Rationality of moduli of elliptic fibrations with fixed monodromy

We prove rationality results for moduli spaces of elliptic K3 surfaces and elliptic rational surfaces with fixed monodromy groups.


Introduction
Let X be an algebraic variety of dimension n over C. One says that X is rational if its function field C(X) is isomorphic to C(x 1 , . . . , x n ). The study of rationality properties of fields of invariants C(X) G = C(X/G) is a classical theme in algebraic geometry. For a finite group G ⊂ PGL n acting on X = P n−1 the problem is referred to as Noether's problem (1916). It is still unsolved for n = 4. Another class of examples is provided by moduli spaces. Birationally, they are often representable as quotients of simple varieties, like projective spaces or Grassmannians, by actions of linear algebraic groups, like PGL 2 . Rationality is known for each of the following moduli spaces: • curves of genus ≤ 6 [18], [32], [20], [21], [31]; • hyperelliptic curves [18], [7]; • plane curves of degrees 4n + 1 and 3n [33], [19]; • Enriques surfaces [24]; • polarized K3 surfaces of degree 18 [32]; • stable vector bundles (with various numerical characteristics) on curves, Del Pezzo surfaces, P 3 [22], [5], [11], [25], [29]; and in many other cases. For excellent surveys we refer to [12] and [33]. We will study rationality properties of moduli spaces of smooth non-isotrivial Jacobian elliptic fibrations over curves π : E → C with fixed global monodromy groupΓ =Γ(E) ⊂ SL 2 (Z). In [8] we developed techniques aimed at the classification of possible global monodromiesΓ. The present paper gives a natural application of these techniques. Let B be an irreducible algebraic family of Jacobian elliptic surfaces. Then the set of subgroupsΓ ⊂ SL 2 (Z) such thatΓ is the (global) monodromy group of some E in this family is finite. Moreover, for every such groupΓ the subset of fibrations with this monodromy is an algebraic (not necessarily closed) subvariety of B.
Generalizing this observation, we introduce (maximal) parameter spaces FΓ of elliptic fibrations with fixed global monodromyΓ (considered up to fiberwise birational transformations acting trivially on the base of the elliptic fibration). These parameter spaces can be represented as quotients of quasi-projective varieties by algebraic groups. In particular, we can consider irreducible connected components of the parameter space FΓ, which we call moduli spaces. Even though these moduli spaces need not be algebraic varieties, we can still make sense of their birational type.
Theorem. LetΓ ⊂ SL 2 (Z) be a proper subgroup of finite index. Then all moduli spaces of (Jacobian) elliptic rational or elliptic K3 surfaces with global monodromyΓ are rational.
Corollary. For allΓ with moduli FΓ of dimension > 0 there exists a number field K such that there are infinitely many nonisomorphic elliptic K3 surfaces over K with global monodromyΓ.
Remark 0.1. Our method shows that many other classes of moduli of elliptic surfaces over P 1 with fixed monodromy are rational or unirational. However, we cannot expect a similar result for all moduli spaces of elliptic surfaces over higher genus curves, since the moduli space of higher genus curves itself is not uniruled (by a result of Harris and Mumford [15]).
We proceed to give a more detailed description of our approach. First of all, we can work not with the monodromy groupΓ itself but rather with its image Γ ⊂ PSL 2 (Z) under the natural projection SL 2 (Z) ։ PSL 2 (Z). Let be the upper half-plane and The natural j-map j : C → P 1 = H/PSL 2 (Z) decomposes as Here M Γ is the j-modular curve corresponding to Γ; it is equipped with a special triangulation, obtained as the pullback of the standard triangulation of S 2 = P 1 (C) (by two triangles with vertices at 0, 1 and ∞) under the map j Γ (which ramifies only over 0, 1 and ∞). We call the obtained triangulation of M Γ a j Γ -triangulation. Let T Γ be the preimage in M Γ of the closed interval [0, 1] ⊂ P 1 . The graph T Γ is our main tool in the combinatorial analysis of Γ.
Denote by χ(E) the Euler characteristic of E. It splits equivalence classes of Jacobian elliptic surfaces (modulo fiberwise birational transformations) into algebraic families. In particular, if C = P 1 then the algebraic variety F r parametrizing (equivalence classes of) Jacobian elliptic surfaces with given χ(E) is irreducible; here we put r = χ(E)/12. Our goal is to analyze the birational type of (irreducible components) F r,Γ ⊂ F r parametrizing fibrations with fixed monodromy groupΓ. It suffices to study parameter spaces F r,Γ corresponding to Γ ⊂ PSL 2 (Z), since every irreducible component of F r,Γ coincides with a component of F r,Γ .
From now on we assume that C = M Γ = P 1 . Denote by R d,Γ the space of rational maps P 1 → P 1 (of degree d) with prescribed ramification (encoded in T Γ ). The spaces F r,Γ are quotients by the action of PGL 2 × H Γ of fibrations over R d,Γ with fibers (Zariski open subsets of) Sym ℓ (P 1 ) (for appropriate d and ℓ). Here PGL 2 acts (on the left) by changing the parameter on the base C = P 1 and H Γ is the group of automorphisms of M Γ = P 1 stabilizing the embedded graph T Γ (acting on the right). The nontriviality of H Γ means that there is a Γ ′ ⊂ PSL 2 (Z) containing Γ as a normal subgroup with H Γ = Γ ′ /Γ. So in most cases in order to prove rationality of F r,Γ it is sufficient to establish it for PGL 2 \R d,Γ , which can be deduced from general rationality results for PGL 2 -quotients (see [9], [18]). To cover all cases we need to set up a rather extensive combinatorial analysis.
Here is a roadmap of the paper. In Section 1 we discuss finite covers M Γ → P 1 in the spirit of Grothendieck's "Dessins d'Enfants" program (see [28], [34] and the references therein) and introduce the invariants GD(Γ), RD(Γ) and ET(Γ). In "ideal" cases ET(Γ) coincides with the number of triangles in the j Γ -triangulation of M Γ (the notation ET(Γ) stands for "Effective Triangles"). In Section 2 we recall basic facts about elliptic fibrations and introduce the invariant ET(E). For an "ideal" elliptic fibration one has ET(Γ) = ET(E). In Section 3 we discuss moduli of elliptic fibrations with fixed monodromy. In Sections 7 and 8 we formulate and prove several rationality results for PGL 2 and related quotients. In Section 5 we classify families of rational elliptic surfaces and elliptic K3 surfaces with different monodromy groups. In Section 4, we study relations between the combinatoric of the graph Γ and the topology of E. And finally, in Section 10 we list (certain) relevant subgroups Γ ⊂ PSL 2 (Z) (represented by trivalent graphs T Γ ). There are too many monodromy groups of elliptic K3 surfaces to be drawn on paper, but we show how to obtain them from our list by simple operations.

Finite covers
Let Γ be a subgroup of finite index in PSL 2 (Z). The latter is isomorphic to a free product Z/3 * Z/2. Consider the map ramified over the points 0, 1, ∞ ∈ P 1 . Denote their preimages in M Γ by A, B and I, respectively. The possible ramification orders are 3 or 1 for A-points, 2 or 1 for B-points and arbitrary for I-points. The points 0, 1 and ∞ subdivide the circle P 1 (R) = S 1 into three segments and, together with the upper and lower hemisphere, define a decomposition of P 1 (C) = S 2 into three triangles. This induces a special triangulation of M Γ with vertices in A, B and I-points which we call the j Γ -triangulation. The preimage of the segment [0, 1] ⊂ P 1 defines a graph T Γ which determines the j Γ -triangulation uniquely. Interior vertices of T Γ are marked by A 6 and ends are marked by either A 2 or B 2 .
Forgetting the markings of T Γ we obtain a connected unmarked topological graph T u Γ with (possibly some) ends and all interior vertices of valency 3 -a trivalent graph. Lemma 1.3. Let X be a compact orientable Riemann surface of genus g(X) and T u ⊂ X an embedding of a connected trivalent graph such that • the set X \ T u is a disjoint union of topological cells; • all interior vertices of T u are trivalent; • the ends of T u are arbitrarily marked by two colors A 2 and B 2 . Then there exist a subgroup Γ ⊂ PSL 2 (Z) and a unique complex structure on X such that X = M Γ and T u = T u Γ . Proof. Assume that we have an embedded graph T u ⊂ X satisfying the conditions above. Mark by A all trivalent vertices and enlarge the graph T u by putting a B-vertex in the middle of any edge bounded by two A-vertices. Put one I-vertex into every connected component of X \ T u and connect all Ivertices with A and B-vertices at the boundary of the corresponding domain. By assumption, every connected component of X \ T u is contractible. Consider the boundary of the individual cell. Every A-vertex of the boundary is connected by edges to B-vertices only. Similarly, the B-vertices are connected by edges only to A-vertices. Hence every triangle of the induced triangulation has vertices colored by three colors: A, B and I. This gives a j Γ -triangulation of X. Following Alexander [1], we observe that a j Γ -triangulation defines a map h : X → P 1 which is cyclically ramified over A, B and I (see [8]). The trivalence of T u implies that h has only 3 or 1-ramifications over 0 ∈ P 1 and only 2 or 1ramifications over 1 ∈ P 1 . Since PSL 2 (Z) = Z/3 * Z/2 there is exactly one subgroup Γ ⊂ PSL 2 (Z) (of finite index) which corresponds to the covering X → P 1 . Any graph T u Γ constructed via a subgroup Γ ⊂ PSL 2 (Z) satisfies the conditions above. Indeed, we have already described the j Γ -triangulation on M Γ . Triangles adjacent to a given I-vertex constitute a contractible cell and the division of M Γ into neighborhoods of I-vertices is a cellular decomposition of M Γ . Hence after removing I-vertices with open edges from them we obtain the preimage of [0, 1]. If we forget the B-vertices which lie between two Avertices we obtain the graph T u Γ . Thus T u Γ ⊂ X = M Γ is the boundary of this cellular decomposition and T Γ is simply T u Γ with an A, B-marking of the ends.
