Moduli of real pointed quartic curves

We describe a natural open stratum in the moduli space of smooth real pointed quartic curves in the projective plane. This stratum consists of real isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$ such that the tangent line at $p$ intersects the curve in two distinct points besides $p$. We will prove that this stratum consists of $20$ connected components. Each of these components has a real toric structure defined by an involution in the Weyl group of type $E_7$.


Introduction
A classical result found in Zeuthen [20] is the classification of smooth real plane quartic curves. The set of real points of such a curve consists of at most four ovals in the real projective plane and the six possible configurations are shown below in Figure 1. The space of real plane quartic curves is the projective space P4,3(R) = P Sym 4 (R 3 ) of dimension 14. The discriminant subspace ∆(R) ⊂ P4,3(R) that consists of singular real quartic curves is of codimension one. It was proved by Klein [9] that each of the six types of smooth real plane quartic curves determines a connected component in the space P4,3(R) − ∆(R). We will be interested in the moduli space Q R = PGL(3, R)\ (P4,3(R) − ∆(R)) whose points represent real isomorphism classes of such quartics. This space also consists of six connected components since the group PGL(3, R) is connected. Figure 1: The six topological types of smooth real plane quartic curves obtained by deforming a union of two ellipses. real plane quartic curve together with a general real point on the curve. By general point we mean that the tangent line at this point intersects the curve in two other, distinct points. In other words the tangent line at this point is not a bitangent or a flex line. These two other points of intersection can be both real or form a pair of complex conjugate points. Our first result is the classification of real quartic curves with a general point, similar to the one given by Zeuthen.  This classification is more subtle then the topological one: the second and fourth picture from the first row are topologically equivalent but represent different components in the moduli space. Something similar occurs in the study of smooth real plane curves of higher degree. For example there exist two smooth real plane curves of degree six with the same configuration of ovals that are non-isomorphic. For a discussion of this phenomenon we refer to [5] Section 4.8.
Our second result concerns the structure of the 20 components of the moduli space. For this we start by extending work of Looijenga [12], [13] on moduli spaces of complex del Pezzo pairs to the real numbers. We focus on del Pezzo surfaces of degree two as these are related to plane quartic curves. A similar treatment could be given for del Pezzo surfaces of other degrees. To formulate our result we introduce some notation. Let Q be a root lattice of type E7 and define the complex adjoint torus T = Hom(Q, C * ). The Weyl group W of type E7 acts by reflections on Q and thus on T. The set of fixed points in T of a reflection in W is called a toric mirror. We denote by T • the complement of all toric mirrors in T for this group action of W . Now we can state our second result. Theorem 1.2. Let (Q • 1 ) R be the moduli space representing real isomorphism classes of pairs (C, p) with C a smooth real plane quartic curve and p ∈ C(R) a general real point. There is an isomorphism of real orbifolds where by definition the right hand side consists of all W -orbits of t ∈ T • such that u · t =t for some involution u ∈ W .
This paper is organized in two parts. The first part consists of Sections 2 through 6 and is dedicated to the proof of Theorem 1.2. The necessary preliminaries on del Pezzo surfaces and involutions in Weyl groups are presented in Sections 2 through 5 and the proof of Theorem 1.2 is presented in Section 6.
The second part consists of Section 7 through 9 and is dedicated to the proof of Theorem 1.1. The main idea of the proof is to use the isomorphism of Equation 1 to study the moduli space (Q • 1 ) R using results on root systems and involutions in Weyl groups on the right hand side. In Section 7 we study the real points of such torus quotients and their connected components for general root systems of type ADE. For type E7 we then prove that there are twenty connected components. In Sections 8 and 9 we relate these twenty components to the pictures of Figure 2. This completes the proof of Theorem 1.1. A possible alternative approach to obtain the classification of Theorem 1.1 would be to consider trigonal curves with a single node in the Hirzebruch surface Σ2. This construction is described in [21]. However this approach would not give a description of the moduli space and its components.

