RATIONALITY OF MODULI OF ELLIPTIC FIBRATIONS WITH FIXED MONODROMY

by

Fedor Bogomolov, Tihomir Petrov and Yuri Tschinkel

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

Contents

0. Introduction . . ................................................ 2 1. Finite covers . . ................................................ 5 2. Elliptic fibrations . . ............................................ 8 3. Moduli spaces . . .............................................. 10 4. Combinatorics . . .............................................. 17 5. Elliptic K3 surfaces with deg(j_{E}) > 1 . . .......................... 20 6. Rational elliptic surfaces with deg(j_{E}) > 1 . . ...................... 24 7. General rationality results . . .................. .................. 25 8. Special rationality results . . .................. .................. 52 9. Rationality of moduli . . ........................................ 55 10. Pictures . . .................................................... 57 References . . .................................................... 61

1991 Mathematics Subject Classification. — Primary 14J27, 14J10; Secondary 14D05.

Key words and phrases. — Elliptic fibrations, Monodromy groups, Moduli spaces.

The first author was partially supported by NSF grant DMS-9801591.

The second author was partially supported by NSF grant DMS-9802154.

The third author was partially supported by NSA and by NSF grant DMS-0100277.

0. 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 nonisotrivial Jacobian elliptic fibrations over curves

π: E → C

with fixed global monodromy group Γ̃ = Γ̃(E) ⊂ SL_{2}(ℤ). In [8] we developed techniques aimed at the classification of possible global monodromies Γ̃. 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 subgroups Γ̃ ⊂ SL_{2}(ℤ) such that Γ̃ is the (global) monodromy group of some E in this family is finite. Moreover, for every such group Γ̃ the subset of fibrations with this monodromy

BΓ̃ := {b ∈ B | Γ̃(E_{b}) = Γ̃}

is an algebraic (not necessarily closed) subvariety of B.

Generalizing this observation, we introduce (maximal) parameter spaces FΓ̃ of elliptic fibrations with fixed global monodromy Γ̃ (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 FΓ̃, 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. Let Γ̃ ⊂ SL_{2}(ℤ) be a proper subgroup of finite index. Then all moduli spaces of (Jacobian) elliptic rational or elliptic K3 surfaces with global monodromy Γ̃ are rational.

Notice that the finite index condition in the theorem is not a restriction since it always holds for nonisotrivial Jacobian elliptic fibrations, considered in this paper.

Corollary. For all Γ̃ with moduli FΓ̃ of dimension > 0 there exists a number field K such that there are infinitely many nonisomorphic elliptic K3 surfaces over K with global monodromy Γ̃.

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 group Γ̃ itself but rather with its image

Γ ⊂ PSL_{2}(ℤ)

under the natural projection SL_{2}(ℤ) ↠ PSL_{2}(ℤ). Let

H = {z ∈ ℂ | Im(z) > 0}

be the upper half-plane and

\overline{H} = H ∪ ℚ ∪ \{∞\}.

The natural j-map

j : C → ℙ^{1} = \overline{H}/PSL_{2}(ℤ)

decomposes as

j = j_{Γ} ∘ j_{E},

where

\begin{aligned}j_{E}: & C & → & M_{Γ} & = & \overline{H}/Γ \\ j_{Γ} : & M_{Γ} & → & ℙ^{1} & = & \overline{H}/PSL_{2}(ℤ). \end{aligned}

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,Γ̃} ⊂ F_{r}

parametrizing fibrations with fixed monodromy group Γ̃. It suffices to study parameter spaces F_{r,Γ} corresponding to Γ ⊂ PSL_{2}(ℤ), since every irreducible component of F_{r,Γ̃} 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 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 combinatorics 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.

1. 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

\overline{H}/Γ = M_{Γ} ⟶^{j_{Γ}} ℙ^{1} = \overline{H}/PSL_{2}(ℤ),

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}.

Notations 1.1. — The graph datum GD(Γ) of T_{Γ} is the formal sum

GD(Γ) := [a_{6}A_{6} + a_{2}A_{2} + b_{2}B_{2}],

where a_{i} (i = 6, 2) is the number of A_{i} vertices and b_{2} is the number of B_{2}-vertices. Denote by τ^{0} = τ^{0}(Γ) the number of vertices of T_{Γ} (including the ends), by τ^{1} = τ^{1}(Γ) the number of edges and by τ^{2} = τ^{2}(Γ) = π_{0}(M_{Γ} \ T_{Γ}).

Remark 1.2. — For given a_{2}, b_{2} there is a unique group with

GD(Γ) = [A_{6} + a_{2}A_{2} + b_{2}B_{2}].

Forgetting the markings of T_{Γ} we obtain a connected unmarked topological graph T_{Γ}^{u} 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_{Γ}^{u}.

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 I-vertices 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 1-ramifications 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_{Γ}^{u} 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 A-vertices we obtain the graph T_{Γ}^{u}. Thus T_{Γ}^{u} ⊂ X = M_{Γ} is the boundary of this cellular decomposition and T_{Γ} is simply T_{Γ}^{u} with an A, Bmarking 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_{Γ}^{u}. The graph T_{Γ}^{u} must contain some 1-skeleton of X. In particular, the map π_{1}(T_{Γ}^{u}) → π_{1}(X) is surjective. Hence T_{Γ}^{u} 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.

Definition 1.7. — Define

\begin{aligned}ET(Γ) & := & 6τ^{0} = 6(a_{6} + a_{2} + b_{2}) \\ Δ(Γ) & := & 6a_{6} + 2a_{2} \end{aligned}

Thus both ET(Γ) and Δ(Γ) depend only on the marking of the ends but not on the embedding of the graph. Observe that Δ(Γ) is the number of triangles in the corresponding j_{Γ}-triangulation of M_{Γ} and that

[PSL_{2}(ℤ) : Γ] = Δ(Γ)/2.

Remark 1.8. — If M_{Γ} arises from an elliptic fibration as in the Introduction then Δ(Γ)/2 equals the number of Dehn twists in Γ around the multiplicative singular fibers.

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 ℕ-valued vector v = (v_{k}), (∑ v_{k} = d), 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 \overline{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

RD(j_{E}) := [v_{1,A}, …, v_{n,A}, v_{n+1,B}, …, v_{n′,B}, \overline{v}_{n′+1}, …, \overline{v}_{n′′}],

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 \overline{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

RD(j_{Γ}) := [v_{0}, v_{1}, v_{∞}]

for the global j_{Γ}-ramification datum.

Example 1.10. — Assume that GD(Γ) = [nA_{6} + A_{2} + B_{2}] is the graph datum of T_{Γ} ⊂ M_{Γ} and let j_{E} : ℙ^{1} → M_{Γ} = ℙ^{1} be a finite cover. Then the j_{E}-ramification datum

RD(j_{E}) = [(2, 3)_{A}, (2, 2, 1)_{B}, (2), (2)]

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}.

2. 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 nonisotrivial 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 jinvariant is a nonconstant rational function on C (nonisotriviality).

It is well known that s^{2} < 0. We define

ET(E) := −24s^{2}.

Lemma 2.1. — We have

ET(E)/2 = −12s^{2} = χ(E) = c_{2}(E).

Proof. — Well known, but we decided to include an argument. Since E is smooth and relatively minimal its canonical bundle K_{E} is induced from a one-dimensional 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

h^{0}(E, Ω^{1}) = h^{1}(E, O) = g

where g is the genus of C. By Riemann-Roch we obtain

χ(O) = 1 − g + h^{0}(E, K_{E}) = χ(E)/12.

We also know that s^{2} + sK_{E} − 2g + 2 = 0 (genus formula). Therefore,

1 − g + h^{0}(E, K_{E}) = deg(K) − 2g + 2 = χ(E)/12

since deg(K) > 2g − 2 and hence h^{1}(C, K) = 0. Further,

sK_{E} = deg(K).

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,

ET(E) = ∑_{p_{i}∈C^{sing}} ET(E_{p_{i}}),

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:

ET  ET
I_{0} I_{0}^{∗} 12
I_{n} 2n I_{n}^{∗} 2n + 12
II 4 IV^{∗}  16
III 6 III^{∗}  18
IV 8 II^{∗}  20

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 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 ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) 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 ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ SL_{2}(ℤ) is a product of 6 elements conjugated to ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) (elementary Dehn twists).

3. Moduli spaces

Every Jacobian elliptic fibration E → ℙ^{1} admits a Weierstrass model Ē. Its geometric realization is given as follows: there exists a pair of sections

\begin{aligned}g_{2} & ∈ & H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r)), \\ g_{3} & ∈ & H^{0}(ℙ^{1}, O_{ℙ^{1}}(6r)) \end{aligned}

such that E is given by

(3.1)  y^{2}z = 4x^{3} − g_{2}xz^{2} − g_{3}z^{3},

inside ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(2r) ⊕ O_{ℙ^{1}}(3r)), subject to conditions – the discriminant Δ = g_{2}^{3} − 27g_{3}^{2} is not identically 0;

– for every point p ∈ ℙ^{1} we have

(3.2)  min(3ν_{p}(g_{2}), 2ν_{p}(g_{3})) < 12,

where ν_{p} is the valuation corresponding to p ∈ ℙ^{1} (see [14] or [13], Section 7).

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

H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r)) ⊕ H^{0}(ℙ^{1}, O_{ℙ^{1}}(6r))

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,Γ̃}, each preserved under the action of PGL_{2}, such that for every u ∈ U′_{r,Γ̃} one has Γ̃(E_{u}) = Γ̃.