Remark 1.4. Graphs which are isotopic in X (modulo diffeomorphisms of X of degree 1) define conjugated subgroups of PSL 2 (Z).
Remark 1.5. Even if we omit the condition of compactness of X we still get a bijection between conjugacy classes of subgroups of finite index of PSL 2 (Z) and embedded trivalent graphs with marked ends.
Remark 1.6. The topology of X restricts the topology of T u Γ . The graph T u Γ must contain some 1-skeleton of X. In particular, the map π 1 (T u Γ ) → π 1 (X) is surjective. Hence T u Γ can be a tree only if X = S 2 . For X = P 1 the connectedness of T u guarantees that all the components of X \ T u are contractible. Hence we can classify graphs in X = P 1 by drawing them on the plane. In general, connectedness of T u is necessary but not sufficient.  Notations 1.9. Let f : C → P 1 be a cover of degree d and p ∈ P 1 a ramification point of f . The local ramification datum is an where v k is the order of ramification of f at a point c k ∈ f −1 (p). A reduced local ramification datum is a vector v obtained from v by omitting all entries v k = 1. The vector v is defined up to permutation of the entries.
For f = j E : C → M Γ = P 1 we have distinguished ramification points, namely those over A-and B-vertices of the graph T Γ ⊂ M Γ . The (global) j E -ramification datum is the vector where the v i,A are local ramification data over A-vertices for i = 1, . . . , n, (resp. v i,B for B-vertices, i = n + 1, . . . , n ′ ) and v i are reduced local ramification data for unspecified other points in M Γ for i > n ′ (distinct from A-and B-vertices of M Γ ).
For f = j Γ : M Γ → P 1 the distinguished (and the only) ramification points are 0, 1, ∞. We write means that deg(j E ) = 5, that j E has ramification points of order 2 and 3 over one point A 2 ∈ T Γ and (2, 2, 1) over one B 2 -point and ramifications of order 2 over two other unspecified points in M Γ .

Elliptic fibrations
In this section we briefly recall some basic facts of Kodaira's theory [23] of elliptic fibrations. For more details we refer to [3], [14] and [35]. Let π : E → C be a smooth non-isotrivial relatively minimal Jacobian elliptic fibration over a smooth projective curve C. This means that: • E is a smooth compact complex projective surface and π is a proper holomorphic map; • the generic fiber of π is a smooth curve of genus 1; • the fibers of E do not contain exceptional curves of the first kind, i.e., rational curves F such that (F 2 ) = −1 (relative minimality); • there exists a (global) zero section s : C → E (Jacobian elliptic fibration); • the j-function which assigns to each smooth fiber π −1 (p) = E p ⊂ E its j-invariant is a non-constant rational function on C (non-isotriviality).
Lemma 2.1. We have Proof. Well known, but we decided to include an argument. Since E is smooth and relatively minimal the canonical bundle K E of E is induced from a onedimensional bundle K on the base C. The sheaf π * K(C) is a subsheaf of K E . Since there are singular fibers we have the following equality where g is the genus of C. By Riemann-Roch we obtain We also know that s 2 + sK E − 2g + 2 = 0 (genus formula). Therefore, Thus s 2 + sK E − 2g + 2 = 0 transforms to s 2 + χ(E)/12 = 0.
Let C sing = {p 1 , . . . , p k } ⊂ C be the set of points on the base corresponding to singular fibers. The topological Euler characteristic χ(E) = c 2 (E) is equal to the sum of Euler characteristics of the singular fibers E p i = π −1 (p i ) (since every generic fiber has Euler characteristic equal to 0). Therefore, where the summation runs over all singular fibers of E and ET(E p i ) is the contribution from the corresponding singular fiber. Since the fibration is Jacobian every singular fiber has a unique representative from Kodaira's list and it is defined by the local monodromy. The possible types of singular fibers and their ET-contributions are: Here I 0 is a smooth fiber, I n is a multiplicative fiber with n-irreducible components. The types II, III and IV correspond to the case of potentially good reduction. More precisely, the neighborhood of such a fiber is a (desingularization of a) quotient of a local fibration with smooth fibers by an automorphism of finite order. The corresponding order is 4 for the case III and 3 in the cases II, IV. The fibers of type I * 0 , (resp. I * n , II * , III * , IV * ) are obtained from fibers I 0 (resp. I n , IV, III, II) (after changing the local automorphism by the involution x → −x in the local group structure of the fibration). We shall call them * -fibers in the sequel.
Remark 2.2. The local invariant ET(E p ) has a monodromy interpretation. Namely, every element of a local monodromy at p ∈ C sing has a minimal representation as a product of elements conjugated to ( 1 1 0 1 ) in SL 2 (Z). The length of this representation equals ET(E p )/2. This explains the equality ET(E * p ) = ET(E p )+12 -the element −1 0 0 −1 ∈ SL 2 (Z) is a product of 6 elements conjugated to ( 1 1 0 1 ) (elementary Dehn twists).

Moduli spaces
Every Jacobian elliptic fibration E → P 1 admits a Weierstrass modelĒ. Its geometric realization is given as follows: there exists a pair of sections such that E is given by
Two pairs (g 2 , g 3 ) and (g ′ 2 , g ′ 3 ) define isomorphic Jacobian elliptic surfaces (E, s) and (E ′ , s ′ ) iff there exits an h ∈ GL 2 (C) transforming (g 2 , g 3 ) into (g ′ 2 , g ′ 3 ) under the natural action of GL 2 on (the GL 2 -linearized) O P 1 (r). We define F r as the set of isomorphism classes of pairs (g 2 , g 3 ) subject to the conditions above.
The parameter space F r has a natural structure of a (categorical) quotient of some open subvariety U r of the sum of two linear GL 2 -representations by the action of GL 2 . Equivalently, F r is a (categorical) quotient of the open subvariety U ′ r = U r /G m of the weighted projective space P 4r,6r (4r + 1, 6r + 1) by the action of PGL 2 .
Lemma 3.1. The variety U ′ r is a disjoint union of locally closed subvarieties U ′ r,Γ , each preserved under the action of PGL 2 , such that for every u ∈ U ′ r,Γ one hasΓ(E u ) =Γ.
Proof. The set of (g 2 , g 3 ) (of bounded degree) such that the j-map decomposes as j Γ • j E is a closed algebraic variety (not necessarily irreducible). Its intersection with the open subvariety defined by the conditions 3.2 is also closed. It contains a finite number of proper locally closed subvarieties corresponding to proper finite index subgroups of Γ (their number is bounded as a function of r). The complement to these subvarieties consists of finitely many irreducible components, each preserved under the action of PGL 2 . For u in any of these components the lift of Γ to the monodromy group of the corresponding fibration E u is constant. The number of such possible lifts is finite. This gives a finite decomposition of U ′ r as claimed. The unstable points of the PGL 2 -action on the weighted projective space correspond to sections g 2 , g 3 with high order of vanishing at some point p. Namely ν p (g 2 ) > 2r, ν p (g 3 ) > 3r. However, the inequality (3.2) implies that 6r < 12. Thus, for r ≥ 2, F r is a PGL 2 -quotient of some open subvariety of the semistable locus P ss 4r,6r (4r + 1, 6r + 1) ⊂ P 4r,6r (4r + 1, 6r + 1). It follows that F r is a quasi-projective algebraic variety. This variety is clearly unirational and in fact rational by [18].
Moreover, for r ≥ 2 we can define a set of subvarieties F r,Γ ⊂ F r such that for every b ∈ F r,Γ the corresponding Jacobian elliptic surface (E b , s) has global monodromy groupΓ.
Remark 3.2. Notice that the maps j E for elliptic fibrations corresponding to different points of the same irreducible component of F r,Γ can have different RD(j E ), even over the A 2 or B 2 -ends of T Γ ⊂ M Γ . Thus, for a given irreducible component, we have the notion of a generic ramification datum RD(j E ) and its degenerations.
The case r = 1, corresponding to rational elliptic surfaces, is more subtle -the subvariety U ′ 1 contains unstable points. The quasi-projective locus of semistable points U ss r ′ is a disjoint union of locally closed PGL 2 -semistable subsets U ss r,Γ ; taking quotients we obtain varieties F 1,Γ parametrizing rational elliptic fibrations with global monodromyΓ.