Involutions in Coxeter groups
In order to make the right-hand side of Equation 1 more explicit we need to understand the conjugacy classes of involutions in the Weyl group of type E7. Weyl groups can be realized as finite Coxeter groups. The classification of conjugacy classes of involutions in a Coxeter group was done by Richardson [16] and Springer [18]. Before this the classification of conjugacy classes of elements of finite Coxeter groups was obtained by Carter [2]. In this section we give a brief review of these results. Definition 2.1. A Coxeter system is a pair (W, S) with W a group presented by a finite set of generators S = {s1, . . . , sr} subject to relations (sisj) m ij = 1 with 1 ≤ i, j ≤ r where mii = 1 and mij = mji are integers ≥ 2. We also allow mij = ∞ in which case there is no relation between si and sj. These relations are encoded by the Coxeter graph of (W, S). This is a graph with r nodes labeled by the generators. Nodes i and j are not connected if mij = 2 and are connected by mij − 2 edges otherwise. If mij = ∞ we connect the vertices by a thick edge. For a Coxeter system (W, S) we define an action of the group W on the real vector space V with basis {es}s∈S. First we define a symmetric bilinear form B on V by the expression B(ei, ej) = 2 cos π mij .
Then for each si ∈ S the reflection: si(x) = x − B(ei, x)ei preserves this form B. In this way we obtain a homomorphism W → GL(V ) called the geometric realization of W . For each subset I ⊆ S we can form the standard parabolic subgroup WI < W generated by the elements {si; i ∈ I} acting on the subspace VI generated by {ei}i∈I . We say that WI (or also I) satisfies the (−1)-condition if there is a wI ∈ WI such that wI · x = −x for all x ∈ VI . The element wI necessarily equals the longest element of (WI , SI ). This implies in particular that WI is finite. Let I, J ⊆ S, we say that I and J are W -equivalent if there is a w ∈ W that maps {ei}i∈I to {ej}j∈J . Now we can formulate the main theorem of [16].
Theorem 2.2 (Richardson). Let (W, S) be a Coxeter system and let J be the set of subsets of S that satisfy the (−1)-condition. Then 1. If c ∈ W is an involution, then c is conjugate in W to wI for some I ∈ J .
2. Let I, J ∈ J . The involutions wI and wJ are conjugate in W if and only if I and J are W -equivalent.
This theorem reduces the problem of finding all conjugacy classes of involutions in W to finding all W -equivalent subsets in S satisfying the (−1)-condition. First we determine which subsets I ⊆ S satisfy the (−1)-condition, then we present an algorithm that determines when two subsets I, J ⊆ S are W -equivalent. If (WI , SI ) is irreducible and satisfies the (−1)-condition then it is of one of the following types A1, Bn, D2n, E7, E8, F4, G2, H3, H4, I2(2p) with n, p ∈ N and p ≥ 4. If (WI , SI ) is reducible and satisfies the (−1)-condition then WI is the direct product of irreducible, finite standard parabolic subgroups (Wi, Si) from (2). The Coxeter diagrams of the (Wi, Si) occur as disjoint subdiagrams of the types in the list of the diagram of (W, S). The element wI is the product of the wI i which act as −1 on the VI i . Now let K ⊆ S be of finite type and let wK be the longest element of (WK , SK ). The element τK = −wK defines a diagram involution of the Coxeter diagram of (WK , SK ) which is non-trivial if and only if wK = −1. If I, J ⊆ K are such that τK I = J then I and J are W -equivalent. To see this, observe that wK wI · I = wK · (−I) = τK I = J. Now we define the notion of elementary equivalence. It is proved in [16] that I and J are W -equivalent if and only if they are related by a chain of elementary equivalences: I = I1 I2 . . . In = J. This provides a practical algorithm to determine all the conjugacy classes of involutions in a given Coxeter group (W, S) using its Coxeter diagram 1. Make a list of all the subdiagrams of the Coxeter diagram of (W, S) that satisfy the (−1)-condition. These are exactly the disjoint unions of diagrams in the list (2). Every involution in W is conjugate to wK with K a subdiagram in this list.