Proof. — The monodromy group of an elliptic fibration does not change under deformations preserving the topological type of singular fibers (it is encoded in the topology of the smooth part). Thus it can change only on algebraic subvarieties where the topological type of singular fibers changes. The monodromy group of E_{u} is upper-semicontinuous under changes of u - it can only drop on a closed subset of the parameter space. For u ∈ U′_{r} the corresponding j-map has a decomposition j_{u} = j_{Γ} ∘ j_{E}. By (3.1),(3.2), j_{u} is nonconstant of degree ≤ 12r. Thus

[PSL_{2}(ℤ) : Γ] = deg(j_{Γ}) ≤ 12r

and the set of possible Γ is finite. Similarly, there are finitely many Γ̃ ⊂ SL_{2}(ℤ) which can appear as monodromy groups of elliptic fibrations parametrized by U′_{r}. Since Γ defines j_{Γ} there are only finitely many possible j_{Γ} which can appear for u ∈ U′_{r}. For every fixed j_{Γ}, a decomposition of j_{u} = j_{Γ} ∘ s, u ∈ U_{r}, where s is a rational map s : ℙ^{1} → ℙ^{1}, deg(s) ≥ 1, determines a 𝔾_{m}-homogeneous set of algebraic equations on the pair (g_{2}, g_{3}) defining the point u ∈ U_{r}. Thus there is a closed algebraic subvariety U_{r,Γ} where j_{u} = j_{Γ} ∘ s. The monodromy of the fibration E_{u} for u ∈ U_{r,Γ} surjects onto Γ unless u is in U_{r,Γ′}, where Γ′ ⊂ Γ. There are finitely many such Γ′ (with the above degree bound) and hence finitely many algebraic varieties U_{r,Γ′} such that for every u ∈ U_{r,Γ} − ⋃ U_{r,Γ′} the monodromy group of E_{u} surjects onto Γ ⊂ PSL_{2}(ℤ).

Let M be an irreducible component of U′_{r,Γ} − ⋃ U′_{r,Γ′}. The monodromy group of E_{u}, u ∈ M is either constant on M or drops when the topology of singular fibers changes. This can occur only on a finite number of closed algebraic irreducible subvarieties M_{i} in M. Since the monodromy group surjects onto Γ it can only drop from the group Γ̃ at a generic point if the map Γ̃ → Γ has a kernel of order 2. If the above map is an isomorphism then the monodormy group is constant on M. If the monodromy group on M does change on M_{i} then Γ̃_{i} (for a generic u ∈ M_{i}) is isomoprphic to Γ and hence does not change on M_{i}. The varieties M and M_{i} are all preserved under the PGL_{2}-action. Thus there is one monodromy group Γ̃ for a generic point of M. Renaming M − ⋂ M_{i} as U′_{r,Γ̃} and M_{i} as U′_{r,Γ̃_{i}} we obtain the algebraic stratification of the lemma.

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_{4r,6r}^{ss}(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,Γ̃} ⊂ F_{r} such that for every b ∈ F_{r,Γ̃} the corresponding Jacobian elliptic surface (E_{b}, s) has global monodromy group Γ̃.

Remark 3.2. — Notice that the maps j_{E} for elliptic fibrations corresponding to different points of the same irreducible component of F_{r,Γ̃} 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_{r}^{ss}′ is a disjoint union of locally closed PGL_{2}-semistable subsets U_{r,Γ̃}^{ss}; taking quotients we obtain varieties F_{1,Γ̃} parametrizing rational elliptic fibrations with global monodromy Γ̃.

Let W′_{1} = U′_{1} − U_{1}^{ss}′ be the complement. It consists of pairs (g_{2}, g_{3}) with

g_{2} = l^{3}f_{2}, g_{3} = l^{4}f_{3},

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 Γ̃ ≠ SL_{2}(ℤ) corresponds to deg(j_{Γ}) ≥ 2. Thus we have to consider two cases:

– deg(j_{Γ}) = deg(j_{E}) = 2;

– deg(j_{Γ}) ≤ 4, deg(j_{E}) = 1.

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} and Γ̃(E_{w}) ≠ SL_{2}(ℤ) then deg(j_{E}) = 1 and one has one of the following graph and ramification data:

\begin{matrix}GD(Γ) & RD(j_{Γ}) \\ [A_{6} + A_{2}] & [(3, 1)_{0}, (2, 2)_{1}, (3, 1)_{∞}] \\ [A_{6} + A_{2} + 2B_{2}] & [(3, 1)_{0}, (2, 1, 1)_{1}, (4)_{∞}] \\ [A_{6} + 3B_{2}] & [(3)_{0}, (1, 1, 1)_{1}, (3)_{∞}] \\ [A_{6} + B_{2}] & [(3)_{0}, (2, 1)_{1}, (2, 1)_{∞}] \end{matrix}

Proof. — The formula j = lf_{2}^{3}/(lf_{2}^{3} − f_{3}^{2}) shows that j_{Γ} has a point with local ramification datum (3, 1) or (3), corresponding to

[PSL_{2}(ℤ) : Γ(E)] = 4 or 3.

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:

[(3, 1)_{0}, (2, 2)_{1}, (2, 2)_{∞}].

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.

Corollary 3.4. — Every irreducible component W′_{1,Γ̃} ⊂ W′_{1} such that Γ̃(E_{w}) ≠ SL_{2}(ℤ) for w ∈ W′_{1,Γ̃} is rational.

Consider an irreducible component F_{r,Γ̃} and the corresponding decomposition j = j_{Γ} ∘ j_{E}. Here

j_{E} = (j_{E,2}, j_{E,3}) : ℙ^{1} → M_{Γ} = ℙ^{1}

is a pair of homogeneous polynomials in 2 variables. Let

G = {(g_{2}, g_{3})} ⊂ H^{0}(ℙ^{1}, O_{ℙ^{1}}(4r)) ⊕ H^{0}(ℙ^{1}, O_{ℙ^{1}}(6r))

be the subset corresponding to smooth elliptic fibrations. Put

J_{Γ} := {j | ∃j_{E} : ℙ^{1} → M_{Γ} s.t. j = j_{Γ} ∘ j_{E}}.

Lemma 3.5. — If j ∈ J_{Γ} ∩ J_{Γ′} with Γ ≠ Γ′ then there exist an h ∈ PSL_{2}(ℤ), a group Γ′′ ⊂ Γ ∩ hΓ′h^{−1} and a map j′′_{E} : ℙ^{1} → M_{Γ′′} such that j = j_{E′′} ∘ j_{Γ′′}.

Proof. — 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 group Γ̃(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 j_{E} = h_{Γ} ∘ j′_{E}, where h_{Γ} ∈ Aut(T_{Γ}) is an automorphism of M_{Γ}, preserving T_{Γ}.

Lemma 3.8. — We have a decomposition

G_{Γ} = ⨆_{k} G_{Γ̃,k}

into a finite union of algebraic irreducible GL_{2}-stable subvarieties such that Γ̃(E_{g}) = Γ̃ for all g ∈ G_{Γ̃,k}.

Proof. — Assume that some g ∈ G_{Γ} belongs to G_{Γ̃,1} ∩ G_{Γ̃,2}, where G_{Γ̃,1}, G_{Γ̃,2} are different (nonconjugated) lifts of Γ into SL_{2}(ℤ). Lemma 3.5 implies that there exists a proper subgroup Γ′′ ⊂ Γ such that g belongs to G_{Γ′′}, contradiction.

Let GΓ̃ = G_{Γ̃,k} be an irreducible component of G_{Γ} as in Lemma 3.8 and g ∈ GΓ̃ 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_{E_{g}}, with ramification datum RD = RD(j_{E_{g}}). We get a map

ϕ_{U} : U_{g} → U_{j_{g}} × (ℙ^{1})^{ℓ},

where U_{g} ⊂ GΓ̃ is a neighborhood of g and U_{j_{g}} ⊂ R(RD) is a neighborhood of the map j_{g} = j_{E_{g}} in the space

R(RD) := {j : ℙ^{1} → ℙ^{1} | RD(j) = RD}

of rational maps with ramification datum RD.

Lemma 3.9. — The map ϕ_{U} is a local (complex analytic) surjection.

Proof. — It suffices to show that the variation of j in the space of maps with RD(j) = RD(j_{E_{g}}) lifts to a variation of the elliptic fibration E_{g}. Observe that M_{Γ} minus the preimages of {0, 1, ∞} carries a natural “projective” local system P L_{Γ} with fiber ℤ ⊕ ℤ modulo ±1. This local system lifts onto a similar local system j^{∗}P L_{Γ} on ℙ^{1}. Since the fundamental group of ℙ^{1} minus j^{−1}({0, 1, ∞}) is a free group the projective local system j^{∗}P L_{Γ} can be lifted to a linear local system with fiber ℤ ⊕ ℤ.

By a theorem of Kodaira, a nonisotrivial elliptic fibration over B is uniquely determined by the nonconstant map j : B → ℙ^{1} = \overline{H}/Γ and the lifting of the induced projective monodromy on ℙ^{1} \ j^{−1}({0, 1, ∞}) with (ℤ ⊕ ℤ)/(ℤ/2) as a fiber to the linear monodromy with ℤ ⊕ ℤ as a fiber. Therefore it suffices to lift the variation of j_{E} to a variation of the corresponding local system with fiber

H_{1}(E_{t}) = ℤ ⊕ ℤ, t ∈ ℙ^{1} − j_{E}^{−1}({0, 1, ∞}).

Any such lifting corresponds to an elliptic fibration with Jacobian map j_{E}. However, its topological type depends on the lifting.

A linear lifting is defined by the choice of local monodromy elements in SL_{2}(ℤ) at every point in the preimage of the corresponding projective monodromy g ∈ PSL_{2}(ℤ). Namely, for every local projective monodromy g ∈ PSL_{2}(ℤ) we have a choice g_{m}, −g_{m} where g_{m} corresponds to the minimal lifting - the lifting with a minimal Betti number for the corresponding singular fiber (see the next section for a more detailed discussion or [23]). The topological type r(E_{g}) is a function of RD(E) and the number of nonminimal liftings l . The existence of a lifting depends only on the product of local liftings (1 or −1), which can be calculated using RD(j) and l. In particular, since it is 1 for E_{g} the same holds for a variation j of j_{E_{g}} (with constant l).

According to Kodaira, a simultaneous lifting of the variation of j_{E} and the variation of the linear system for E_{g} as above is equivalent to the existence of a variation of the elliptic surface E_{g}. This completes the proof of the lemma.

Corollary 3.10. — Let F′_{r,Γ̃} ⊂ F_{r,Γ̃} be an (irreducible) component with generic ramification datum RD. Then F′_{r,Γ̃} surjects (rationally) onto the quotient of the variety of rational maps R(RD) by H_{Γ}.