′ be the complement. It consists of pairs (g 2 , g 3 ) with where l is a linear form (vanishing at a point p and) coprime to f 2 , f 3 and deg(f 2 ) = 1, deg(f 3 ) = 2. For w ∈ W ′ 1 we have deg(j) ≤ 4. The case of Γ = SL 2 (Z) corresponds to deg(j Γ ) ≥ 2. Thus we have to consider two cases: The first case does not occur since j −1 (0) has ramification of type (3, 1) (by the assumption that f 2 is coprime to l and that 3ν p (g 2 ) < 12). Thus the j-map cannot be decomposed even locally into a product of two maps. The second case leads to Lemma 3.3. If w ∈ W ′ 1 andΓ(E w ) = SL 2 (Z) then deg(j E ) = 1 and one has one of the following graph and ramification data: Proof. The formula j = lf 3 2 /(lf 3 2 − f 2 3 ) shows that j Γ has a point with local ramification datum (3,1)  Since only two more branch points are allowed and one of them is 1 (with local ramifications 1 or 2), the Euler characteristic computation gives the ramification data listed in the statement plus one more: However, this datum is impossible for topological reasons (the only possible graph datum is [A 6 + A 2 ] and there is a unique embedded graph T Γ with this datum).
If deg(j) = 3 then one has a cyclic point of order 3, leading to the data above.
Consider an irreducible component F r,Γ and the corresponding decomposition j = j Γ • j E . Here is a pair of homogeneous polynomials in 2 variables. Let be the subset corresponding to smooth elliptic fibrations. Put The monodromy group and its image in PSL 2 (Z) are uniquely determined by the smooth part of the elliptic fibration. Therefore, in any smooth family of elliptic surfaces Γ(generic fiber) ⊇ Γ(special fiber).
Since Γ is defined modulo conjugation by elements in SL 2 (Z) the claim follows.
Corollary 3.6. We have a decomposition G = G Γ into a finite (disjoint) union of algebraic GL 2 -stable subvarieties such that for all g = (g 2 , g 3 ) ∈ G Γ the monodromy groupΓ(E g ) ⊂ SL 2 (Z) is a subgroup of a central Z/2-extension of Γ.
Remark 3.7. For a given g ∈ G Γ the map j E is not unique. Let j E and j ′ E be two such maps. Then Lemma 3.8. We have a decomposition into a finite union of algebraic irreducible GL 2 -stable subvarieties such that Γ(E g ) =Γ for all g ∈ GΓ ,k .
Let GΓ = GΓ ,k be an irreducible component of G Γ as in Lemma 3.8 and g ∈ GΓ its generic point. It determines a set of * -fibers on the base P 1 . We denote their number by ℓ. Choose (one of) the j Eg , with ramification datum RD = RD(j Eg ). We get a map of rational maps with ramification datum RD.
Lemma 3.9. The map φ U is a local (complex analytic) surjection.
Proof. First observe that on M Γ there is a projective local system (Z⊕Z)/(Z/2) which induces a projective local system on an open part of P 1 . The obstruction to the extension of this system to a linear Z ⊕ Z-system is an integer modulo 2 which depends only on the topological type of the projective system. Therefore, it doesn't change under a small variation of maps j with fixed RD. Now it suffices to apply Kodaira's main theorem which guarantees the existence and uniqueness of an elliptic fibration with a given linear Z ⊕ Z-system. Proof. Since both F ′ r,Γ and R(RD) are algebraic varieties the local complex analytic surjection from Lemma 3.9 extends to an algebraic correspondence. Moreover, two decompositions of the map j as j = j Γ • j E differ by an element in H Γ . This gives a map to the quotient space, which is a (global) rational surjection.
Proposition 3.11. Every irreducible component F r,Γ contains an open part F ′ r,Γ with the following properties: • F ′ r,Γ is a quotient of an algebraic variety U ′ r,Γ,ℓ by the (left) action of PGL 2 and (right) action of a subgroup H Γ of Aut(T Γ ); • U ′ r,Γ,ℓ admits a fibration with fiber (an open subset of ) Sym ℓ (P 1 ) and base the variety R r,Γ of maps f : P 1 → M Γ with fixed local ramification data over A 2 and B 2 -points of T Γ ⊂ M Γ ; • the action of PGL 2 on U ′ r,Γ,ℓ is induced from the standard PGL 2 -action on P 1 ; • the group Aut(T Γ ) is a subgroup of PGL 2 (acting on M Γ ).
Proof. Elliptic surfaces parametrized by a smooth irreducible variety have the same ET(E), which depends on the number ℓ of * -fibers in E, on the degree of j E and on the ramification properties over the ends of T Γ . Once ℓ is fixed, for any given j E , the * -mark can be placed over arbitrary ℓ-points of P 1 . Their position defines a unique surface E. This implies that U ′ r,Γ,ℓ is fibered with fibers (birationally) isomorphic to Sym ℓ (P 1 ) = P ℓ . The ramification properties of j E remain the same on the open part of U ′ r,Γ,ℓ (since the number of *fibers remains the same). Thus the base of the above fibration is the space of rational maps f : P 1 → P 1 = M Γ with fixed ramification locus. Any such map defines an elliptic surface E with given Γ (see [8]). The PGL 2 -action on U ′ r,Γ,ℓ identifies points corresponding to isomorphic surfaces E. Additional nontrivial isomorphisms correspond to exterior automorphisms of Γ, coming from the action on M Γ , i.e., automorphisms of the graph T Γ .
Remark 3.12. If the PGL 2 × Aut(T Γ )-action on U ′ r,Γ,ℓ is almost free then the rationality of PGL 2 \U ′ r,Γ,0 /Aut(T Γ ) implies the rationality the corresponding quotients for all ℓ. In the other cases the degree of j E is small and they are handled separately (see Section 9).
Most of the graphs T Γ have trivial automorphisms. In particular, any nontrivial automorphism acts on the ends of the graph. In general, automorphisms of the pair (M Γ , T Γ ) correspond to elements of Γ ′ /Γ where Γ ′ ⊂ PSL 2 (Z) is a maximal subgroup with the property that Γ is a normal subgroup of Γ ′ .
Lemma 3.13. The group Aut(T Γ ) acts freely on the set of ends and end-loops.
If h stabilizes an end or an end-loop of T Γ then it stabilizes the unique adjacent vertex and its other end. Any element of PGL 2 (C) preserving a closed interval is the identity.
Corollary 3.14. For r ≤ 2, the only possible groups Aut(T Γ ) are cyclic, dihedral or subgroups of S 4 . More precisely, for graphs with one end Aut(T Γ ) = 1 and graphs with two ends Aut(T Γ ) is a subgroup of Z/2.

Lemma 3.15. Let
R := {f : P 1 → P 1 } be the space of rational maps with ramifications over exactly 0 and ∞. Then R is a G m -fibration over the product of symmetric spaces Sym m i (P 1 ).
Proof. Indeed any two cycles c 1 and c 2 of fixed degree are equivalent on P 1 . Therefore, there is a rational function f on P 1 with . The function f is defined modulo multiplication by a constant. The space of cycles c 1 = i n i p i is a product of symmetric powers Sym m (P 1 ) where m is the number of equal n i .

Combinatorics
In this section we investigate relations between ET(E) and ET(Γ). We keep the notations of the previous sections.
Here α 1 and α 2 equal the number of points over A 2 -ends of T Γ with ramification multiplicity 1 (mod 3) and 2 (mod 3), respectively, β 1 is the number of odd ramification points over the B 2 -ends and ℓ is the number of * -fibers of E.
Proof. The summand deg(j E )∆(Γ) corresponds to multiplicative fibers of E. The next summands are the contributions of those singular fibers of E which are in the preimage of A 2 or B 2 -ends of T Γ . If the ramification order at a point p over a B 2 -end is even then the corresponding fiber with minimal ET is smooth and hence does not contribute to ET(E). If it is odd then the fiber with minimal ET is of type III and we have to add 6β 1 . Similarly, for the preimages of A 2 -ends and * -twists.
is the number of independent closed loops of T Γ ⊂ M Γ .
The following simple procedures produce new graphs: • If T 1 and T 2 are (unmarked) trivalent graphs we can join T 1 and T 2 along two edges. For the resulting graph T ′ we have If T i are marked and the marking of the ends of T ′ is induced from the marking of the corresponding ends of T 1 and T 2 then • We can glue an end p of T 1 to an edge of T 2 . In this case The change of ∆ depends on the marking of the end: Remark 4.5. Any connected graph T can be uniquely decomposed into a union of a saturated graph and a union of trees.
Proof. Every vertex of T Γ has either one or three incoming edges. Therefore, the number of edges i is the number of vertices with i-edges). Thus τ 0 = τ 0 1 + τ 0 3 is even and since ET(Γ) = 6τ 0 we are done.