2. Find out which subdiagrams of a given type are W -equivalent by using chains of elementary equivalences.
Example 2.4 (E7). We use the procedure described above to determine all conjugacy classes of involutions in the Weyl group of type E7. This result will be used many times later on. Since W7 contains the element −1 the conjugacy classes of involutions come in pairs {u, −u}. We label the vertices of the Coxeter diagram as in Figure 3 It turns out that all involutions of a given type are equivalent with the exception of type A 3 1 : here there are two non-equivalent involutions as seen in Figure 4. The types of involutions that occur are

Del Pezzo surfaces
In this section we review the theory of del Pezzo surfaces over the real and the complex numbers. For del Pezzo surfaces over the complex numbers two excellent references for the proofs in this section are [15] or [6].

Complex del Pezzo surfaces
Definition 3.1. A del Pezzo surface Y is a smooth, complex projective surface whose anticanonical system | − KY | is ample. The degree of Y is the self-intersection number: d = KY · KY of the canonical class in the Picard group Pic(Y ) of Y . It is an integer with 1 ≤ d ≤ 9.
A del Pezzo surface can be realized as the blowup of a configuration of points in the projective plane. This is expressed by the following theorem. 2. The smooth quadric P 1 × P 1 in which case d = 8.
From now on we only consider del Pezzo surfaces of the first kind. Exhibiting a del Pezzo surface as a blowup π : Y → P 2 fixes a basis of the Picard group Pic(Y ). This basis consists of the classes Ei = π −1 (Pi) with 1 ≤ i ≤ r of the exceptional curves over the blown up points and the class E0 of the strict transform of a general line in P 2 . The anticanonical class expressed in this basis of Pic(Y ) is given by It is represented by the strict transform of a cubic in P 2 through the points B = {P1, . . . , Pr}. We also write Pic 0 (Y ) for the orthogonal complement of −KY in Pic(Y ). From the description of a del Pezzo surface as a blowup of P 2 it follows that the Picard group Pic(Y ) is isomorphic to the hyperbolic lattice Z1,r of rank r + 1 and signature (1, r). It has a basis {e0, . . . , er} with inner product defined by the relations An isomorphism φ : Z1,r → Pic(Y ) is called a marking of the del Pezzo surface Y if it maps the element k = −3e0 + e0 + . . . + er to the canonical class KY of Pic(Y ). An isomorphism (Y, φ) ∼ = (Y , φ ) of marked del Pezzo surfaces is an isomorphism F : Y → Y such that the following diagram commutes.
Exhibiting a del Pezzo surface as a blowup π : Y → P 2 is equivalent to adding a marking to Y ; from the marking φ we recover the blowup map by blowing down the exceptional curves φ(ei) for 1 ≤ i ≤ r. This determines a set B = {P1, . . . , Pr} of r points in general postition in P 2 . If two marked del Pezzo surfaces are isomorphic then the corresponding point sets B and B are related by an element of PGL(3, C). As a consequence the elements of the space This implies that DP d is isomorphic to an open subset of (A 2 ) r−4 .

The Cremona action of the Weyl group
The stabilizer of the element k = −3e0 + e1 + . . . + er in the orthogonal group O(Z1,r) is a finite Coxeter group of type A1A2, A4, D5, E6, E7, E8 for r = 3, . . . , 8. A set of generators S = {s1, . . . , sr} for Wr is given by the reflections in the simple roots These root span a lattice Qr that is precisely the orthogonal complement of k ⊥ in Z1,r. The group Wr acts on marked del Pezzo surfaes by composing with the marking: w · (X, φ) = (X, φ • w −1 ) for w ∈ Wr. This action is simply transitive so that the orbit space DP d = Wr\ DP d is a coarse moduli space for del Pezzo surfaces of degree d.