Proof. — Since both F′_{r,Γ̃} 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,Γ̃} contains an open part F′_{r,Γ̃} with the following properties:

– F′_{r,Γ̃} is a quotient of an algebraic variety U′_{r,Γ̃,ℓ} by the (left) action of PGL_{2} and (right) action of a subgroup H_{Γ} of Aut(T_{Γ});

– U′_{r,Γ̃,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,Γ̃,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,Γ̃,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,Γ̃,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,Γ̃,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,Γ̃,ℓ} is almost free then the rationality of PGL_{2}\U′_{r,Γ̃,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.

Proof. — Consider j_{Γ} : M_{Γ} → ℙ^{1}. Then h ∈ Aut(T_{Γ}) is any element in PGL_{2}(ℂ) such that j_{Γ}(hz) = j_{Γ}(z) for all z ∈ T_{Γ} ⊂ M_{Γ}. 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 c_{1} = f^{−1}(0) and c_{2} = f^{−1}(∞). If c_{1}, c_{2} do not intersect then deg(f) = deg(c_{1}) = deg(c_{2}). 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}.

4. Combinatorics

In this section we investigate relations between ET(E) and ET(Γ). We keep the notations of the previous sections.

Lemma 4.1. — Let j : E → C be an elliptic fibration. Then

ET(E) = deg(j_{E})Δ(Γ) + 8α_{2} + 4α_{1} + 6β_{1} + 12ℓ. (4.1)

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})Δ(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.

Corollary 4.2. — In particular,

ET(E) ≤ deg(j_{E}) ET(Γ) + 12ℓ,

with equality if

\begin{aligned}2α_{2} + α_{1} & = & a_{2} · deg(j_{E}) \\ β_{1} & = & b_{2} · deg(j_{E}). \end{aligned}

Definition 4.3. — We call T_{Γ} saturated if all vertices of T_{Γ}^{u} are trivalent and a tree if it is contractible.

Remark 4.4. — For saturated graphs Δ(Γ) = 12rk π_{1}(T_{Γ}), where

rk π_{1}(T_{Γ}) = rk H_{1}(T_{Γ})

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

ET(T′) = ET(T_{1}) + ET(T_{2}) + 12.

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

Δ(T′) = Δ(T_{1}) + Δ(T_{2}) + 12.

– We can glue an end p of T_{1} to an edge of T_{2}. In this case

ET(T′) = ET(T_{1}) + ET(T_{2}).

The change of Δ depends on the marking of the end:

Δ(T′) = \begin{cases}Δ(T_{1}) + Δ(T_{2}) + 6 & if p = B_{2} \\ Δ(T_{1}) + Δ(T_{2}) + 4 & if p = A_{2}. \end{cases}

Remark 4.5. — Any connected graph T can be uniquely decomposed into a union of a saturated graph and a union of trees.

Lemma 4.6. — ET(Γ) is divisible by 12.

Proof. — Every vertex of T_{Γ} has either one or three incoming edges. Therefore, the number of edges

τ^{1} = \frac{1}{2}(τ_{1}^{0} + τ_{3}^{0}),

(τ_{i}^{0} is the number of vertices with i-edges). Thus τ^{0} = τ_{1}^{0} + τ_{3}^{0} 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
Δ(Γ) \{ \begin{matrix}= 6k & if all ends are B_{2}, \\ > 6k & otherwise. \end{matrix}

Lemma 4.8. — For all Γ one has

Δ(Γ) ≥ ET(Γ)/2 + 6(rk H_{1}(T_{Γ}) − 1).

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 Δ(Γ) will change by 6k + 6. Finally, the ratio Δ(Γ)/ET(Γ) only increases if we change B_{2}- to A_{2}-markings for some ends. Indeed, Δ(Γ) increases without changing ET(Γ).

Corollary 4.9. — If Δ(Γ) = ET(Γ)/2 then T_{Γ} is a concatenation of a loop L and some trees. Moreover, all the ends of T_{Γ} are of type B_{2}.

Proposition 4.10. — Let E → C be an elliptic fibration with ET(E) < 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.

Corollary 4.11. — For every elliptic fibration E → ℙ^{1} one has

ET(E) ≥ ET(Γ).

Further, if deg(j_{E}) = 2 and j_{E} is ramified over only one B_{2}-point then

ET(E) ≥ 2 ET(Γ) − 12.

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.

5. Elliptic K3 surfaces with deg(j_{E}) > 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

general families : ET(E) − 12ℓ = deg(j_{E}) ET(Γ),
special families : ET(E) − 12ℓ < deg(j_{E}) ET(Γ).

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 5.1. — There are no special families of elliptic K3 surfaces with

ET(Γ) = 48, 36.

Proof. —

– If ET(Γ) = 48 then Δ(Γ) ≥ 18 and deg(j_{E}) ≤ 2. However, deg(j_{E}) = 2 contradicts Corollary 4.11 (ET(E) ≥ 96 − 24 > 48).

– If ET(Γ) = 36 and Δ(Γ) > 16 then deg(j_{E}) = 2, contradicting to 4.11. We are left with Δ(Γ) = 16, 14, 12 for deg(j_{E}) = 3 and Δ(Γ) = 12 for deg(j_{E}) = 4.

– If deg(j_{E}) = 4 then T_{Γ} is a tree with GD(Γ) = [2A_{6}+4B_{2}]. By Lemma 4.1, all ramifications over the B_{2}-ends are even, which contradicts C = ℙ^{1} (compute χ(C)).

– If deg(j_{E}) = 3 then T_{Γ} is a tree (by 4.8) and

GD(Γ) = [2A_{6} + a_{2}A_{2} + (4 − a_{2})B_{2}]

with a_{2} ≤ 2. We have

48 ≥ ET(E) ≥ 3(12 + 2a_{2}) + 4α_{1} + 8α_{2} + 6β_{1},

where β_{1} ≥ 2 (since deg(j_{E}) is odd there is odd ramification over some B_{2}- end). Therefore, a_{2} = 0 and consequently, β_{1} ≥ 4, contradiction.

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,Γ̃}) then

ET(Γ) deg(j_{E}) GD(Γ)  RD(j_{E})
24  4  [2A_{6} + 2B_{2}]  [(2, 2)_{B}, (2, 2)_{B}, (2), (2)]
24  3  [2A_{6} + 2B_{2}]  [(2, 1)_{B}, (2, 1)_{B}, (2), (2)]
24  3  [2A_{6} + A_{2} + B_{2}] [(3)_{A}, (2, 1)_{B}, (2)]
12  6  [A_{6} + A_{2}]  [(3, 3)_{A}, (3, 3)_{A}, (2), (2)]
12  5  [A_{6} + A_{2}]  [(3, 1, 1)_{A}]
12 5 ≤ d ≤ 8 [A_{6} + B_{2}]  [β = (β_{i})_{B}, (2)_{B}^{d′}],

where

β_{i} ∈ ℕ, ∑ β_{i} = d, #odd β_{i} ≤ 8 − d

and

d′ = 2d − #nonzero β_{i}.

Proof. — Follows from Lemma 4.1. First observe that Δ(Γ) ≤ 16, which implies that a_{6} = 2 and a_{2} ≤ 2. If a_{2} = 2 then Δ(Γ) = 16 and

α_{1} = α_{2} = β_{1} = 0.

Hence both A_{2}-ends have a 3-cyclic ramification and the cover corresponds to a subgroup Γ′ ⊂ Γ of index 3. This excludes GD(Γ) = [2A_{6} + 2A_{2}]. If deg(j_{E}) = 4 then Δ(Γ) = 12 which implies that all preimages of B_{2}-ends have even ramification. The description of all other ramification data follows similarly from Lemma 4.1. Notice that the (omitted) possibilities

ET(Γ) deg(j_{E}) GD(Γ) RD(j_{E})
12  6  [A_{6} + A_{2}] [(6)_{A}, (3, 3)_{A}]
12  5  [A_{6} + A_{2}] [(3, 2)_{A}]

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,Γ̃}) then

deg(j_{E}) GD(Γ)  RD(j_{E})
j_{1}  4  [A_{6} + A_{2} + 2B_{2}] [(1, 1, 1, 1)_{A}, (2, 2)_{B}, (2, 2)_{B}, (2), (2)]
j_{2}  4  [A_{6} + A_{2} + 2B_{2}] [(3, 1)_{A}, (2, 2)_{B}, (2, 2)_{B}] + ∗
j_{3}  4  [A_{6} + A_{2} + 2B_{2}] [(3, 1)_{A}, (2, 2)_{B}, (2, 1, 1)_{B}, (2)]
j_{4}  3  [A_{6} + 2A_{2} + B_{2}] [(3)_{A}, (1, 1, 1)_{A}, (2, 1)_{B}]
j_{5}  3  [A_{6} + A_{2} + 2B_{2}] [(1, 1, 1)_{A}, (2, 1)_{B}, (2, 1)_{B}, (2), (2)]
j_{6}  3  [A_{6} + A_{2} + 2B_{2}] [(3)_{A}, (1, 1, 1)_{B}, (2, 1)_{B}, (2)]

or GD(Γ) = [A_{6} + 3B_{2}] and

deg(j_{E}) RD(j_{E})
j_{7}  8  [(2, 2, 2, 2)_{B}, (2, 2, 2, 2)_{B}, (2, 2, 2, 2)_{B}, (2), (2)]
j_{8}  6  [(2, 2, 2)_{B}, (2, 2, 2)_{B}, (2, 2, 1, 1)_{B}, (2), (2)]
j_{9}  6  [(2, 2, 2)_{B}, (2, 2, 2)_{B}, (2, 2, 2)_{B}, (2)] + ∗
j_{10}  5  [(2, 2, 1)_{B}, (2, 2, 1)_{B}, (2, 2, 1)_{B}, (2), (2)]
j_{11}  4  [(2, 1, 1)_{B}, (2, 1, 1)_{B}, (2, 2)_{B}, (2), (2)]
j_{12}  4  [(2, 1, 1)_{B}, (2, 2)_{B}, (2, 2)_{B}, (2)] + ∗
j_{13}  3  [(1, 1, 1)_{B}, (2, 1)_{B}, (2, 1)_{B}]
j_{14}  3  [(2, 1)_{B}, (2, 1)_{B}, (2, 1)_{B}, (2)] + ∗

or ET(Γ) = 12 and GD(Γ) = [2A_{2}] with deg(j_{E}) = 4 − 10, 12.

(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

GD(Γ) ≠ [A_{6} + 3B_{2}].

