ = (( N − M − N X i =1 M i ) = M ( N − − N X i =1 k i ) . N SUBFIELDS OF THE FUNCTION FIELD OF A GENERAL SURFACE IN P Set k = 1 we have ρ = M s, where s = ( N − N X i =0 k i ) . One has immediately Proposition 3.1. (1) ρ ≥ with the only one exception k = 2 , k = 3 . (2) If N > then one has s > with the only exception k = 2 , k = 3 k = 5 . In this case K S = 29 λ .(3) If N = 2 then K S = (( k − k − − λ. (4) If N > then we have ρ > N. Proof. Only the last part needs a comment. When N = 3 with k = 2 ,k = 3 k = 5, we have ρ = 29 > . Otherwise we have ρ ≥ Y i k i , it follows (it is a very rough estimate) ρ > N ! ≥ N. (cid:3) Remark 3.2. One has(1) K S = λ ⇐⇒ k = 2 and k = 3 . (2) If k = 2 and k = k then ρ = k − . (3) If k = 3 and k = 4 then ρ = 5 and K S = F + 2 F . Theorem 3.3. Let C be a very general smooth plane curve of degree d ≥ .Then there is no birational immersion κ from C into any elliptic surface S with p g = q = 0 of Kodaira dimension if Pic( S ) is torsion free.Proof. We keep the notation of the previous section. Let π : S → P be theelliptic fibration, and let N be the number of the multiple fibers.The proof is given by contradiction. We assume that κ : C → S is abirational immersion where C is a smooth plane curve of degree d > . We will apply Proposition 1.2 and Lemma 2.1. We first observe that since d ≥ S is M ( S ) ≤ N ≤ d < ( d + 1)( d + 2)2 − . It follows that κ ( C ) cannot be rigid in S. Arguing as in Proposition 1.3 weget again using Proposition 1.2( d + 1)( d + 2)2 − ( d − d − < M ( S ) − deg κ ∗ ( K S )2 . That is 3 d < N − deg κ ∗ ( K S )2 = 17 + N − ρ deg κ ∗ ( λ )2 . We have κ ∗ ( K S ) = ρλ. Since a positive multiple of λ is F moves linearly,so we have deg κ ∗ ( K S ) ≥ ρ. Finally we get3 d < 17 + N − ρ . When N = 2, ρ = ( k − k − − d < 19. So d ≤ . When N > ρ/ > N , and it follows 3 d < − N . So d < . Now we have only to consider the case d = 6 and N = 2. This is possibleonly for ρ = 1 since 3 d < − ρ , and by Proposition 3.1 this happensprecisely when N = 2 , k = 2 , k = 3 . We have κ ∗ ( K S ) = O C ( p ) for p ∈ C. Set κ ( C ) = D , then we have D · K S = 1 and D · F = 6 . The composition π ◦ κ : C → P has degree6. Since the only rational maps of degree 6 of a smooth curve C ⊂ P areobtained by projecting from a point of x ∈ P we obtain O C (6 p ) = κ ∗ O S ( F )is the g that gives the embedding of C → P . But this implies that p is aflex point of the maximal order 6 on C , which contradicts the fact that C is a very general plane curve of degree 6 . (Alternatively intersecting C withthe multiple fibers we find on C a 3-tangent and a bitangent to 2 flexes). (cid:3) Remark 3.4. One can prove that a very general curve of genus ≥ (in thesense of moduli) cannot be birationally immersed in any elliptic surface with p g = q = 0 of Kodaira dimension if Pic( S ) is torsion free:We consider the map f = π ◦ κ : C → P . Let α be the degree of f . Thenwe have α = κ ( C ) · F = M ( κ ( C ) · λ ) . We remark that the points p ∈ C suchthat κ ( p ) ∈ F i are ramification points of f with multiplicity a multiple of k i . Let g be the genus of C . From the Hurwitz formula we have g − ≥ − α + N X i =1 ( α − α/k i ) = α (( N − − N X i =1 /k i ) . Suppose N > . Clearly α ≥ M. Since the k i are pairwise relatively prime,we get: g − ≥ M ( N − − ( 12 + 13 + 15 + 17 ) − ( N − 4) 111 ) > ( N − 4) 10 M Now since k ≥ and k i > i − we get M > N ( N − ≥ N ( N − N − . Therefore we have g − > ( N − 4) 10 M > N ( N − N − N − . It implies that N ≤ max (4 , √ g ) . By Propositions 1.3, 2.4, and the proof ofTheorem 3.3, we have the inequality g − ≤ g − N + 8 . It implies that g − ≤ max(12 , √ g ) . This gives g ≤ . If g = 7 then the above argument implies that N = 4 . But this cannothappen because it contradicts the Hurwitz formula g − ≥ · · · − ( 12 + 13 + 15 + 17 )) . Proof of Main Theorem Now we are ready to prove our main Theorem (Theorem 0.1 in Intro-duction). We will give a proof by contradiction. Assume that a dominantrational map f : X Y exists.From Section 3.5 in [11] we may assume:(1) p g ( Y ) = q ( Y ) = 0 and Y is simply connected.(2) Pic( X ) = Z [ L ] , where L is the hyperplane section of X . N SUBFIELDS OF THE FUNCTION FIELD OF A GENERAL SURFACE IN P From the classification theory of algebraic surfaces, we have two casesfor Y : either Y is of general type, or Y is an elliptic surface with Kodairadimension 1 . Lemma 4.1. Let C be a general hyperplane section of X . If f : X Y is dominant then f C : C → Y is birational onto its image.Proof. Since C is a general hyperplane of X , we may assume that a generalpoint of Y belongs f C ( C ) . We recall that the Jacobian of a general hyper-plane section C is simple. Then if f C : C → f C ( C ) is not birational thenthe normalization of f C ( C ) is rational. Since Y is not ruled it follows that f is not dominant. Therefore it contradicts the assumption. (cid:3) Proof. of Theorem 0.1 . Assume by contradiction that for a very generalsurface of degree d we have a dominant rational map to Y. We get a birationalimmersion from a very general plane curve of degree d into a surface Y with p g = q = 0. The surface Y is either of general type or an elliptic surface withKodaira dimension 1 . In the first case we have d ≤ Y is an elliptic surfacethen we get d < d = 5 is the only remaining case. After a resolution of the singularitiesof f , we get Z p / / g ❅❅❅❅❅❅❅❅ X f ~ ~ Y. Let E be the exceptional divisor. 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II . Osaka J.Math. 44 (2007), 723–764. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu,Daejon 305-701, Korea E-mail address : [email protected] Dipartimento di Matematica, Universit`a di Pavia via Ferrata 1, 27100 Pavia,Italia E-mail address ::