We now describe this action on the set of blown up points in P 2 . Suppose (X, φ) is a marked del Pezzo surface and π : X → P 2 is the corresponding blowing up map with B ⊆ P 2 the set of blown up points. If φ is another marking of X then φ = φ • w for some element w ∈ Wr. The element w defines a birational transformation ρ(w) of P 2 in the following way: first blow up P 2 in the points of B. Then blow down the exceptional curves φ (ei) = φ(w · ei) for 1 ≤ i ≤ r. This determines a new set of points B and blowup map π : X → P 2 corresponding to φ such that the following diagram commutes.
In this way we obtain a homomorphism of the Weyl group Wr to the group of birational transformations of P 2 ρ : Wr → Bir(P 2 ) We can calculate this representation on the set S of simple reflections. The element ρ(si) with 1 ≤ i ≤ r − 1 acts on Zr,r by so that it corresponds to the transposition of the points Pi and Pi+1. The element sr gives a more interesting transformation. It acts on Z1,r by Geometrically this means that ρ(sr) is obtained by first blowing up P1, P2 and P3 and then blowing down the strict transforms of the lines connecting them. This birational transformation ρ(sr) is called the standard triangular Cremona transformation based in P1, P2 and P3. A simple calculation shows that s7(2e0 − e1 − e2 − e3) = e0 so that the image of a conic through P1, P2, P3 under the standard triangular Cremona transformation is a line. If we assume that these points are then ρ(sr) is given by (x : y : z) → (yz : xz : xy). To summarise: the group Wr acts on (P 2 ) r − ∆ by permuting the points and by standard triangular Cremona transformations centered in triples of distinct points.

Del Pezzo surfaces of degree two
Suppose that Y is a del Pezzo surface of degree two so that it is isomorphic to the blowup of the projective plane P 2 in 7 points. The anticanonical system of Y defines a morphism It is a double cover of P 2 branched along a smooth quartic curve C ⊂ P 2 . Conversely a smooth quartic C = {f (x, y, z) = 0} determines a del Pezzo surface Y of degree two by the formula Consequently every del Pezzo surface Y has a special involution that corresponds to the deck transformation of the double cover Y → P 2 . This is called the Geiser involution. In terms of Equation 6 this involution is given by If we choose a marking φ : Pic(Y ) → Z1,7 then the induced involution on the lattice Z1,7 is given by It fixes the element k and acts as −1 on k ⊥ so that it corresponds to the central element −1 ∈ W (E7). An element e ∈ Z1,7 that satisfies e·e = −1 and e·k = −1 is called exceptional.
The set E of exceptional elements forms a single W (E7)-orbit and consists of the 56 elements 1. ei with 1 ≤ i ≤ 7, the class of the exceptional divisor Ei.
, the class of the strict transform of the cubic Ki through 6 points with a node at a seventh point.
The elements of E come in 28 pairs (ei, ki), (lij, cij) whose elements are interchanged by the involution ρ. The geometric meaning of this is as follows. An exceptional element E ∈ Pic(Y ) corresponds to a exceptional curve on the del Pezzo surface Y and there are 56 of these. The two elements of a pair (E, ρY (E)) are mapped to a single bitangent of the quartic curve by the anticanonical map. This accounts for all 28 bitangents of a smooth plane quartic curve.