First observe that deg(j_{E}) ≤ 6, since Δ(Γ) ≥ 8. If deg(j_{E}) ≥ 5 then, by 4.1, GD(Γ) = [A_{6}+A_{2}+2B_{2}]. If deg(j_{E}) = 6 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

GD(Γ) = [A_{6} + 3A_{2}],
= [A_{6} + 2A_{2} + B_{2}],
= [A_{6} + A_{2} + 2B_{2}]

and 3 ≤ deg(j_{E}) ≤ 4. 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 Δ(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

RD(j_{E}) = [(3, 1)_{A}, (3, 1)_{A}, (2, 2)_{B}],

which corresponds to a group covering, contradiction.

Similarly, if GD(Γ) = [A_{6} + A_{2} + 2B_{2}] and deg(j_{E}) = 4 then Δ(Γ) = 8 and 4α_{1} + 8α_{2} + 6β_{1} ≤ 16. We have α_{1} ≥ 1 and 8α_{2} + 4α_{1} = 16 or 4. In the first case, both B_{2}-ends are completely ramified, and we get j_{1}. The second case splits into subcases: β_{1} = 0 or 2, leading to j_{2}, resp. j_{3}. If deg(j_{E}) = 3, then if GD(Γ) = [A_{6} + 2A_{2} + B_{2}] then exactly one of the A_{2}-ends has cyclic ramification. It follows that β_{1} = 1, which leads to j_{4}. If GD(Γ) = [A_{6} + A_{2} + 2B_{2}] there are two subcases: there is cyclic ramification over the A_{2}-end or not. In the first subcase, possible RD(j_{E}) include [(2, 1)_{B}, (2, 1)_{B}], which is excluded as it gives a group covering. The other case leads to j_{6}. In the second subcase, we get j_{5}.

Consider the case T_{Γ} = A_{6} + 3B_{2}. Here Δ(Γ) = 6 and

ET(E) ≥ 6 deg(j_{E}) + 6n,

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.

In deg(j_{E}) = 4 we could have further RD:

RD(j_{E}) = [(2, 2)_{B}, (2, 1, 1)_{B}, (2, 2)_{B}, (2)],
= [(2, 2)_{B}, (2, 1, 1)_{B}, (2, 1, 1)_{B}, (2), (2)],
= [(2, 2)_{B}, (2, 1, 1)_{B}, (2, 1, 1)_{B}, (3)]

but they are obtained as degenerations of j_{12} and j_{13}.

The only GD(Γ) which allow deg(j_{E}) ≥ 12 are [A_{2}+B_{2}] and [2A_{2}]. The first case corresponds to PSL_{2}(ℤ) (which we don’t consider). The second case corresponds to subgroups Γ ⊂ PSL_{2}(ℤ) of index 2. For a generic E in each moduli space the ramification datum RD(j_{E}) is one of the following:

RD(j_{E}) = [(3, …, 3_{n_{1}}, 1, …, 1)_{A}, (3, …, 3_{n_{1}}, 1, …, 1)_{A}, (2)^{d}] + ∗,

where n_{1}, n_{2}, d are nonnegative integers such that

deg(j_{E}) − (n_{1} + n_{2}) ≤ 4,

3n_{1}, 3n_{2} ≤ deg(j_{E}) and, d ≤ 2(deg(j_{E} − (n_{1} + n_{2} + 1))).

(In particular, d ≤ 4).

6. Rational elliptic surfaces with deg(j_{E}) > 1

Lemma 6.1. — There are no special families of rational elliptic surfaces with ET(Γ) = 24.

Proof. — If deg(j_{E}) = 2 then j_{E} cannot be ramified over more than one B_{2}-end (otherwise it is a group covering). Therefore, we can apply Corollary 4.11 and get ET(E) > 2 · 24 − 12 > 24, contradiction (to 2.1). Thus deg(j_{E}) = 3 or 4 and a_{6} = 1. Moreover, Δ(Γ) ≤ 8. This leaves the cases:

\begin{aligned}GD(Γ) & = & [A_{6} + 3B_{2}], \\ & = & [A_{6} + A_{2} + 2B_{2}]. \end{aligned}

In the first case deg(j_{E}) = 3 is impossible, and deg(j_{E}) = 4 leads to

RD(j_{E}) = [(2, 2)_{B}, (2, 2)_{B}, (2, 2)_{B}, (2, 2)_{B}]

which corresponds to a group covering. In the second case deg(j_{E}) ≠ 4 (since Δ(Γ) = 8) and deg(j_{E}) = 3 implies that β_{1} ≥ 2 and ET(E) ≥ 36, contradiction.

Lemma 6.2. — If j_{E} is special (and generic for the corresponding irreducible component of F_{1,Γ̃}) then

deg(j_{E}) GD(Γ) RD(j_{E})
j_{15}  6  [2A_{2}]  [(3, 3)_{A}, (3, 3)_{A}, (2), (2)]
j_{16}  4  [2A_{2}]  [(3, 1)_{A}, (3, 1)_{A}, (2), (2)]
j_{17}  3  [2A_{2}]  [(3)_{A}, (1, 1, 1)_{A}, (2), (2)]
j_{18}  4  [A_{6} + B_{2}] [(2, 2)_{B}, (2), (2), (2), (2)]
j_{19}  3  [A_{6} + B_{2}] [(2, 1)_{B}, (2), (2)]
j_{20}  3  [A_{6} + A_{2}] [(3)_{A}, (2), (2)]

Proof. — If a_{6} ≥ 1 then deg(j_{E}) = 4 or 3. In the first case a_{2} = 0 and GD(Γ) = [A_{6} +B_{2}] and we have complete ramification over the B_{2}-end. This gives j_{18}. In the second case the ramification over B_{2} is (2, 1)_{B} and we get j_{19}. If GD(Γ) = [A_{6}+A_{2}] then deg(j_{E}) = 3 and α_{1} = α_{2} = 0, leading j_{20}.

It remains to consider GD(Γ) = [2A_{2}]. We apply the same formulas as in the proof of Lemma 5.3, with the inequality

deg(j_{E}) − n_{1} − n_{2} ≤ 2.

We have deg(j_{E}) ≤ 6 and α_{1} = α_{2} = 0. Notice that deg(j_{E}) = 5 is impossible.

7. 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:

Theorem 7.2. — A unirational surface is rational.

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 by ℭ̃_{n}, 𝔇̃_{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 by

B̃, T̃, NT̃

the corresponding subgroups in GL_{2} (or SL_{2}).

Let V be an n-dimensional vector space, G̃ ⊂ 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.

Corollary 7.5 (of Theorem 7.2). — For all n ≤ 3 the space G\ℙ(V) is rational.

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.

Definition 7.8. — A finite group G̃ ⊂ GL_{n} = GL(V) is called imprimitive if there exists a decomposition V = ⊕_{α}V^{α} such that for all α and g̃ ∈ G̃ there is an α′ with g̃V^{α} = V^{α′}. Otherwise, G is called primitive.

Remark 7.9. — There are 29 types of primitive subgroups of GL_{4}. For some of them, like

𝔄_{6}, 𝔄_{7}, PSL_{2}(F_{7}), 𝔖_{6},

rationality of the quotient is still unknown.

Theorem 7.10. — [26] For every primitive solvable subgroup G ⊂ PGL_{4} the quotient G\ℙ^{3} is rational.

Remark 7.11. — In [26] it is shown that

G\ℙ^{3} ∼_{G} G′\X_{3},

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

i_{1} : x := ( \begin{matrix}a & b \\ c & −a + d \end{matrix} ) ↦ ( \begin{matrix}−a + d & −b \\ −c & a \end{matrix} )

and

i_{2} : y ↦ y · det(y)^{−1}.

are two commuting involutions. Another set of independent generators of ℂ(a, b, c, d) is given by {a, b, c, det(x)} (write d = (det(x) + bc + a^{2})/a). Now the involutions take the form

i_{1} : (a, b, c) → (−a, −b, −c)

and

i_{2} : det(x) → det(x)^{−1}

and we can write down independent generators of the field of invariants. If

D := \frac{det(x) + 1}{det(x) − 1}

then

\begin{matrix}i_{2} : & D & ↦ & −D \\ i_{1} : & (a, b, c, D) & ↦ & (−a, −b, −c, −D). \end{matrix}

This finishes the proof.

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 af-action (almost free) if St_{gen} is trivial.

We use a more precise version of Theorem 7.7:

Theorem 7.14. — [17] Let ρ : PGL_{2} → PGL(V) be a representation and ρ̃ a lifting of ρ to a representation of GL_{2} → GL(V). Let

G′′ := St_{gen}(GL_{2}, V) and G := GL_{2}/G′′.

If the central ℭ_{2} ⊄ G′′ then

ℙ(V) ∼_{G} G × S,

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

E ∼_{G} X × 𝔸^{r}

with trivial G-action on the affine space 𝔸^{r}.

Proof. — Denote by O the G-orbit through a generic point in E. Shrinking (equivariantly) X, if necessary, we may assume that the map

π : H^{0}(X, E) → H^{0}(O, E|_{O})

is surjective. With our assumptions, there exists a basis s_{1}, ..., s_{r} such that for each j, the G-orbit of s_{j} projects isomorphically onto its image in X and generates a trivial 1-dimensional G-equivariant sub-bundle of the restriction E|_{O} of E to the orbit O. It follows that E|_{O} = ⊕_{j=1}^{r}G · s_{j}. In particular, H^{0}(O, E) contains the trivial G-module M generated by s_{1}, ..., s_{r}. Moreover, M generates H^{0}(O, E|_{O}) over every point of O. Since π is a map of G-modules and G is reductive H^{0}(X, E) contains a submodule M′ such that π(M′) = M (as G-modules). The elements of M′ generate E over a generic point of X. A basis s′_{1}, ..., s′_{r} of M′ gives the desired splitting of the action.

Corollary 7.16. — Let G be a reductive group and

E′′ → E′ → X

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

E′′ ∼_{G} E′′_{s′} × 𝔸^{r′}

(where r′ = rk E′), with trivial G-action on 𝔸^{r′}.

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 and ρ̃ : G̃ → E a G̃-action lifting ρ. Consider a generic orbit G · x ⊂ X and the linear action of G̃ on the space of sections H^{0}(X, E).