Example 4.7. If T Γ is a tree with k + 2 vertices then ET(Γ) = 12k + 12 Proof. A direct computation shows that for saturated graphs one has an equality. Suppose that T Γ is a concatenation of a saturated graph T sat and a tree T tree . The number of ends drops by one and the number of A 6 vertices increases by 1. Thus the tree will add 12k + 12 to ET(Γ) but ∆(Γ) will change by 6k + 6. Finally, the ratio ∆(Γ)/ET(Γ) only increases if we change B 2 -to A 2 -markings for some ends. Indeed, ∆(Γ) increases without changing ET(Γ).  Then: • M Γ = P 1 and T Γ is a tree without A 2 -ends and with ET(Γ) > 24; • deg(j E ) = 2 and it is ramified in all (B 2 ) ends of T Γ (and, possibly, some other points); • E has 1 or 2 singular fibers of type I n .
Proof. From 4.1 and 4.8 we conclude that rk H 1 (T Γ ) = 0 which implies that T Γ is a tree and M Γ = P 1 . By Lemma 2.1 and our assumption, ET(Γ) > 24, which implies that deg(j E ) ≤ 2. If deg(j E ) = 1, we apply Corollary 4.2 and get a contradiction to the assumption. For deg(j E ) = 2 combine Definition 1.7 and (4.1): ET(E) = ET(Γ) + 4a 2 + 4α 1 + 8α 2 + 6β 1 − 12. Since α 1 , resp. β 1 is twice the number of unramified A 2 , resp. B 2 -ends, and α 2 is the number of ramified A 2 -ends we see that if at least one of them is not zero, then ET(E) ≥ ET(Γ). The claim follows.  Proof. If deg(j E ) = 2 and C = P 1 then j E ramifies in two points. If neither of these points is B 2 then, by Lemma 4.1, ET(E) ≥ 2 ET(Γ). If both of these points are B 2 -points then the covering j E corresponds to a subgroup Γ ′ of index 2 in Γ and C = M Γ ′ , contradiction. Otherwise, the claimed inequality follows from Lemma 4.1.
In this section we assume that C = P 1 , that j E > 1 and that Γ is a proper subgroup of PSL 2 (Z). We consider In Section 3 we showed that the main building block in the construction of moduli space of elliptic surfaces with fixed Γ is the space of rational maps j E : C → M Γ of fixed degree and ramification restrictions over certain points.
For a general family there are no such restrictions and the corresponding moduli spaces are rational by classical results of invariant theory for actions of PGL 2 and its algebraic subgroups (see Section 7). For special families the corresponding space of rational maps is more complicated.
Lemma 5.2. If T Γ is not a tree and j E is special (and generic for the corresponding irreducible component of F 2,Γ ) then Proof. Follows from Lemma 4.1. First observe that ∆(Γ) ≤ 16, which implies that a 6 = 2 and a 2 ≤ 2. If a 2 = 2 then ∆(Γ) = 16 and are degenerations of the listed cases (see Remark 3.2).
Lemma 5.3. If T Γ is a tree and j E is special (and generic for the corresponding irreducible component of F 2,Γ ) then (In the above tables, + * means that there exists a moduli space of elliptic surfaces with the same RD(j E ) and with an additional * -fiber over an unspecified point.) Proof. Assume that ET(Γ) = 24 and T Γ is a tree with then j E has to be completely ramified over all ends and no other ramifications are allowed by Euler characteristic computation. Therefore, it is a group-covering and can't be j E . If deg(j E ) = 5 then there are two odd ramifications over B 2 -ends, and by 4.1, ET(Γ) > 48.
We are left with If there are at least two A 2 -ends without 3-cyclic ramification points over them then ET(E) > 48 (see 4.1). The first case is impossible: deg(j E ) = 4 does not occur (the degree is not divisible by 3), if deg(j E ) = 3 and there is at most one 3-cyclic ramification over an A 2 -end then, by 4.1, ET(E) > 48, contradiction. Consider the second case and deg(j E ) = 4. Then ∆(Γ) = 10 and 4α 1 + 8α 2 + 6β 1 ≤ 8. Since α 1 ≥ 2 we have α 2 = β 1 = 0 and α 1 = 2. The only possible which corresponds to a group covering, contradiction.
Consider the case T Γ = A 6 + 3B 2 . Here ∆(Γ) = 6 and where n is a number of points with odd ramification over B 2 -vertices. It follows that 48 ≥ 6 deg(j E ) + 6β 1 and β 1 ≥ 3 if deg(j E ) is odd and the number of odd ramifications over each B 2 -end is congruent to deg(j E ) modulo 2.
If deg(j E ) = 8 then all preimages of B 2 -vertices are 2n-ramified. If deg(j Γ ) is odd then ET(E) ≥ 6 deg(j Γ ) + 18, which excludes deg(j Γ ) = 7. Now assume deg(j Γ ) = 6. The number of possible odd ramifications over any B 2 -end is even by 4.1 and it cannot exceed 2. There are two possibilities listed above. Assume that deg(j E ) = 5. The minimal possible ramifications are (2, 2, 1) over all B 2 -ends. Since 10 − 6 = 4 we can add two more points.
Lemma 6.2. If j E is special (and generic for the corresponding irreducible component of F 1,Γ ) then We have deg(j E ) ≤ 6 and α 1 = α 2 = 0. Notice that deg(j E ) = 5 is impossible.

General rationality results
Notations 7.1. We will denote by S n the symmetric group on n letters, by A n the alternating group, by D n the dihedral group and by C n = Z/n the cyclic group. In particular, S 2 = C 2 = Z/2 and D 2 = Z/2×Z/2 (sometimes we prefer the notation S 2 over C 2 to stress that the action is by permutation). We write Gr(k, n) for the Grassmannian of k-planes in a vector space of dimension n and V d for the space of binary forms of degree d. We will denote by GL 2 , PGL 2 , G m etc. the corresponding complex algebraic groups. For a group G, we denote by Z g the centralizer of g ∈ G and by Z G its center. We denote by M 2 = V 1 ⊕ V 1 the space of 2 × 2-matrices. We write V V −→ X or simply V −→ X for a locally trivial (in Zariski topology) fibration V over X with generic fiber V . We will often write G-map (etc.), instead of G-equivariant map.
We say that two algebraic varieties X and X ′ are birational, and write X ∼ X ′ , if C(X) = C(X ′ ). A variety X of dimension n is rational if X ∼ A n , k-stably rational if X × A k ∼ A n+k and stably rational if there exists such a k ∈ N. We say that X is unirational if X is dominated by A n . The first basic result, a theorem of Castelnuovo from 1894, is: Already in dimension three, one has strict inclusions rational stably rational unirational (see the counterexamples in [16], [2], [10], [6]). There is a very extensive literature on rationality for various classes of varieties. We will use the following facts: Lemma 7.3. Let S → B be a ruled surface with base B and π : C → S a conic bundle over S. Assume that the restriction of π to a generic P 1 ⊂ S is a conic bundle with at most three singular fibers. Then C ∼ A 2 × B.
Lemma 7.4. Let π : C → S be a conic bundle over an irreducible variety S and Y ⊂ C a subvariety such that the restriction of π to Y is a surjective finite map of odd degree. Then C has a section and C ∼ S × A 1 .
Let G be an algebraic group. A (good) rational action of G is a homomorphism ρ rat : G → Bir(X) such that there exists a birational model X ′ of X with the property that ρ rat extends to a (regular) morphism G × X ′ → X ′ . We consider only rational actions. We write X ∼ G Y for a G-birational (= G-equivariant birational) isomorphism between X and Y . We will denote by G\X a model for the field of invariants C(X) G .
Let E → X be a vector bundle. A linear action of G on E is a rational action which preserves the subspace of fiberwise linear functions on E. In particular, there is a linear G-action on regular and rational sections of E.
We are interested in rationality properties of quotient spaces for the actions of PGL 2 , its subgroups and products of PGL 2 with finite groups. The finite subgroups of PGL 2 are C n , D n , A 4 , S 4 , A 5 .
We denote byC n ,D n etc. their lifts to GL 2 (as central C 2 -extensions). We denote by B, T = C * , N T the upper-triangular group, the standard maximal torus and the normalizer of this torus in PGL 2 and byB ,T, NT the corresponding subgroups in GL 2 (or SL 2 ).
Let V be an n-dimensional vector space,G ⊂ GL(V ) a subgroup and G its projection to PGL(V ), acting naturally on P(V ). Determining the rationality of quotients G\P(V ) (at least for finite groups) is known as Noether's problem. Theorem 7.6. [29], [36] A quotient of P(V ) by a (projective) action of a connected solvable group, a torus or a finite abelian subgroup of a torus is rational.
A fundamental rationality result is the following theorem of Katsylo: Theorem 7.7. [17] For any representation V of GL 2 or PGL 2 the quotient PGL 2 \P(V ) is rational.
In general, the quotients need not be rational (see Saltman's counterexamples in [30]). We now describe some partial results for n = 4, which we will use later on. Remark 7.11. In [26] it is shown that where X 3 is the Segre cubic threefold and G ′ is a quotient of G. The problem is then reduced to the (easy) case of imprimitive actions.
We will also need to consider quotients by nonlinear actions.
Lemma 7.12. The quotient of GL 2 (or PGL 2 ) by the involution i : x → x −1 is rational.