Real del Pezzo surfaces
We review some results of Wall [19] on real del Pezzo surfaces. Other references on this subject are Kollár [10] and Russo [17]. A real del Pezzo surface is a pair (Y, χY ) with Y a complex del Pezzo surface and χY : Y → Y a real form on Y . The real points Y (R) of Y are the fixed points under χY . The action of χ induces an involution χ * Y on the Picard group Pic(Y ) which preserves the canonical class K and the intersection product. By fixing a marking φ : Z1,r → Pic(Y ) we obtain an involution of the lattice Z1,r by the formula The involution χ preserves the element k ∈ Z1,r. As we have seen such an involution corresponds to an involution u in the Weyl group Wr. The conjugacy class of this involution in Wr is an important invariant of the real structure on Y . A real del Pezzo surface Y of degree two is the double cover of the projective plane P 2 ramified over a smooth real plane quartic curve C ⊂ P 2 so that We fix the sign of f so that f > 0 on the orientable interior part of C(R) ⊂ P 2 (R). By using the deck transformation ρY of the cover we see that there are two real forms of Y : These real forms satisfy: In [19] Wall determines the correspondence between the conjugacy classes of the u ∈ W (E7) and the topological type of Y (R). The results are shown in Table 1. We use the notation kX for the disjoint union and # k X for the connected sum of k copies of a real surface X. From this table we see that except for the classes of D4 and A 3 1 the conjugacy class of u ∈ W (E7) determines the topological type of the real plane quartic curve C(R).

Moduli of del Pezzo pairs
In this section we study del Pezzo surfaces obtained by blowing up r points on a fixed plane singular cubic. The strict transform of this cubic is a singular anti-canonical curve on the del Pezzo surface. This is the situation studied by Looijenga for del Pezzo surfaces of degree two in [12] and for general del Pezzo surfaces in [13]. In this article we restrict ourselves to the case of del Pezzo surfaces of degree two. Definition 4.1. A del Pezzo pair of degree two is a pair (Y, Z) consisting of a del Pezzo surface Y of degree two and a singular anti-canonical curve Z ⊂ Y . We denote the moduli space of del Pezzo pairs of degree two by DPP2. By adding a marking to the del Pezzo surface Y we obtain a marked del Pezzo pair (Y, Z, φ) with φ : Z1,7 → Pic(Y ) a marking of Y . The moduli space of marked del Pezzo pairs is denoted by DPP2.
The smooth points of an irreducible plane cubic admit a group law. For smooth cubics this is well known. A similar construction for the group law can be applied to singular irreducible cubics as follows. Let Z be a irreducible plane cubic curve and let O be an inflection point of Z. The map is a bijection and defines a group law on Z ns (C). For a nodal cubic it is well known that there is an isomorphism of groups: Pic 0 (Z) ∼ = C * . It is unique up to multiplication by an element of Aut(C * ) ∼ = {±1}. Similarly for a cuspidal cubic we have an isomorphism Pic 0 (Z) ∼ = C that is unique up to multiplication by an element of Aut(C) ∼ = C * . A useful property of the group law is the following.
Irreducible cubic with a node and B ⊂ Z ns .

II
Irreducible cubic with a cusp and B ⊂ Z ns .
Irreducible cubic with a node that coincides with a blown up point or reducible cubic consisting of a conic and a line intersecting in two points. III Irreducible cubic with a cusp that coincides with a blown up point or reducible cubic consisting of a conic and a tangent line.   Suppose that (Y, Z, φ) is a marked del Pezzo pair and π : Y → P 2 is the corresponding blowup map. The image π(Z) is a plane cubic through the seven points B = {P1, . . . , P7}.
In Table 2 we distinguish four cases according to the type (nodal or cuspidal) of ZB and the location of the points B. We will use the symbols used in Kodaira's classification of the singular fibers of an elliptic pencil to denote the type of the curve Z.
These four types of Z ⊂ Y each define a stratum in the moduli space of del Pezzo pairs. The stratum of type I1 where ZB is an irreducible nodal cubic and B ⊂ Z ns B is generic and defines an open subset DPP • 2 ⊂ DPP2. For now we assume that Z is of type I1 and we identify Z with ZB so that we can make use of the group law on the singular cubic ZB. By composing the marking φ : Z1,7 → Pic(Y ) with the restriction homomorphism Pic(Y ) → Pic(Z) we obtain a map that assigns ei → is an inflection point of Z. Restricting this map to the root lattice Q < Z1,7 induces a homomorphism χ ∈ Hom(Q, Pic 0 (Z)) characterized by the relations Proof. From the construction of χ and Proposition 4.2 we see that . . ,Pi, . . . , P7 are conconic so that the condition that the points are in general position is equivalent to χ(α) = 0 for all roots α ∈ R.