Assume that G̃ is reductive and V is a linear representation of G̃ which is contained in H^{0}(X, E). Then there exists an affine open X′ ⊂ X such that the vector bundle E → X′ admits a G̃-map onto a G̃-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-vector bundle and O a G-orbit through a generic point. Assume that H^{0}(O, E) (the restriction of the space of sections to O) contains V as a submodule. We obtain a G-map

v : H^{0}(O, E) → V^{∗}

(the dual module, considered as a vector bundle over a point).

Lemma 7.18. — There exists a G-stable Zariski open U ⊂ X and a rational G-map of H^{0}(U, E) → V^{∗} extending v.

Proof. — A generic orbit O has a G-equivariant neighborhood U, with U/G affine, such that

\xymatrix{H^{0}(U, E) \ar@{->>}[r] & H^{0}(O, E).}

The module H^{0}(U, E) is a direct sum of finite dimensional irreducible G-modules. 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

X × ℙ(V_{2d}) ∼_{PGL_{2}} X × ℙ(V_{2d}),

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

PGL_{2}\X × ℙ(V_{2d})

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 action ρ̃ of GL_{2} lifting ρ. Assume that X contains a PGL_{2}-orbit Y ∼ PGL_{2} such that the GL_{2}-module H^{0}(Y, V_{Y}) contains V_{d}, for some odd d. Then

ℙ(V) ∼_{PGL_{2}} PGL_{2} × S,

(with trivial PGL_{2}-action on S). Otherwise, V is induced from a GL_{2}-vector bundle on PGL_{2}\X.

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

\xymatrix{H^{0}(X, V) \ar@{->>}[r] & H^{0}(Y, V_{Y}).}

Since H^{0}(Y, V_{Y}) is an algebra over H^{0}(Y, O_{Y}) = ⊕_{d≥0}V_{2d}, it contains V_{1} as a submodule. We obtain a PGL_{2}-equivariant surjective map

ℙ(V) → ℙ(V_{1}) = ℙ^{1}.

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_{2}-module. The semisimplicity 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.

Lemma 7.22. — Let V be a representation of G of dimension ≥ 2 (with G acting on the left). Then V ⊕ V is a G × GL_{2}-space (with right GL_{2}-action) and

V ⊕ V ∼_{G×GL_{2}} \xymatrix{V \ar[d]^{M_{2}=V_{1}⊕V_{1}} \\ Gr(2, V),}

a vector bundle with fibers 2 × 2-matrices (with right GL_{2}-action).

Proof. — Consider the map

\begin{matrix}V ⊕ V & → & Gr(2, V) \\ (v, v′) & ↦ & ⟨v, v′⟩, \end{matrix}

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_{2}-action on the fibers is the right multiplication on matrices:

(v, v′) ↦ (av + bv′, cv + dv′).

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).

Corollary 7.23. — Assume that the action of G′ on Gr(2, V) has a slice S so that Gr(2, V) ∼ S × G′. Let V_{S} be the restriction of V to S (this makes sense by Corollary 7.16). Then

G\V/GL_{2} ∼ G′\V_{S}.

Remark 7.24. — The group G′′ acts as scalars on V = V_{1} ⊕ V_{1} (it commutes with GL_{2}).

Lemma 7.25. — Assume that we are in the situation of Corollary 7.23, G = GL_{2} and H ⊂ GL_{2} has finite image in PGL_{2}. Then G\V/H is rational.

Proof. — By Corollary 7.24, the slice S is 3-stably rational, since

S × PGL_{2} ∼ Gr(2, V)

and Gr(2, V) is rational. The quotient of V_{S} by a fiberwise linear action is birational to (M_{2}/H) × S (every vector bundle admits an H-equivariant trivialization over an open subset of S). There is a left action of 𝔾_{m}^{2} ⊂ GL_{2} on M_{2} = V_{1} ⊕ V_{1} which commutes with H. Thus M_{2}/H is (birationally) a three-dimensional variety with an af-action of 𝔾_{m}. The quotient (a surface) is unirational, hence rational (by Theorem 7.2), and

G\(V ⊕ V)/H ∼ S × (V_{1} ⊕ V_{1})/H ∼ S × ℂ^{3}.

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.

Lemma 7.26. — We have

St_{gen}(N_{T} × N_{T}, ℙ(M_{2})) = ℭ_{2}.

Proof. — Indeed N_{T} contains

\begin{aligned}𝔾_{m} = \{t\} & : & (x, y) & ↦ & (tx, t^{−1}y), \\ i & : & (x, y) & ↦ & (y, x). \end{aligned}

The corresponding actions on ℙ(M_{2}) are

(a, b, c, d) ↦ (t_{1}t_{2}a, t_{1}^{−1}t_{2}b, t_{1}t_{2}^{−1}c, t_{1}^{−1}t_{2}^{−1}d)

and

\begin{matrix}i_{1} : & a → c, & b → d \\ i_{2} : & a → b, & c → d, \end{matrix}

respectively. A matrix (a, b, c, d) ∈ M_{2} can be transformed to (1, 1, 1, d) by a unique element of 𝔾_{m} × 𝔾_{m}, the 𝔖_{2} × 𝔖_{2}-orbit of which consists of two elements (for d, d^{−1}).

Corollary 7.27. — The group N_{T} × ℂ^{∗} acts almost freely on ℙ(M_{2}). There is an open, N_{T} × N_{T}-stable subvariety U ⊂ ℙ(M_{2}) such that

\xymatrix{U \ar[d]^{ℂ^{∗}×ℂ^{∗}} \\ ℂ^{∗} ⊂ ℙ^{1},}

with a transitive action of ℂ^{∗}×ℂ^{∗} ⊂ N_{T}×N_{T} on the fibers. The diagonal subgroup

𝔖_{2}^{Δ} ⊂ 𝔖_{2} × 𝔖_{2} = (N_{T} × N_{T})/(ℂ^{∗} × ℂ^{∗})

acts on each fiber as an involution x ↦ x^{−1}. The factor 𝔖_{2} = (𝔖_{2} × 𝔖_{2})/𝔖_{2}^{Δ} acts on the base ℂ^{∗} ⊂ ℙ^{1} as an involution without fixed points, on the first factor in the fiber as x → x^{−1}, and as identity on the second factor.

Corollary 7.28. — Let 𝔇 ⊂ N_{T} be a dihedral subgroup such that 𝔇\N_{T} = ℂ^{∗}. Then the ℂ^{∗}-bundle

C = 𝔇\ℙ(M_{2}) → N_{T}\ℙ(M_{2})

is induced from the ℂ^{∗}-bundle

𝔇\ℙ(M_{2})/N_{T} → N_{T}\ℙ(M_{2})/N_{T} = ℙ^{1}

and is hence birationally trivial.

Proof. — Indeed, the left and the right actions of N_{T} commute. By Lemma 7.26, St_{gen}(N_{T} × N_{T}, ℙ(M_{2})) = ℭ_{2}, which implies that the bundle is induced.

Lemma 7.29. — For every dihedral group 𝔇 and every H ⊂ N_{T} the conic bundle

C_{H} = 𝔇\ℙ(M_{2})/H → N_{T}\ℙ(M_{2})/H,

has a section.

Proof. — The quotient 𝔇\U/H from Corollary 7.27 admits a fibration

\xymatrix{𝔇\U/H \ar[d]^{ℂ_{𝔇}^{∗}×ℂ_{H}^{∗}/𝔖_{2}} \\ ℙ^{1}/𝔖_{2}.}

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_{Δ}^{∗} ⊂ ℂ^{∗} × ℂ^{∗}. Isomorphisms ℂ_{H}^{∗} → ℂ^{∗} and ℂ_{𝔇}^{∗} → ℂ^{∗} induce a birational fiberwise isomorphism

\xymatrix{C_{H} = & 𝔇\ℙ(M_{2})/H \ar[d] \\ & N_{T}\ℙ(M_{2})/H} ∼ \xymatrix{𝔖_{2}\ℙ(M_{2})/𝔖_{2} \ar[d] & = C_{0} \\ N_{T}\ℙ(M_{2})/𝔖_{2}}

and it suffices to consider 𝔇 = 𝔖_{2}, H = 𝔖_{2}. In this case, an alternative equivariant completion of U is given by

U ⊂ \xymatrix{ℙ_{1}^{1} × ℙ_{2}^{1} × ℙ_{3}^{1} \ar[d] \\ ℙ_{3}^{1}} ,

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

ℙ^{1} × ℙ^{1}/𝔖_{2} × 𝔖_{2} = ℙ^{1} × ℙ^{1}

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 to 𝔄̃_{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 to 𝔄̃_{5}. Then G is either

1. a finite subgroup of ℂ^{∗},

2. a dihedral group or

3. 𝔄̃_{4}, 𝔖̃_{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,

ℂ^{∗} × ℙ^{1} × W′ ∼_{G′} (ℂ^{∗} × ℙ^{1}) × W′,

with trivial action of G′ on ℂ^{∗} × ℙ^{1}. Thus

𝔇\V ∼ (𝔇′\W′) × (ℂ^{∗} × ℙ^{1})

and we can apply induction.

We turn to the last case. An irreducible representation of 𝔄̃_{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 of 𝔖̃_{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

W = Sym^{3}(V_{1}) = V_{1}^{χ} ⊕ V_{1}^{−χ},

as a 𝔄̃_{4}-representation, where V_{1}^{χ} = V_{1} ⊗ χ, V^{−χ} = V_{1} ⊗ χ^{−1} and

χ : 𝔄_{4} → ℤ/3 ⊂ ℂ^{∗}

is the cubic character. A pair of (generic) points

p_{χ} ∈ ℙ^{1} = ℙ(V^{χ}), p_{−χ} ∈ ℙ^{1} = ℙ(V^{−χ})

defines a line ℙ^{1} ⊂ ℙ(W). This shows that

ℙ(W) ∼𝔖̃_{4} [figure: L; ℙ^{1} × ℙ^{1}] ,

where 𝔖_{4} acts on the base, 𝔄_{4} acts linearly on the fiber L and 𝔖_{2} = 𝔖̃_{4}/𝔖_{4} acts as an involution on the fiber L. Thus 𝔖̃_{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:

l_{i} : (x_{i}, y_{i}) − (y_{i}, x_{i}) ⊂ ℙ^{1} × ℙ^{1}

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_{q}^{2} := 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_{Δ}^{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_{q}^{2} = 𝔇_{2}\ℙ^{2}, the image of ℙ_{Δ}^{1} intersects ℂ^{∗} in one point. Therefore, the images of ℙ_{Δ}^{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_{q}^{2} is ramified exactly over a union of four lines. If suffices to observe that every conic bundle over ℙ_{q}^{2} 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_{q}^{2} 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} of 𝔄̃_{4}. If V is faithful for 𝔄̃_{4} then there is an α_{0} ∈ A such that V^{α_{0}} is a three-dimensional irreducible faithful representation of 𝔄̃_{4} and

V ∼𝔄̃_{4} V^{α_{0}} × (⊕_{α≠α_{0}}V^{α})

with trivial action of 𝔄̃_{4} on ⊕_{α≠α_{0}}V^{α} (by 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 of 𝔖̃_{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.