Proof. The involution decomposes as a product i = i 1 • i 2 , where A (rational) slice for the action of G is a subvariety S ⊂ X such that the general G-orbit intersects S in exactly one point. (The slice S need not be a rational variety. To avoid confusion, we will always refer to S as a slice.) A subvariety Y ⊂ X is called a (G, H)-slice (where H ⊂ G is a subgroup) if G · Y ∼ X and gy ∈ Y implies that g ∈ H. Clearly, G\X ∼ H\Y . Moreover, if f : X → X ′ is a G-equivariant morphism and Y ′ is a (G, H)-slice in X ′ then f −1 (Y ′ ) is a (G, H)-slice in X.
Notations 7.13. For (a reductive group) G acting (rationally) on X we denote by St gen = St gen (G, X) the generic stabilizer (defined up to conjugacy). The action is called an afaction (almost free) if St gen is trivial.
We use a more precise version of Theorem 7.7: Theorem 7.14. If the central C 2 ⊂ G ′′ then where S is a rational variety (with trivial G-action). If C 2 ⊂ G ′′ then • either the PGL 2 -action on P(V ) has no slice and G\P(V ) is rational • or P(V ) ∼ G G × S, where the slice S is a rational variety (with trivial G-action).
We now explain some general techniques in the study of rationality of quotient varieties.
Lemma 7.15. Let E → X be a vector bundle of rank r = rk (E). Let G be an (affine) reductive group acting on E such that the generic orbit of G in E projects isomorphically onto a generic orbit of G in X. Then a G-equivariant sequence of vector bundles such that the generic G-orbit of E ′ projects isomorphically onto its image. Choose a generic G-equivariant section s ′ of E ′ → X and denote by E ′′ s ′ the restriction of E ′′ to this section. Then Proposition 7.17. Let X be a variety with an action ρ : G → X of a linear algebraic group G. Let E → X be a vector bundle andρ :G → E aG-action lifting ρ. Consider a generic orbit G · x ⊂ X and the linear action ofG on the space of sections H 0 (X, E).
Assume thatG is reductive and V is a linear representation ofG which is contained in H 0 (X, E). Then there exists an affine open X ′ ⊂ X such that the vector bundle E → X ′ admits aG-map onto aG-representation V * .
If the action of G on X is almost free we may think of X as being (birational to) a principal fibration over the quotient G\X with fiber G. If G is affine we may assume that X and G\X are also affine. Let us also recall a standard general construction of G-maps: if the ring C[X] is a direct sum of G-modules then any G-submodule V ⊂ C[X] defines a G-map X → Spec(V ). We also have a vector bundle version of the above construction: let E → X be a G-

Proof. A generic orbit O has a G-equivariant neighborhood
The module H 0 (U, E) is a direct sum of finite dimensional irreducible Gmodules. We can now take any submodule V ⊂ H 0 (U, E) which surjects isomorphically onto a submodule in H 0 (O, E).
Lemma 7. 19. If X has an af -action of PGL 2 then with diagonal PGL 2 -action on the left and trivial PGL 2 -action on P(V 2d ) on the right.
Proof. We know that C[PGL 2 ], as a PGL 2 -module, is sum of all even modules V 2d . This gives a PGL 2 -map s : X → P(V 2d ). The quotient is a projective bundle over the quotient PGL 2 \X, with a section obtained from s. Therefore, it is birational to the product (PGL 2 \X) × P(V 2d ), which gives the claimed PGL 2 -isomorphism.
Corollary 7.20. Let X be a variety with an af -action of PGL 2 . Then X is a (PGL 2 , N T )-slice in X × P(V 2 ) (with diagonal PGL 2 -action).
Lemma 7.21. Assume that X has an af -action ρ of PGL 2 . Let V V −→ X be a vector bundle over X with an actionρ of GL 2 lifting ρ. Assume that Proof. Let Y be an orbit such that H 0 (Y, V Y ) contains V d , for some odd d. Shrinking X, if necessary, gives a surjective map of GL 2 -modules We obtain a PGL 2 -equivariant surjective map Since the stabilizer of a point in P 1 is solvable, we get a slice S ⊂ P(V), as claimed.
Assume that there is an orbit Y ∼ PGL 2 such that V Y contains only even weight GL 2 -submodules. Then the central C 2 ⊂ GL 2 acts trivially on V Y . If follows that V Y is a trivial PGL 2 -bundle, and H 0 (Y, V Y ) a trivial PGL 2module. The semi-simplicity of the PGL 2 -action implies that H 0 (X, V) contains H 0 (Y, V Y ) as a submodule. Shrinking X if necessary, we can find linearly independent PGL 2 -invariant sections, whose specializations to Y generate H 0 (Y, PGL 2 ). Therefore, V is lifted from the quotient PGL 2 \X.
Proof. Consider the map defined on the open, G × GL 2 -invariant subset of noncollinear pairs (v, v ′ ) ∈ V ⊕ V (with fibers consisting of pairs spanning the same 2-space). The GL 2action on the fibers is the right multiplication on matrices: Assume that G is reductive and denote by G ′′ := St gen (G, Gr(2, V )) and by G ′ := G/G ′′ the quotient group of G which acts effectively on Gr(2, V ).
The group PGL 2 acts on P(M 2 ) on both sides. We will need an explicit description of the action for some of its subgroups. Proof. Indeed N T contains The corresponding actions on P(M 2 ) are  Here C * D ×C * H is the quotient of the fiber C * ×C * of U → C * by the intersection of D, H with the diagonal C * ∆ ⊂ C * ×C * . Isomorphisms C * H → C * and C * D → C * induce a birational fiberwise isomorphism and it suffices to consider D = S 2 , H = S 2 . In this case, an alternative equivariant completion of U is given by with an action of S 2 × S 2 , where the first S 2 acts as an involution on the first two factors and identity on the base while the complementary S 2 acts only on the base. Thus the quotient is a conic bundle over the complement in to the branch locus of the quotient map. Here the left (resp. right) S 2 acts as an involution on the left (resp. right) P 1 and the branch locus is exactly the union of four lines. By Lemma 7.3, this conic bundle has a section (it is nonsingular on a pencil of lines minus at most two points).
Lemma 7.30. Let G be a subgroup of SL 2 , not equal toÃ 5 , and V a linear representation of G. Then G\P(V ) is rational.
Proof. For G = SL 2 this is a theorem of Katsylo [17]. We now consider proper subgroups G SL 2 . If G is solvable and connected then rationality for the quotient follows from a theorem of Vinberg [36]. For compact G the proof is similar to the dihedral case described below. Assume now that G is finite and not equal toÃ 5 . Then G is either 1. a finite subgroup of C * , 2. a dihedral group or 3.Ã 4 ,S 4 . The first case is easy. For dihedral groups all irreducible representations of G have dimension ≤ 2 and the corresponding quotients are rational by Theorem 7.2. Let V be a faithful representation of a dihedral group D (otherwise, we are reduced to a quotient group). Thus V = W ⊕ W ′ , where dim W = 2 and dim W ′ ≥ 1. Denote by G ′ = G/C ′ the quotient acting faithfully on W ′ (C ′ is a cyclic group). We have W ∼ D C * × P 1 , with trivial action of D on C * and trivial action of C ′ on P 1 . By Lemma 7.15, with trivial action of G ′ on C * × P 1 . Thus and we can apply induction.
We turn to the last case. An irreducible representation ofÃ 4 is either a character, or a faithful two-dimensional representation, or a three-dimensional representation, trivial on the center (a faithful representation of A 4 ). An irreducible representation ofS 4 is either a faithful two-dimensional representation, a faithful four-dimensional representation W := Sym 3 (V 1 ) or a representation of S 4 (of dimension ≤ 3).
For irreducible representations of dimension ≤ 3 rationality for the quotient follows from Theorem 7.2. We turn to W . Recall that is the cubic character. A pair of (generic) points defines a line P 1 ⊂ P(W ). This shows that where S 4 acts on the base, A 4 acts linearly on the fiber L and S 2 =S 4 /S 4 acts as an involution on the fiber L. ThusS 4 \P(W ) is a conic bundle over the rational surface S 4 \(P 1 × P 1 ). We now analyze the geometry of this bundle in more detail. Consider the action D 2 ⊂ S 4 on P 1 × P 1 and on P 2 = Sym 2 (P 1 ). Every involution i ∈ D 2 has two invariant points x i , y i . Consider the graphs P 1 connecting the points (x i , y i ) − (y i , x i ). Their set is equal to P 1 and there is a graph: consisting of points (x, i(x)). The line l i is exactly the subset of i-invariant points in P 1 ×P 1 . The action of D 2 is free outside the three lines l i , i ∈ D 2 , i = 1. There are exactly 6 points which are invariant under D 2 . The corresponding action on P 2 can be described as follows. There are three points corresponding to (x i , y i ) which are stable under D 2 and three lines (images of l i ) so that the action is free on the torus C * × C * (the complement in P 2 to the union of l i ). The group D 2 acts on C * × C * as a translation by the subgroup of points of order 2.