After fixing an isomorphism Pic 0 (Z) ∼ = C * we can identify the space with the complex torus T = Hom (Q, C * ). This identification is not canonical but is unique up to multiplication by an element of Aut(C * ) ∼ = {±1} which acts on T. The Weyl group W of type E7 acts on T by its natural action on Q and we denote the complement of the toric mirrors for this action by T • .
Theorem 4.4 (Looijenga). Let (Y, Z, φ) be a marked del Pezzo pair of degree two with Z a nodal anti-canonical curve. The association extends to an isomorphism of orbifolds The left hand side is the open stratum of the moduli space of marked del Pezzo pairs of degree two with Z of type I1. Similarly we have an isomorphism of orbifolds We construct an inverse to the map of Equation 10 by constructing seven points on a fixed nodal cubic Z. Fix an isomorphism C * → Z ns by choosing one of the three inflection points O on Z as a unit element. The group law then satisfies titjt k = 1 if and only if the corresponding points Pi, Pj, P k on Z ns are colinear. Since the seven points should satisfy (9) they must also satisfy the equality where we consider χ as an element of Hom(Q ⊗ Z C, C * ). This determines the seven points uniquely up to addition of an inflection point of Z ns (or equivalently multiplication by a third root of unity of C * ). Blowing up these seven points determines a marked del Pezzo surface Y and the pullback of Z under the blowup map defines a nodal anti-canonical curve on Y isomorphic to Z.
To conclude this section we obtain explicit descriptions of the standard triangular Cremona transformation centered in three points on an irreducible plane nodal cubic ZB in terms of the coordinate t ∈ C * ∼ = Z ns (C). The Cremona map ρ(s7) centered in the points P1, P2, P3 of Z with coordinate t maps Z to another nodal cubic Z which can be mapped back to Z with new coordinate t by an element of PGL(3, C). If ti, tj, t k , t1, t2, t3 ∈ Z are distinct points lying on a conic, then t i , t j , t k lie on a line by the properties of the standard triangular Cremona transformation so that Similarly, the standard triangular Cremona transformation maps the line L12 to t 3 , so that for a point ti on L12: From these formulas we compute which determines t up to multiplication by a third root of unity. These formulas can also be derived by computing the action of s7 ∈ W on Equation 11.

Strata of smooth pointed quartic curves
We have seen in Section 3.3 that the moduli space DP2 of del Pezzo surfaces of degree two and the moduli space Q of plane quartic curves are isomorphic. In this section we relate the moduli space DPP2 of del Pezzo pairs of degree two and its strata to the moduli space of smooth pointed plane quartics Q1. We first define this latter space Q1.
Definition 5.1. Let k be the field of real or complex numbers. A pointed plane quartic curve is a pair (C, p) with C a plane quartic curve and p ∈ C(k). The space Γ of smooth pointed quartics curve is defined by The group PGL(3, k) acts on Γ and the quotient represents isomorphism classes of smooth pointed plane quartics.
To a pointed quartic (C, p) we can associate a del Pezzo pair (Y, Z) in the following way. The del Pezzo surface Y of degree two is defined by in weighted projective space. The morphism defined by the anti-canonical map is realized by the projection map ψ : Y → P 2 given by: Every anti-canonical curve on Y is the pullback under ψ of a line in P 2 . We define Z = ψ −1 TpC to be the pullback of the tangent line to C at p. It is a singular anti-canonical curve on Y of arithmetic genus 1. Its Kodaira type is determined by the type of the intersection divisor D = (C · TpC) defined below.
be a divisor on a curve C with the pi distinct and ordered in such a way that d1 ≥ . . . ≥ dr. The type of D is the r-tuple d = (d1, . . . , dr).