Lemma 7.31. — Let V be a representation of G ⊊ SL_{2}, with G ≠ 𝔄̃_{5}. Then G\ Gr(2, V) is rational.

Proof. — The relevant groups G can be subdivided as follows:

1. G is a subgroup of the normalizer of a maximal torus;

2. G an infinite subgroup of a Borel subgroup;

3. G = 𝔄_{4}, 𝔄̃_{4};

4. G = 𝔖_{4}, 𝔖̃_{4}. Let V = W ⊕ W′ be a reducible representation of G. Then (birationally)

\xymatrix{Gr(2, V) \ar[d]^{Hom(ℂ_{x}^{2},W)} \\ Gr(2, W)}

(where x is a point on the base). In particular, if dim W ≤ 2 then

Gr(2, V) ∼_{G} Hom(W′, W),

with linear G-action on Hom(W′, W). This reduction suffices for the relevant infinite groups (for example, for connected solvable G we can apply Lemma 7.6). Further,

– if St_{gen}(G, Gr(2, V)) = 1 then (birationally)

G\ Gr(2, V) → G\ Gr(2, W),

a vector bundle.

– if St_{gen}(G, Gr(2, V)) = ℭ ⊂ Z_{G} (a cyclic subgroup) then (birationally)

G\ Gr(2, V)
ℭ\ Hom(ℂ_{x}^{2},W)
G\ Gr(2, W).

We now consider 𝔄_{4}, 𝔄̃_{4}, 𝔖_{4}. The rationality of G\ Gr(2, V) for irreducible representations of these groups follows from the fact that all of them have dimension ≤ 3. Assume now that V = W ⊕ W′, with W irreducible of dimension 3. The classification of these representations implies that the action of the center must be trivial. Then, birationally,

\xymatrix{Gr(2, V) \ar[d]^{Hom(ℂ_{x}^{2},W′)} \\ ℙ^{2} = ℙ(W^{∗}).}

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 of 𝔖̃_{4}. Let W be its unique irreducible representation of dimension four (as in Lemma 7.30). We claim that 𝔖̃_{4}\ Gr(2, W) is rational. Indeed, as 𝔄̃_{4}-modules, we have

W = W^{χ} ⊕ W^{−χ},

where W^{χ}, W^{−χ} are two copies of the standard representation of 𝔄̃_{4} of dimension 2 and χ (resp. −χ) indicates the eigenspace decomposition for the nontrivial character

χ : 𝔄_{4} → ℤ/3 ⊂ ℂ^{∗}.

Further,

Gr(2, W) ∼ Hom(W^{χ}, W^{−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

Hom(W^{χ}, W^{−χ}) = Sym^{2}(W^{−χ}) ⊕ C_{1},

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

\begin{aligned}f & : & Hom(W^{χ}, W^{−χ}) & → & C_{1}, \\ & & s & ↦ & (x, s(y)) − (s(x), y), \end{aligned}

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}. Thus 𝔖̃_{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 of 𝔖̃_{4}\ Gr(2, W), and more generally, 𝔖̃_{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 a 𝔖̃_{4}-equivariant homogeneous rational map f : Gr(2, nW) → V′ sending the generic 𝔖̃_{4}-orbit in W to the generic 𝔖̃_{4}-orbit in V′. Notice that the center ℭ_{2} acts as a scalar on Hom(W, V′). We have (birationally)

(7.1)  [figure: 𝔖̃_{4}\ Gr(2, V); 𝔖_{2}\ Hom(ℂ_{x}^{2},V′); ∼; ℂ^{∗}×; ℙ(Hom(ℂ_{x}^{2},V′)); 𝔖̃_{4}\ Gr(2, nW); 𝔖_{4}\ Gr(2, nW)]

(with rational bases). The projective bundle on the right has a section. Indeed,

(7.2)  [figure: Hom(ℂ_{x}^{2},V′); Gr(2, nW)]

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_{x}^{2}, 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.

We proceed to describe possible SL_{2}, resp. PGL_{2}-actions on Grassmannians. (If all weights in V are of the same parity then Gr(2, V) carries the PGL_{2}-action, otherwise the SL_{2}-action.)

Lemma 7.32. — Let V be a faithful SL_{2}-representation of dimension ≥ 3. Then

V  St_{gen}
dim ≥ 5 1
V_{4}  ℭ_{2}
V_{3}  𝔇_{2}
V_{2} ⊕ V_{0} ℭ_{2}
V_{2}  N_{T}
V_{1} ⊕ V_{1} ℂ^{∗}
V_{1} ⊕ V_{0}  B̃

Further,

– Gr(2, V_{4}) has a (PGL_{2}, N_{T})-slice S = Sym^{2}(ℙ^{2}) with an af-action of N_{T}/ℭ_{2}, (where ℭ_{2} is the center of N_{T});

– Gr(2, V_{3}) has a (PGL_{2}, 𝔄_{4})-slice birational to ℙ^{1}, with 𝔄_{4} acting on ℙ^{1} as ℭ_{3}.

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.

If V = V_{2} ⊕ V′, with dim V′ > 2, then Gr(2, V) is (birationally) a fibration over Gr(2, V_{2}), with fibers Hom(ℂ^{2}, V′) so that St_{gen} = 1 if dim V′ > 3. If V = V_{2} ⊕ V_{1} then St_{gen} is the same as the (generic) stabilizer of the NT̃-action on Hom(V′, V_{1}), V′ ∈ Gr(2, V_{2}) = ℙ^{2}, hence trivial. For V = V_{2} ⊕ V_{0}, St_{gen} = ℭ_{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 functionals - which is zero.

Lemma 7.33. — The quotient PGL_{2}\ Gr(2, V) is 2-stably rational.

Remark 7.34. — For even d ≥ 10, PGL_{2}\ Gr(2, V_{d}) is rational by [32].

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

PGL_{2}\ Gr(2, V) × ℂ^{2} ∼_{G} N_{T}\ Gr(2, V).

The claim follows from Lemma 7.31. It remains to consider:

1. Gr(2, V_{4}),

2. Gr(2, V_{3}),

3. reducible V.

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)

N_{T}\X ∼ ℂ^{∗} × N_{T}\ℙ^{3},

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.

Proposition 7.35. — Let G, H be finite solvable subgroups of PGL_{2}. Then

G\PGL_{2}/H

is rational.

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 g̃V^{α} = V^{α′} for all g̃ ∈ 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

M_{2} = V_{1} ⊕ V′_{1} = (C_{1} ⊕ C′_{1}) ⊕ (C_{2} ⊕ C_{2})′.

We claim that (birationally)

G\ℙ(M_{2})/H

ℙ^{2} = G\Sym^{2}(ℙ^{1})

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)

\xymatrix{G\ℙ(M_{2})/H \ar[d]^{ℂ^{∗}=N_{T}/H} \\ N_{T}\ℙ(M_{2})/H.}

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 twodimensional eigenspaces for the ℂ^{∗}-action on M_{2}. Thus

ℙ(M_{2})/H

ℙ^{2} = ℙ^{1} × ℙ^{1}/𝔖_{2}

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
N_{T}×N_{T}
ℙ^{1} − 3 pts,

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

\xymatrix{G\ℙ(M_{2})/H \ar[d]^{ℂ^{∗}} \\ U \ar[d]^{ℂ^{∗}} \\ ℙ^{1} − 3 pts.}

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 to 𝔄̃_{5}. Then

GL_{2}\(V ⊕ V)/H

is rational.

Proof. — First of all, V_{1} ⊕ V_{1}/H is rational. Next, by Lemma 7.22,

V ⊕ V ∼_{GL_{2}×GL_{2}} \xymatrix{V \ar[d]^{M_{2}=V_{1}⊕V_{1}} \\ Gr(2, V).}

First we assume that V has odd weight. The Grassmannian Gr(2, V) carries the action of PGL_{2}. If we restrict the bundle V to a generic PGL_{2}-orbit O in Gr(2, V) then the corresponding module H^{0}(O, V_{O}) contains V_{1} as a submodule. By Lemma 7.18, this gives an equivariant map

V → V_{1} ⊕ V_{1}

with a 1-transitive action of GL_{2} on the target. Thus

(7.3)  GL_{2}\V/H ∼ H\ Gr(2, V)

(with the same subgroup H ⊂ GL_{2} appearing on the left). Indeed, GL_{2} ⊂ (V_{1} ⊕ V_{1}) = M_{2} and multiplication by H on the right gives an orbit x · H. This orbit is a (GL_{2} × H, H^{x} × H)-slice (with H^{x} = xHx^{−1}) and it is stabilized exactly by H^{x} × H, acting doubly transitively on the set H^{x} · x. It follows that every point x′ ∈ x · H is a (H^{x} × H, H^{x})-slice of the orbit x · H. The quotient H\ Gr(2, V) is rational by 7.31.

Assume that V has even weight. If the PGL_{2}-action is af then

GL_{2}\V/H ∼ (PGL_{2}\ Gr(2, V)) × (ℂ^{∗}\(V_{1} ⊕ V_{1})/H).

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_{T}-action 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 NT̃ × 𝔖_{2} × H (where NT̃ ⊂ GL_{2}). The action of NT̃ × 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}.

Proposition 7.37. — Let

X ⟶^{L} Y = ∏_{j∈J} ℙ(V_{d_{j}})

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

X × X ∼_{GL_{2}×𝔖_{2}} V_{d} ⊕ V_{d}

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

ℙ(V_{d}) ∼_{PGL_{2}} S × PGL_{2},

for a rational slice S (with trivial PGL_{2}-action). We have a PGL_{2} ×ℂ^{∗} ×𝔖_{2}-action on

ℂ × PGL_{2} × S × ℂ × PGL_{2} × S.