The quotient P 2 q := D 2 \P 2 is a nonsingular variety isomorphic to P 2 (indeed the only possible singularities come from the three D 2 -invariant points in P 2 but the quotient by the local action is nonsingular). The diagonal P 1 ∆ ⊂ P 1 ×P 1 projects onto a conic C ⊂ P 2 , which is invariant under D 2 . The conic C intersects the "vertical" and "horizontal" subgroups in C * × C * ⊂ P 2 in two points and does not intersect the line at infinity.
Thus in P 2 q = D 2 \P 2 , the image of P 1 ∆ intersects C * in one point. Therefore, the images of P 1 ∆ and of l i are lines (since pairwise intersections of the l i are equal to 1) and the (C 2 ) 3 -covering P 1 × P 1 → P 2 q is ramified exactly over a union of four lines. If suffices to observe that every conic bundle over P 2 q has a section. Indeed, let p be the intersection point of two lines l i and l i ′ and consider the pencil of lines in P 2 q through p. Each line in this pencil intersects the ramification locus in at most three points and we can apply Lemma 7.3. Now we turn to reducible representations V = ⊕ α∈A V α ofÃ 4 . If V is faithful forÃ 4 then there is an α 0 ∈ A such that V α 0 is a three-dimensional irreducible faithful representation ofÃ 4 and Lemma 7.15). If V is faithful for A 4 then V contains a faithful irreducible three-dimensional representation of A 4 and we can apply the same argument. In all other cases V is a sum of one-dimensional representations and we are reduced to Case 1.
Finally, consider reducible representations V ofS 4 . If V is faithful then it contains either a faithful irreducible two-dimensional representation or the faithful representation W . Again, we apply Lemma 7.15 as before. If V is faithful for S 4 then it contains a faithful irreducible representation of dimension ≤ 3 and we conclude as above. In all other cases V is a sum of one-dimensional representations.
The G-action is equivalent to a G-action on a vector bundle G\ Gr(2, V ) G\ Gr(2, W ) = G\P 2 .
Finally, let us consider the case ofS 4 . Let W be its unique irreducible representation of dimension four (as in Lemma 7.30). We claim thatS 4 \ Gr(2, W ) is rational. Indeed, asÃ 4 -modules, we have where W χ , W −χ are two copies of the standard representation ofÃ 4 of dimension 2 and χ (resp. −χ) indicates the eigenspace decomposition for the nontrivial character χ : , with a linear A 4 -action (since the center acts trivially) and a permutation S 2 inverting the map w ∈ Hom(W χ , W −χ ). More precisely, W −χ = (W χ ) * and where C 1 corresponds to skew symmetric maps and A 4 acts on C 1 by χ. The involution S 2 = S 4 /A 4 acts on C 1 and on Sym 2 (W −χ ) as t → t −1 . In particular, if C * × C * is the diagonal group acting on Sym 2 (W −χ ) ⊕ C 1 then S 2 acts as X → s −1 X, where s ∈ C * × C * and X ∈ Sym 2 (W −χ ) ⊕ C 1 . There is an equivariant map with an effective action of S 3 = S 4 /D 2 on the target C 1 , which to a subspace s ∈ C 2 ⊂ W χ ⊕ W −χ assigns the value of the 2-form (x, s(y)) − (s(x), y). The fiber of f is D 2 -birational to Sym 2 (W χ ) = P 2 . We have already seen in the proof of Lemma 7.30 that D 2 \P 2 = P 2 . ThusS 4 \ Gr(2, W ) is a C * -bundle over a P 2 -fibration over S 2 \C 1 . It is clear that this P 2 -fibration is trivial. The quotient conic bundle is nondegenerate over a product of P 2 with an open subvariety in C 1 /S 3 . Hence it has a section. Rationality ofS 4 \ Gr(2, W ), and more generally,S 4 \ Gr(2, W ⊕ · · · ⊕ W ), follows (the latter is a vector bundle over the former).
Assume now that V = nW ⊕ V ′ , where dim V ′ ≥ 1, and n ∈ N. Since the S 4 -action on Gr(2, nW ) is af there is aS 4 -equivariant homogeneous rational map f : Gr(2, nW ) → V ′ sending the genericS 4 -orbit in W to the generic S 4 -orbit in V ′ . Notice that the center C 2 acts as a scalar on Hom(W, V ′ ). We have (birationally) (with rational bases). The projective bundle on the right has a section. Indeed, is an equivariant quotient bundle of the trivial bundle with fiber Hom(W, V ′ ). The map f defines an S 4 -equivariant section s(f ) in the projective bundle in (7.1). The (equivariant) linear projection Hom(W, V ′ ) → Hom(C 2 x , V ′ ) maps s(f ) to an equivariant section of the bundle in (7.2). Thus s(f ) projects onto a section of the bundle on the right in (7.1), making it birationally trivial.
(If all weights in V are of the same parity then Gr(2, V ) carries the PGL 2action, otherwise the SL 2 -action.)
Proof. Consider first irreducible representations V = V d = Sym d (V 1 ) and assume that the stabilizer of a generic line P 1 ⊂ P(V ) contains a nontrivial cyclic group C. Then C fixes at least two points in this P 1 . Any orbit of C on P 1 is a union of a zero-cycle C · x and a zero-cycle supported in the fixed points. In particular, the subvariety of points in P ( V d ) which are fixed by C has dimension ≤ d/|C|. The dimension of the variety of C-fixed lines in P(V ) is therefore ≤ 2d/|C|. The subvariety of distinct cyclic subgroups C ⊂ PGL 2 has dimension 2 and dim Gr(2, V d ) = 2d − 2. Since d/|C| ≤ d/2 the inequalities 2d − 4 > 2d/2 and d − 4 > 0 imply the result. Assume that V = ⊕ j∈J V d j , |J| ≥ 2 and that St gen = 1. Then d j ≤ 2, for all j ∈ J. Indeed, the stabilizer of a generic P 1 through a generic point p ∈ P(V d ) is a subgroup of the stabilizer of p, which stabilizes some generic line in the tangent space at p. This group is trivial for d > 2 and equal to C 2 for d = 2.
In the remaining cases d j = 0 or 1, for all j ∈ J. If V contains at least three copies of V 1 then the argument above shows that the action is af . Similarly, if V = V 1 ⊕ V 1 then St gen = C * and if V = V 1 ⊕ V 1 ⊕ V 0 then St gen = 1. For V 1 ⊕ 3V 0 , the generic stabilizer is the same as for three linear functionalswhich is zero.
Proof. By Lemma 7.32, if dim V ≥ 5 then the St gen = 1 and we can apply Lemma 7.19 and Corollary 7.20 to conclude that The claim follows from Lemma 7.31. It remains to consider: In the first case, St gen (PGL 2 , Gr(2, V 4 )) = S 2 , with normalizer N T ⊂ PGL 2 . We claim that the subset X ⊂ Gr(2, V 4 ) of S 2 -invariant points is a (PGL 2 , N T )slice. Indeed, there is a Zariski open subset U ⊂ X such that the stabilizer of each point in U is exactly S 2 . In particular, g · U intersects U only if g ∈ N T . Consider the P 2 ⊂ P(V 4 ) consisting of S 2 -invariant subschemes containing 4 points. Any line in U joins a pair of points in this P 2 . Therefore, we have a (birational) N T -isomorphism of U and Sym 2 (P 2 ). The stabilizer of a generic point in X is a central subgroup in N T whose action on P 2 is equivalent to a linear action on C 2 . (Indeed, Sym 2 (V 1 ) = C ⊕ W 2 , where C is the trivial representation -the invariant symmetric form -and W 2 is a faithful two-dimensional representation of N T /S 2 ). Thus instead of X with the N T -action we can consider C 2 × C 2 with the (N T /S 2 ) × S 2 -action (where the second S 2 interchanges the factors). In particular, (by linearity) and is hence rational.
In the second case, Gr(2, V 3 ) has a surjection of degree 2 onto P(V 4 ). The connected component of the preimage of the (PGL 2 , S 4 )-slice P 1 in P(V 4 ) is a (PGL 2 , A 4 )-slice, isomorphic to P 1 . The quotient is rational.
If V is reducible and the PGL 2 -action on the Grassmannian has nontrivial stabilizer then dim V < 5. Rationality follows since dim Gr(2, V ) ≤ 4 and the generic orbit has dimension at least 2. Proof. The action is birational to the (projective) action of G × H on P(M 2 ), where G acts on the right and H on the left. The groups G, H are either: • cyclic; • dihedral or • A 4 , S 4 . The case of primitive solvable groups is covered by Theorem 7.10, [26]. If V is reducible then there is a nontrivial action of C * on G\P(V )/H, leading to rationality. This covers the case when either G or H is cyclic.
We claim that if V is irreducible and imprimitive (for the G × H-action) then either G or H is dihedral. By definition, V := M 2 = ⊕ α V α , such that gV α = V α ′ for allg ∈ G × H. Moreover, by irreducibility, all V α must have the same dimension, = 1 or 2. Notice that imprimitivity for an action of a group G ′ implies imprimitivity for the induced action of every subgroup G ′′ ⊂ G ′ (with the same decomposition of V ). We now claim that the actions of A 4 × A 4 , and consequently of A 4 × S 4 and S 4 × S 4 are primitive. Indeed, A 4 × A 4 contains D 2 × D 2 as a normal subgroup, for which the imprimitive structure is either a sum of two subspaces of dimension 2 or four subspaces of dimension 1, corresponding to the choice of a subgroup S 2 ⊂ D 2 . The first possible imprimitive structure for D 2 × D 2 does not extend to one for A 4 × A 4 (which has no index 2 subgroups). The second structure is also impossible: A 4 rotates the subgroups S 2 ⊂ D 2 , hence there is no A 4 -invariant imprimitive structures for D 2 × D 2 .