There are four possibilities for the type of D corresponding to the types for Z in Table  2. Similarly we obtain four strata in the space Γ. The strata Γ bit and Γ flex where the point Table 3: Strata in the space of pointed quartics p is respectively a bitangent and an inflection point have codimension one and the stratum Γ hflex where p is a hyperflex has codimension two in the space Γ.
6 Moduli of real del Pezzo pairs of degree two Let (C, p) be a smooth real pointed plane quartic curve. By the results of the previous section and Section 3.4 we can associate to (C, p) a real del Pezzo pair (Y, Z) with real form χ − Y such that Y − (R) in nonorientable. The real form restricts to Z which is a real curve of arithmetic genus 1 on Y and Z ns (R) = ∅. If (C, p) is in the open stratum Γ • then the tangent line TpC intersects C in two other distinct points which can both be real or a pair of complex conjugate points. In both cases the curve Z is of type I1 (it has a single node).
Since Z ns (R) = ∅ there are two possibilities for the real form induced by χ − Y on Z ns (C) ∼ = C * . Either it maps: t →t and Z ns (R) ∼ = R * or t →t −1 and Z ns (R) ∼ = S 1 . An example of both is given in Figure 5.
where T • denotes the complement in T = Hom(Q, C * ) of the mirrors of the action of the Weyl group W of type E7.
Proof. A lot of work has already been done in the proof of Theorem 4.4. First we need to show that the element χ : Q → C * we associate to (C, p) is a real point of W \T • . By definition this means that w · χ =χ for some element w ∈ W . The real structure χ − Y of Equation 3.4 acts on Pic 0 (Y ) ∼ = Q as an involution u ∈ W . Since Y − (R) is nonorientable we see from Table 1 that this involution is of type 1, A1, A 2 1 , A 3 1 or D4. The action of the restriction of χ − Y to Pic 0 (Y ) ∼ = C * is one of t →t ±1 so the element χ satisfies u · χ =χ ±1 . Since the Weyl group W of type E7 contains −1 we can rewrite this as ±u · χ =χ so that χ is indeed a real element of W \T • .
Conversely, let χ be a real point of (W \T • )(R). By Proposition 7.6 we can assume that u·χ =χ ±1 with u ∈ W an involution of type 1, A1, A 2 1 , A 3 1 or D4. As in the proof of Theorem 4.4 we fix a real nodal cubic Z in P 2 and an isomorphism Z ns (C) ∼ = C * by choosing a real inflection point. The real form of Z is then equivalent to one of t →t ±1 . As in Equation 4 the element χ determines seven points in C * by the formula ti = χ(ei − e 0 3 ) which we interpret as points on Z ns (C). Since χ is real these points satisfy where the involution u ∈ W acts by the Cremona action of the Weyl group as a birational involution of P 2 . This involution lifts to an anti-holomorphic involution of the del Pezzo surface Y obtained by blowing up the seven points. These two construction are inverse to each other.

Reflection groups and real tori
In this section we study the connected components of the space (W \T • )(R) where T is the complex torus T = Hom(Q, C * ) for Q a root lattice of type ADE. For type E7 this space has 20 connected components which we describe explicitly as quotients of real subtori of T.