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

ℂ × ℙ^{3} × ℂ × ℙ^{3}
ℙ^{1}×ℙ^{1}
Gr(2, V_{3})

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

[figure: ℙ^{1}×ℂ×ℙ^{1}×ℂ; ℙ^{1}]

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:

\xymatrix{X \ar[d]^{L} \\ S × PGL_{2}}

and, using Lemma 7.21, reduce to either a vector bundle over

PGL_{2}\PGL_{2} × PGL_{2}/𝔖_{2},

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_{s}^{1} = ℙ^{1} of ℙ^{4}, will be a set of six lines ℙ_{g,h}^{1}, labeled by a pair of generators g, h ∈ 𝔇_{2} (which act trivially on ℙ_{s}^{1} ⊂ ℙ^{4}). More precisely, the line ℙ_{g,h}^{1} is the set given by (x : gx : hx : ghx) ∈ (ℙ^{1})^{4}, for x ∈ ℙ^{1}. The map ℙ_{g,h}^{1} → ℙ_{s}^{1} = ℙ_{t,s}^{1}/𝔇_{2} has degree 4. Thus ℙ_{g,h}^{1} 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

Y × Y = ℙ(V_{d}) × Y′ × ℙ(V_{d}) × Y′

such that the PGL_{2}-action is af and

ℙ(V_{d}) × Y′ × ℙ(V_{d}) × Y′ ∼_{PGL_{2}×𝔖_{2}} (Y′ × Y′) × (ℙ(V_{d}) × ℙ(V_{d}))

(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

X := PGL_{2}\(ℙ_{1}(V_{2}) × ℙ_{2}(V_{2}) × ℙ_{1}(V_{2}) × ℙ_{2}(V_{2}))/𝔖_{2}

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

X → PGL_{2}\ℙ_{1}(V_{2}) × ℙ_{1}(V_{2})/𝔖_{2}

and the PGL_{2} × 𝔖_{2}-equivariant map of degree 6

\begin{aligned}pr : & ℙ(V_{2}) × ℙ(V_{2}) & → & ℙ(V_{4}) \\ & (Q_{1}, Q_{2}) & ↦ & Q_{1} · Q_{2}. \end{aligned}

The space ℙ(V_{4}) has a (PGL_{2}, 𝔖_{4})-slice ℙ_{s}^{1} (the 𝔇_{2}-invariant polynomials). The zeroes of a (polynomial) p ∈ ℙ_{s}^{1} form an orbit under 𝔇_{2}. The preimage pr^{−1}(ℙ_{s}^{1}) ⊂ 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_{g}^{1} ⊂ 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_{g}^{1} → ℙ_{s}^{1} has degree two and its fibers coincide with orbits of h (since g acts trivially on ℙ_{g}^{1}). The action of h is given by

h : {(x, gx), (hx, ghx)} ↦ {(hx, ghx), (x, gx)}.

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_{g}^{1}. The line ℙ_{g}^{1} 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_{s}^{1}). Each ℙ_{g}^{1} is a (PGL_{2} × 𝔖_{2}, 𝔇_{4} × 𝔖_{2})-slice of ℙ^{2} × ℙ^{2}. Therefore,

X ∼ 𝔇_{4}\ℙ^{1} × ℙ^{2} × ℙ^{2}/𝔖_{2}.

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 of 𝔇̃_{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

ℙ^{1} × ℙ^{2} × ℙ^{2} ∼_{𝔇_{4}×𝔖_{2}} [figure: V′⊕V′; ℙ^{1}]

(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_{g}^{1} 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_{g}^{1} coincide with fibers of the map ℙ_{g}^{1} → ℙ_{s}^{1}. On the other hand, i acts nontrivially on ℙ_{s}^{1}. We claim that

𝔇_{4}\(V′ ⊕ V′) × ℙ^{1}/𝔖_{2}

𝔇_{4}\(V′ × ℙ^{1})

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

𝔇_{4}\(V′ ⊕ V′) × ℙ^{1}/𝔖_{2}

𝔇_{4}\((V′ ⊕ V′)/V′_{inv}) × ℙ^{1}/𝔖_{2}

is a vector bundle with base a quotient of the vector bundle (V′ ⊕ V′/V′_{inv}) → ℙ_{g}^{1} by 𝔇_{4} × 𝔖_{2}. The variety (V′ ⊕ V′/V′_{inv}) × ℙ^{1} has a fiberwise (scalar) ℂ^{∗}-action commuting with the 𝔇_{4} × 𝔖_{2}-action. Since every ℂ^{∗}-action has a slice,

X′ := 𝔇_{4}\((V′ ⊕ V′)/V′_{inv}) × ℙ^{1}/𝔖_{2},

is rational by dimensional reasons: X′/ℂ^{∗} is a unirational, therefore, rational surface and

X′ ∼ (X′/ℂ^{∗}) × ℂ^{∗}.

Proposition 7.39. — Let X be V ⊕ V, where V = V_{d} is an irreducible GL_{2}- representation, ℓ > 0 and H ⊂ SL_{2} with H ≠ 𝔄̃_{5}. Then

GL_{2}\X × ℙ(V_{ℓ})/H

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

(ℙ^{1} × ℙ(V_{ℓ}) ∼_{𝔄_{4}} ℙ^{1} × ℙ(V_{ℓ})

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.

Proposition 7.40. — Let X = (⟶^{L} Y)^{2}, where Y = ∏_{j∈J} ℙ(V_{d_{j}}) and ℓ > 0. Then

GL_{2}\X × ℙ(V_{ℓ})/𝔖_{2}

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

GL_{2}\X × ℙ(V_{ℓ})/𝔖_{2} ∼ (ℂ^{∗}\ℙ(V_{ℓ})) × (ℂ^{2}/ℂ^{∗} × 𝔖_{2}),

a rational variety.

8. 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.

Let R(r_{0}, r_{1}, r_{∞}) be the space of rational maps f : ℙ^{1} → ℙ^{1} with local ramification data (vectors) r_{0}, r_{1}, r_{∞} over the points 0, 1, ∞.

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. — In these cases the map f = f_{0}/f_{∞} is given by coprime polynomials satisfying the equations:

– f_{0}^{2} − f_{∞}^{2} = g_{1}^{2}g′_{1};

– f_{0}^{2} − f_{∞}^{2} = g_{1}^{3}g′_{1};

– f_{0}^{3} − f_{∞}^{3} = g_{1}^{2}g′_{1}, where g′_{1} is an arbitrary polynomial. The first equation leads to

(f_{0} − f_{∞})(f_{0} + f_{∞}) = g_{1}^{2}g′_{1}

and, by coprimality, to

\begin{aligned}f_{0} − f_{∞} & = & g_{11}^{2}g′_{11}, \\ f_{0} + f_{∞} & = & g_{12}^{2}g′_{12}, \end{aligned}

with arbitrary g_{11}, g′_{11}, g_{12}, g′_{12} (satisfying the obvious degree conditions) — a union of rational varieties.

The second case is analogous. Consider the third case: since f_{0}^{3} − f_{∞}^{3} is a square we obtain

\begin{aligned}f_{0} − f_{∞} & = & g_{1}^{2} \\ f_{0} − ζf_{∞} & = & g_{2}^{2} \\ f_{0} − ζ^{2}f_{∞} & = & g_{3}^{2} \end{aligned}

(where ζ^{3} = 1) and we need to solve

\frac{2ζ}{1 + ζ} g_{1}^{2} + \frac{1 − ζ}{1 + ζ} g_{2}^{2} = g_{3}^{2}.

Now we apply the parametrization as above.

Corollary 8.2. — Let R(r_{0}, r_{1}, r_{∞}) be as in 8.1. Then

PGL_{2}\R(r_{0}, r_{1}, r_{∞})

is rational.

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_{2}- representations). By the theorem of Katsylo 7.14, the corresponding quotients are rational.

Remark 8.3. — Only the first case with g′_{1} = 1 can admit a nontrivial action of H_{Γ} (which necessarily is ℤ/3). But even in this case the action of ℤ/3 is linear and it commutes with the action of GL_{2} on pairs of polynomials. Lemma 7.31 implies rationality.

Lemma 8.4. — Every irreducible component of the variety R of rational maps f : ℙ^{1} → ℙ^{1} of degree 5 and prescribed global ramification datum

RD(f) = [(2, 2, 1)_{0}, (2, 2, 1)_{1}, (2, 2, 1)_{∞}, (2), (2)]

is rational.

Proof. — Changing the variables (fixing two ramification points over 1 ∈ ℙ^{1} as 0, ∞), we can write f = F_{1}/F_{2} where

\begin{aligned}F_{1}(x) & = & f̂_{1}(x)^{2}â_{1}(x)^{2}b̂_{1}(x) \\ F_{2}(x) & = & f̂_{2}(x)^{2}â_{2}(x)^{2}b̂_{2}(x) \end{aligned}

where f̂_{1}, f̂_{2}, â_{1}, â_{2}, b̂_{1}, b̂_{2} are linear forms in x. Since the leading coefficients of F_{1} and F_{2} are equal we can assume that they are both equal to 1 and write f̂_{1}(x) = x + f_{1}, …, b̂_{2}(x) = x + b_{2}, with some nonzero constants f_{1}, …, b_{2}. Since we have one free parameter (under the action of PGL_{2}) we can assume that b_{1} = 1. Thus

f̂_{1}(x)^{2}â_{1}(x)^{2}â_{2}(x) − f̂_{2}(x)^{2}b̂_{1}(x)^{2}b̂_{2}(x) = ∑_{i} g_{i}x^{i} = c_{1}x^{2}(x + c_{2})

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:

X_{1} = a_{1} + f_{1}, Y_{1} = a_{1}f_{1}, X_{2} = f_{2} + a_{2}, Y_{2} = f_{2}a_{2}, b_{1}, b_{2}.