It remains to consider the case when both G and H are dihedral. On V 1 there is a unique imprimitive structure, corresponding to the eigenspaces C 1 , C 2 of the elements of G. In particular, there is an imprimitive structure on We claim that (birationally) is a conic bundle degenerating precisely over the image of the diagonal and the subvarieties in P 2 with nontrivial stabilizers. Indeed, since H ⊂ N T (a C 2 -extension of C * ), (birationally) The quotient C * \P(M 2 ) is (birationally) a fibration over P 1 × P 1 , with S 2 acting by permutation, where the coordinate P 1 s are the projectivizations of the two-dimensional eigenspaces for the C * -action on M 2 . Thus is a conic bundle nondegenerate outside a conic (the image of the diagonal in P 1 × P 1 ). The G-action commutes with the N T -action and is effective on the base. This proves the claim.
We have G ⊂ N T and G\P 2 → N T \P 2 is a conic bundle. Since the left and right actions of N T commute, G\P 2 contains an open subvariety U × C * where the restriction of the conic bundle is nondegenerate. Here C * = G\N T and U is a subset of P 1 = N T \P 2 . Therefore the conic bundle has at most 2 singular fibers on any completion of the fiber C * ⊂ U × C * . Rationality follows.
We can now describe some open subvariety in the quotient G\P(M 2 )/H explicitly. Consider the action of C * ⊂ N T on both sides C * \P(M 2 )/C * . With respect to this action P(M 2 ) is birationally equivalent to a trivial C * × C *fibration over P 1 . Now we add the action of S 2 on both sides. The product S 2 × S 2 acts on the base P 1 . The group S 4 contains a normal subgroup D 2 ⊂ N T and the action of each S 2 ⊂ D 2 inverts the respective C * action. Thus (birationally) N T \P(M 2 )/NT where the deleted points are the ramification points of the map P 1 → P 1 /D 2 . In particular, there is an open U such that By Lemma 7.3, the conic bundles are trivial.
Finally, the conic bundles on P 2 /S 4 and P 2 /A 4 have sections. Indeed, both A 4 and S 4 contain dihedral subgroups of index 3 (D 2 , resp. D 4 ). The image of the section in the conic bundle over D 2 \P 2 (resp. D 4 \P 2 ), has odd degree in the conic bundles over A 4 \P 2 and S 4 \P 2 , respectively. We apply Lemma 7.4. Proposition 7.36. Let V be an irreducible GL 2 -representation and H ⊂ SL 2 a finite group, not equal toÃ 5 . Then First we assume that V has odd weight. The Grassmannian Gr ( Assume that V has even weight. If the PGL 2 -action is af then If it is not af , then, by Lemma 7.32, V = V 4 or V 2 . For V = V 4 we have the (PGL 2 , N T )-slice X = Sym 2 (P 2 ) with the N Taction which we can replace by C 2 × C 2 with a (N T /C 2 ) × C 2 -linear action. In particular, we identify the quotient with a quotient of C 2 ⊕ C 2 ⊕ V 1 ⊕ V 1 by a linear action of NT × S 2 × H (where NT ⊂ GL 2 ). The action of NT × H on V 1 is transitive with stabilizer C 2 × H. Hence it is equivalent to the action of D 2 × H on C 2 ⊕ C 2 ⊕ V 1 , which is a C 2 -vector bundle (permutation of the anti-invariant part of S 2 -action) over C 2 × V 1 , with D 2 × H action. The latter quotient is rational. For V = V 2 the action is transitive on Gr(2, V ) = P 2 and the quotient has dimension 2 -rationality follows.
We will also need a more general result for H = S 2 .
be a GL 2 -homogeneous line bundle. If at least one d j = 2 then GL 2 \X × X/S 2 is rational.
Proof. Case 1. |J| = 1. If d = d 1 is even or if d is odd and the line bundle has odd degree on P(V d ) then and we apply Proposition 7.36. If the line bundle has even degree then it is trivial and GL 2 acts as PGL 2 × C * . If the PGL 2 -action on P(V d ) is af we have for a rational slice S (with trivial PGL 2 -action). We have a PGL 2 × C * × S 2action on The quotient variety is a vector bundle over PGL 2 \PGL 2 × PGL 2 /S 2 (rational by Lemma 7.12). The claim follows. If the PGL 2 -action is not af , then V = V 3 or V 1 . For V 1 the quotient is rational by dimensional reasons. For V 3 we have a projection commuting with both actions. Recall that Gr(2, V 3 ) has P 1 as a (PGL 2 , A 4 )slice, with A 4 effectively acting as a cyclic group C 3 = A 4 /D 2 on P 1 (the group D 2 acts trivially on the (PGL 2 , A 4 )-slice P 1 ⊂ P 4 and similarly for Gr(2, V 3 ), see Lemma 7.32). Thus the quotient is the same as for the bundle under the action of A 4 × S 2 . In particular, it is a vector bundle over a P 2 = D 2 \P 1 × P 1 /S 2 -fibration over P 1 = P 1 /C 3 , hence is rational.
Case 2. |J| ≥ 2. If at least one d j is odd and > 1 or if all d j = 1 and |J| > 2, then there is a slice S and the PGL 2 -action is af . We can write Y as (the total space of the) line bundle: and, using Lemma 7.21, reduce to either a vector bundle over when L is trivial on PGL 2 , or to GL 2 \GL 2 × GL 2 /S 2 otherwise. In both cases the base is rational by Lemma 7.12.
If d j = 1 for every j ∈ J and |J| = 2 then the there is a map (P 1 ) 4 → P(V 4 ) = Sym 4 (P 1 ) = P 4 (of degree 24, mapping 4 points to a form of degree 4). The preimage in (P 1 ) 4 of the (PGL 2 , S 4 )-slice P 1 s = P 1 of P 4 , will be a set of six lines P 1 g,h , labeled by a pair of generators g, h ∈ D 2 (which act trivially on P 1 s ⊂ P 4 ). More precisely, the line P 1 g,h is the set given by (x : gx : hx : ghx) ∈ (P 1 ) 4 , for x ∈ P 1 . The map P 1 g,h → P 1 s = P 1 t,s /D 2 has degree 4. Thus P 1 g,h is a (PGL 2 , D 2 )-slice of (P 1 ) 4 and the quotient of a vector bundle L⊕L −→ P 1 by a linear action of D 2 is rational.
Assume that all d i are even. Then L is (birationally) trivial. Unless |J| = 2 and d 1 = d 2 = 2, there is a decomposition of such that the PGL 2 -action is af and (with trivial PGL 2 -action on P(V d )), by Lemma 7.19. The quotient is birational to a vector bundle over PGL 2 × C * \X ′ × X ′ /S 2 , where X ′ is the trivial line bundle over Y ′ .
We have reduced to |J| = 1 treated in Case 1 or to |J| = 2 and d 1 = d 2 = 2, treated in Lemma 7.38.
Lemma 7.38. The quotient is rational, where P 1 (V 2 ) and P 2 (V 2 ) are different copies of P 2 = P(V 2 ) and S 2 acts by permutation.
Proof. Consider the projection The space P(V 4 ) has a (PGL 2 , S 4 )-slice P 1 s (the D 2 -invariant polynomials). The zeroes of a (polynomial) p ∈ P 1 s form an orbit under D 2 . The preimage pr −1 (P 1 s ) ⊂ P 2 × P 2 consists of 3 lines, each invariant under D 2 . Indeed, the ordered pair (Q 1 , Q 2 ) corresponds to a choice of a generator g ∈ D 2 such that x, g(x) are zeroes of Q 1 and h(x), hg(x) are zeroes of Q 2 . Thus the line P 1 g ⊂ P 2 × P 2 consists of tupels {(x, gx), (hx, ghx)}, where x is an arbitrary point in P 1 and (x, gx) = Q 1 , (hx, ghx) = Q 2 . The map P 1 g → P 1 s has degree two and its fibers coincide with orbits of h (since g acts trivially on P 1 g ). The action of h is given by Thus h(Q 1 , Q 2 ) = (Q 2 , Q 1 ) and the action of h coincides with the restriction of the permutation action on P 2 × P 2 to P 1 g . The line P 1 g is invariant under D 4 × S 2 (considered as a subgroup of (PGL 2 × S 2 )). The group S 4 permutes the lines in pr −1 (P 1 s ). Each P 1 g is a (PGL 2 × S 2 , D 4 × S 2 )-slice of P 2 × P 2 . Therefore, The space P 2 × P 2 contains a subspace C 2 × C 2 with a linear action of D 4 × S 2 . Indeed, the action of D 4 on P 1 corresponds to the irreducible representation ofD 4 on C 2 = V . Under the D 4 -action, one has a decomposition Sym 2 (V ) = V ′ ⊕V ′′ , where dim V ′ = 2, dim V ′′ = 1 and the action of D 4 on P 2 is equivalent to the linear action on V ′ . The additional S 2 permutes the P 2 and hence acts by permutation on V ′ ⊕ V ′ . Thus (a vector bundle). Consider the effective action of (the nonabelian group) D 4 × S 2 on P 1 . It has a normal subgroup D 2 ×S 2 with generators g, h, k and an element i, i 2 = 1 which commutes with g, k and acts on h as ihi = gh. The stabilizer of a generic point on P 1 g is a normal abelian subgroup generated by g, hk. Thus D 4 × S 2 acts on P 1 effectively through the quotient D 4 / g, hk = D 2 . The action of this D 2 on P 1 is almost free. Indeed, the action of k coincides with the action of h and permutes Q 1 , Q 2 . Thus the orbits of h and k on P 1 g coincide with fibers of the map P 1 g → P 1 s . On the other hand, i acts nontrivially on P 1 s . We claim that is a vector bundle. Indeed, consider the subspace V ′ inv ⊂ V ′ ⊕ V ′ of invariant vectors (under the permutation). The action of D 4 ×S 2 on ((V ′ ⊕V ′ )/V ′ inv )×P 1 is almost free. Hence × P 1 has a fiberwise (scalar) C * -action commuting with the D 4 × S 2 -action. Since every C * -action has a slice, is rational by dimensional reasons: X ′ /C * is a unirational, therefore, rational surface and X ′ ∼ (X ′ /C * ) × C * . Then is rational (where H acts trivially on P(V ℓ )).