Write the equations on the coefficients g_{j} as

\begin{aligned}2X_{1} + b_{1} & = & 2X_{2} + b_{2} \\ Y_{1}^{2}b_{1} & = & Y_{2}^{2}b_{2} \\ Y_{1}^{2} + 2X_{1}Y_{1}b_{1} & = & Y_{2}^{2} + 2X_{2}Y_{2}b_{2}. \end{aligned}

Since b_{1} = 1, for a fixed b_{2} we get

\begin{aligned}2X_{1} + 1 & = & 2X_{2} + b_{2} \\ Y_{1} & = & ±\sqrt{b_{2}}Y_{2} \\ b_{2}Y_{2} + 2\sqrt{b_{2}}X_{1} & = & Y_{2} + 2X_{2}b_{2}. \end{aligned}

This is a union of two (affine) lines. After a rational covering (\sqrt{b_{2}}) our surface is (rationally) a ℙ^{1}-bundle over ℙ^{1}, a rational surface.

Lemma 8.5. — Every irreducible component of the variety R of rational maps f : ℙ^{1} → ℙ^{1} of degree 4 and ramification datum

RD(f) = [(2, 2)_{0}, (2, 1, 1)_{1}, (2, 1, 1)_{∞}]

is a rational surface.

Proof. — Using the PGL_{2}-action on the preimage ℙ^{1} we can assume that the points (2, 2) are +1, −1, respectively, and that the point of degree 2 (in the local ramification datum (2, 1, 1)) over 0 is ∞. Thus we can write

(x^{2} − 1)^{2} − c(x + c_{1})(x + c_{2})(x + c_{3})^{2} = g_{2}(x),

where g_{2} is an arbitrary polynomial of degree 2 and c is some constant. We get two equations

c = 1,
c_{1} + c_{2} + 2c_{3} = 0.

Thus we have a (rational) surjection of ℙ^{2} onto R.

Lemma 8.6. — Every irreducible component of the variety R of rational maps f : ℙ^{1} → ℙ^{1} of degree 4 with ramification datum

RD(f) = [(2, 2)_{0}, (3, 1)_{1}, (2, 1, 1)_{∞}, (2), (2)]

is a rational curve.

Proof. — A generic map with this ramification datum is given by the equation f = f_{1}/f_{2}, where

f_{1} = (x^{2} − 1)^{2}, f_{2} = (x + c_{1})(x + c_{2})(x + c_{3})^{2}

and

f_{1} − f_{2} = (x^{2} − 1)^{2} − c(x + c_{1})(x + c_{2})(x + c_{3})^{2} = g_{1}(x),

where g_{1}(x) is linear. Thus c = 1 and

c_{1} + c_{2} + 2c_{3} = 0,
c_{1}c_{2} + 2c_{1}c_{3} + 2c_{2}c_{3} + c_{3}^{2} = 0,

clearly rational.

Lemma 8.7. — The irreducible component of the variety R of rational maps f : ℙ^{1} → ℙ^{1} of degree 3 with ramification datum

RD(f) = [(2, 1)_{0}, (2, 1)_{1}, (2, 1)_{∞}, (2)]

is a rational curve.

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,Γ̃} as quotients (by the left PGL_{2} and right H_{Γ}-action)

PGL_{2}\U′_{r,Γ̃,ℓ}/H_{Γ}.

Here

U′_{r,Γ̃,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}.

First we consider general families:

ET(E) − 12ℓ = deg(j_{E}) ET(Γ).

For d = (d_{1}, ..., d_{k}) ∈ ℕ^{k} we put

ℙ^{d} := ∏_{j=1}^{k} ℙ(V_{d_{j}}).

Recall that R_{Γ} is (birationally) the total space of a line bundle over the space

ℙ^{d} × ℙ^{d′},

where ∑_{j=1}^{k} d_{j} = ∑_{j=1}^{k′} d′_{j}.

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

PGL_{2}\ℙ^{d} × ℙ^{d′},

which is the theorem of Katsylo 7.14.

Case 2. d = d′ and k ≥ 2. By Corollary 3.14, H_{Γ} = 𝔖_{2} (permutation of the factors). This case is covered by Proposition 7.37.

Case 3. d = d′ = (d). This case is covered by Proposition 7.36.

Now we discuss the special families:

ET(E) − 12ℓ < deg(j_{E}) ET(Γ).

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.

10. 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

48 = ET(E) ≥ ET(Γ)

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

Figure 1. The tree T_{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.

Figure 2. The loop L.

This gives the following list of graphs:

Figure 3. The case ET(Γ) = 12.

There is only one saturated graph from the list above which has no outer loops (Figure 4).

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) = 24: Again, we have only one topological tree T_{24} with ET(T) = 24:

Figure 5. The tree T_{24}.

Case ET(Γ) = 36: There are only 3 saturated graphs without end-loops (modulo equivalent embedding into the sphere):

Figure 6. The case ET(Γ) = 36.

Any other graph is either a tree or a sum of a saturated graph T′ with ET(T′) = 0, 12, 24 with trees (with complementary ET). There is only one topological tree T_{36} with ET(T_{36}) = 36.

Figure 7.

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:

Figure 8.

This gives 8 graphs T_{Γ} in the first case and 4 in the second case.

Case ET(Γ) = 48: We have one tree T_{48} with ET(Γ) = 48:

Figure 9. The tree T_{48}.

Here is the list of all saturated graphs with ET(Γ) = 48.

Figure 10. Saturated graphs in the case ET(Γ) = 48.

References

[1] J. Alexander, Note on Riemann spaces, Bull. AMS 26, 370–372, (1920).

[2] M. Artin, D. Mumford, Some elementary examples of unirational varieties which are nor rational, Proc. Lond. Math. Soc. 25, 75–95, (1972).

[3] W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Ergebnisse der Math. und ihrer Grenzgebiete, vol. 4, Springer-Verlag, 1984.

[4] G. Belyi, On Galois extensions of the maximal cyclotomic field, Izv. AN USSR 43:2, 269–276, (1979).

[5] H. Boden, K. Yokogawa, Rationality of moduli spaces of parabolic bundles, Journ. Lond. Math. Soc. (2) 59, no. 2, 461–478, (1999).

[6] A. Beauville, J.-L. Colliot-Thélèlene, J.-J. Sansuc, P. Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. 121, 238–318, (1985).

[7] F. Bogomolov, Rationality of the moduli space of hyperelliptic curves of arbitrary genus, CMS Conf. Proceedings 6, AMS, 17–37, (1986).

[8] F. Bogomolov, Yu. Tschinkel, Monodromy of elliptic surfaces,

math.AG/0002168.

[9] F. Bogomolov, P. Katsylo, Rationality of some quotient varieties, Mat. Sb. (N.S.) 4, 584–589, (1985).

[10] H. Clemens, Ph. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95, 281–356, (1972).

[11] L. Costa, R. Miró-Roig, On the rationality of moduli spaces of vector bundles on Fano surfaces, Journ. Pure Appl. Algebra 137, no. 3, 199–220, (1999).

[12] I. Dolgachev, Rationality of fields of invariants, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–16, Proc. Sympos. Pure Math., 46, Part 2, AMS 1987.

[13] R. Friedman, Algebraic surfaces and holomorphic vector bundles, Springer Verlag, 1998.

[14] R. Friedman, J. Morgan, Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Math. und ihrer Grenzgebiete, vol. 27, Springer-Verlag, 1994.

[15] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 no. 1, 23–88, (1982).

[16] V. Iskovskikh, Yu. I. Manin, Three-dimensional quartics and counterexamples to Luroth’s problem, Math. Sb. 86, 140–166, (1971).

[17] P. Katsylo, Rationality of fields of invariants of reducible representations of the group SL_{2}, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5, 77–79, [18] P. Katsylo, Rationality of the moduli spaces of hyperelliptic curves, Izv. Akad.

Nauk SSR Ser. Math. 48, 705–710, (1984).

[19] P. Katsylo, Rationality of moduli varieties of plane curves of degree 3k, Math. Sb. (N.S.) 136, no. 3, 377–383, (1989).

[20] P. Katsylo, Rationality of the variety of moduli of curves of genus 5, Math. Sb. 182, no. 3, 457–464, (1991).

[21] P. Katsylo, Rationality of the moduli variety of curves of genus 3, Comm. Math. Helv. 71, no. 4, 507–524, (1996).

[22] A. King, A. Schofield, Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.) 10, no. 4, 519–535, (1999).

[23] K. Kodaira, On compact analytic surfaces II, III, Ann. of Math. (2) 77, 563– 626, (1963); 78, 1–40, (1963).

[24] Sh. Kondo, The rationality of the moduli space of Enriques surfaces, Comp. Math. 91, no. 2, 159–173, (1994).

[25] P. E. Newstaead, Rationality of moduli spaces of stable bundles, Math. Ann. 215, 251–268, (1975).

[26] I. Kolpakov-Mirochnichenko, Yu. Prokhorov, Rationality of fields of invariants of some four-dimensional linear groups and an equivariant construction related to the Segre cubic, Math. Sborn. 182, 1430–1445, (1991).

[27] W.-P. Li, Relations between moduli spaces of stable bundles over ℙ^{2} and rationality, Journ. Reine Angew. Math. 484, 201–217, (1997).

[28] P. Lochak, L. Sch, eds., Geometric Galois Actions, LMS Lecture Notes, vol. 242, Cambridge Univ. Press, 1997.

[29] T. Maeda, An elementary proof of the rationality of moduli spaces for rank 2 vector bundles on ℙ^{2}, Hiroshima Journ. of Math. 20, no. 1, 103–107, (1990).

[30] D. J. Saltman, Noether’s problem over an algebraically closed frield, Inv. Math. 77, 71–84, (1984).

[31] N. I. Shepherd-Barron, Invariant theory for S_{5} and the rationality of M_{6}, Compositio Math. 70, no. 1, 13–25, (1989).

[32] N. I. Shepherd-Barron, The rationality of some moduli spaces of plane curves, Compositio Math. 67, no. 1, 51–88, (1988).

[33] N. I. Shepherd-Barron, Rationality of moduli spaces via invariant theory, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), 153–164, Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989.

[34] L. Schneps, ed., The Grothendieck’s Theory of Dessins d’Enfants, LMS Lecture Notes, vol. 200, Cambridge Univ. Press, 1994.

[35] I. Shafarevich, et al., Algebraic Surfaces, Proc. Steklov Inst. Math. 75, III-VI, 52–84, 162–182, (1965).

[36] E. B. Vinberg, Rationality of the field of invariants of a triangular group, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 115 no. 2, 23–24, (1982).