Proof. If ℓ is even and the action of GL 2 or a quotient of GL 2 by a central subgroup is af then we apply Lemma 7.19 combined with Proposition 7.36, resp. 7.37. If ℓ is odd and the action is af then there exists a slice, which is a rational variety, by Lemma 7.31 resp. 7.30. Rationality follows. Now we assume that the action is not af . This means that d ≤ 4. The subcases with d ≤ 2 are trivial since the action on the corresponding Grassmannian is transitive. If ℓ is odd, then the PGL 2 -action on Gr(2, V ) × P(V ℓ ) has a rational slice and our claim follows.
If d = 3, the action of PGL 2 on Gr(2, V 3 ) has a (PGL 2 , A 4 )-slice P 1 . For even ℓ > 0 the action of A 4 on P ℓ is faithful and it lifts to a linear representation of of A 4 . Further, A 4 -acts on P 1 is through a cyclic quotient. Thus with trivial A 4 -action on the P 1 on the right. This implies that the quotient is equivalent to P 1 × (P ℓ /A 4 ) × (V 1 ⊕ V 1 )/C * × H, a product of rational varieties.
If d = 4, the action of PGL 2 on Gr(2, V 4 ) has a (PGL 2 , N T )-slice X ′ . The action of N T on P(V ℓ ) is linear and the quotient of X × P ℓ is a vector bundle over the quotient of X, which is rational.
is rational (where S 2 acts trivially on P(V ℓ ) and by permutation on X).
Proof. The same argument as in the proof of Proposition 7.39 shows that it suffices to assume that the action on X is not af . This happens only if Y = P 2 or P 1 . The case Y = P 2 reduces to Proposition 7.39 (Grassmannian). If Y = P 1 then the action of PGL 2 on P 1 × P 1 is transitive and a rational variety.

Special rationality results
In this section we collect rationality results for spaces of rational maps P 1 → P 1 with prescribed (special) ramification over exactly three distinguished points (0, 1, ∞) and unspecified ramifications over other points.
Proposition 8.1. Assume that (r 0 , r 1 , r ∞ ) satisfies one of the following: • all entries of the vectors r 0 , r ∞ are even and some fixed number of entries of r 1 is even; • all entries of the vectors r 0 , r ∞ are even and a fixed number of entries of r 1 is divisible by 3; • all entries of the vectors r 0 , r ∞ are divisible by 3 and all entries of r 1 are even. Then R(r 0 , r 1 , r ∞ ) is a finite union of irreducible rational varieties.
Proof. We have established an explicit parametrization of R(r 0 , r 1 , r ∞ ) as a direct sum of spaces of polynomials (with different weights as irreducible GL 2representations). By the theorem of Katsylo 7.14, the corresponding quotients are rational. is rational.
Since we have one free parameter (under the action of PGL 2 ) we can assume that b 1 = 1. Thuŝ with arbitrary constants c 1 , c 2 . We get a system of equations on the coefficients g j : g 4 = 0, g 1 = 0, g 0 = 0.
Remark that the coefficients of g are symmetric functions on pairs (f 1 , a 1 ) and (f 2 , a 2 ). To parametrize R we introduce the following variables: Write the equations on the coefficients g j as This is a union of two (affine) lines. After a rational covering ( √ b 2 ) our surface is (rationally) a P 1 -bundle over P 1 , a rational surface. is a rational surface.
Proof. Reduces easily to the rationality of a cuspidal cubic curve. 9. Rationality of moduli Theorem 9.1. Any connected component of a moduli space of rational or K3 elliptic surfaces with fixed monodromy group is rational.
Proof. In Proposition 3.11 we have identified (Zariski open subsets of) the corresponding moduli spaces F r,Γ as quotients (by the left PGL 2 and right H Γ -action) PGL 2 \U ′ r,Γ,ℓ /H Γ . Here U ′ r,Γ,ℓ ∼ PGL 2 ×H Γ Sym ℓ (P 1 ) × R Γ and R Γ = {f : P 1 → P 1 } is the space of rational maps (with prescribed ramification). For elliptic rational or K3 surfaces ℓ ≤ 3 and H Γ is either trivial, cyclic, dihedral or a subgroup of S 4 (see Corollary 3.14). The actions if PGL 2 and H Γ commute and H Γ acts only on R Γ .  Recall that R Γ is (birationally) the total space of a line bundle over the space Case 1. d = d ′ . Then, by 3.14, H Γ = 1 and rationality of PGL 2 \R Γ (in all cases) follows from the rationality of which is the theorem of Katsylo 7.14.
We use the classification of these families established in Section 5. All families listed in Lemma 5.2 are covered by Propositions 7.37 and the Theorem 7.14. Consider the families listed in Lemma 5.3: Lemma 7.30 covers the cases j 1 , j 4 , j 5 , j 6 , j 13 . The case j 2 , j 8 and j 12 are covered by Proposition 8.1, j 3 by Lemma 8.6, j 7 , j 9 , j 10 by 8.1 and 8.3, j 11 by Lemma 8.5. The case j 14 is covered by Lemma 8.7. Finally, the families j 15 and j 16 (listed in Lemma 6.2) are covered by Proposition 7.37 and the remaining families j 17 − j 20 by Theorem 7.14.
Remark 9.2. Our methods extend to some moduli spaces of elliptic surfaces with higher Euler characteristic. In particular, the results of Section 8 imply that any moduli space of Jacobian elliptic surfaces over P 1 such that a generic surface in this space has only singular fibers of multiplicative type is rational. However, we expect that there are nonrational moduli spaces already for Euler characteristic 36.

Pictures
In this section we give a combinatorial description of monodromy groups of elliptic K3 surfaces. More precisely, we describe a simple procedure which allows to enumerate all the possible graphs Γ with given ET(Γ). Let E → P 1 be an elliptic K3 surface. We have shown in Section 4 that and that ET(Γ) is divisible by 12. Thus ET(Γ) equals 12, 24, 36 or 48 and all possible Γ ⊂ PSL 2 (Z) are described by connected trivalent graphs T Γ with ≤ 8 edges embedded into S 2 , with an arbitrary bicoloring of the ends.
Case ET(Γ) = 12 : There is only one tree T 12 with ET(T 12 ) = 12 The ends of T 12 can be either A or B-vertices. To obtain all possible graphs T Γ with ET(Γ) = 12 we just need to attach to T 12 a single loop L. This gives the following list of graphs: There is only one saturated graph from the list above which has no outer loops ( Figure 4). This graph will be a basic building block in the construction of graphs with ET(Γ) > 12 -we will attach trees and loops to its edges. The edges are numbered to simplify the count of all possible outcomes.
Case ET(Γ) = 36: There are only 3 saturated graphs without end-loops (modulo equivalent embedding into the sphere):  The number of possible markings of the tree or loops at the ends is 81 but due to the symmetry of the graph the actual number of graphs T Γ corresponding to different placement of loops at the end and markings is smaller: there are 34 different T Γ of this type.
The number of markings of T 36 is 16 but due to its symmetry the number of different graphs T Γ is 7. (Recall that two graphs T Γ give the same Γ modulo conjugation if they are isotopic in a S 2 ).
The graphs of tree type with one end loop are topologically equivalent to: There are 8 possible markings of the above graph and they all give different T Γ with ET(Γ) = 36. We have 12 different T Γ with 2 end-loops, 6 with 3 end-loops and one with 4 end-loops.
All topological graphs which are sums of a loop and a tree can be obtained by placing a loop into a tree. Thus there are two types:  Here is the list of all saturated graphs with ET(Γ) = 48.