Reflection groups and root systems
We start by recalling some facts about reflection groups and root systems. Our main reference is [1] Chapter VI. Let V be a real, finite dimensional vector space of dimension n with an inner product (·, ·). For every nonzero α ∈ V we define the reflection sα ∈ O(V ) by The mirror Hα is defined as the fixed point locus of the reflection sα. A root system R ⊂ V is a finite set of nonzero vectors called roots that satisfy the following properties R4 2 (β,α) (α,α) ∈ Z for all α, β ∈ R. A system of simple roots ∆ = {α1, . . . , αr} ⊆ R is a basis for V such that every root is an integral linear combination r i=1 ciαi of simple roots of the same sign. It is known that such a simple system always exists. From now on we assume we have fixed a system of simple roots ∆ ⊂ R. For every root α ∈ R we define the coroot α ∨ by The set of coroots R ∨ is again a root system (the coroot system) with corresponding coroot lattice Q ∨ = ZR ∨ . A root system R is called irreducible if it is non-empty and cannot be decomposed as an orthogonal direct sum R = R1 ⊕ R2 of two non-empty root systems R1 and R2. Let R be an irreducible root system. We define the highest rootα of R with respect to ∆ as the unique root such that r i=1 ci is maximal. We also define α0 = −α. The Weyl group W is the group generated by the reflections sα with α ∈ R or equivalently by the simple reflection sα with α ∈ ∆. It is a finite group and acts simply transitively on the connected components of V \ ∪Hα which are called chambers. The fundamental chamber C with respect to a given system of simple roots ∆ is defined by Its closureC is a fundamental domain for the action of W on V .
The affine Weyl group Wa is the group generated by the affine reflections s α,k with α ∈ R and k ∈ Z defined by The mirror of s α,k is the affine hyperplane H α,k = {x ∈ V ; (α, x) = k}. The affine Weyl group Wa is the semidirect product of W by the coroot lattice: Wa = Q ∨ W . This allows us to write s α,k = t(kα ∨ )sα where t(kα ∨ ) denotes translation over kα ∨ in V . The group Wa acts simply transitively on the connected components of the space V • = V \ ∪H α,k which are called alcoves. The fundamental alcove A is the simplex given by We want to determine the stabilizer StabW a (x) of an x ∈ V in the affine Weyl group. Since all points in the orbit Wa · x have conjugate stabilizers, we can assume that x ∈Ā. The stabilizer StabW a (x) is the group generated by the reflections in the mirrors H α,k that contain x. It is a Weyl group with root system R(x) and system of simple roots ∆(x) given by These root systems can be reducible, even if the root system R is irreducible.

The extended affine Weyl group
The coweight lattice P ∨ is defined by and contains Q ∨ as a subgroup of finite index. It has a basis { ∨ 1 , . . . , ∨ r } dual to the basis of simple roots of R, so that (αi, ∨ j ) = δij. The extended affine Weyl group W a is defined as the semidirect prodct P ∨ W with P ∨ acting on V by translations. We will prove that W a is the extension of Wa by a finite subgroup of the automorphisms of the fundamental alcove.
Let ni = (α, ∨ i ) be the coefficient of αi in the highest rootα. For notation it is convenient to define ∨ 0 = 0 ∈ P ∨ and n0 = 1. The fundamental alcove A is the open n-simplex with vertices { ∨ i /ni} r i=0 . Let J be the set of indices 0 ≤ i ≤ r such that ni = 1. The vertices ∨ i with i ∈ J or equivalently: R( ∨ i /ni) ∼ = R are called special. Put R0 = R and let w0 be the longest element of W with respect to the basis of simple roots in equation {αi} r i=1 . We also define for every i ∈ J \ {0} the root system Ri generated by the simple roots {α1, . . . ,αi, . . . , αr}.
Let wi be the longest element of the Weyl group W (Ri) with respect to the basis of simple roots (15). For every i ∈ J we now define the following element of the extended affine Weyl group γi = t( ∨ i )wiw0. Observe that γi(0) = ∨ i and γ0 = w 2 0 = 1. Proposition 6 from [1] §2 VI states that we have equality {w ∈ W a ; w(A) = A} = {γi}i∈J (16) and we can identify the group (16) with the finite Abelian group P ∨ /Q ∨ by assigning to γi the class of i mod Q ∨ where i ∈ J. We see that the group P ∨ /Q ∨ acts simply transitively on the special points. Since the affine Weyl group Wa acts simply transitively on the alcoves it follows from the above that we have an isomorphism by assigning (t(Z)w, γi) → t(Z + w ∨ i )wwiw0. The extended affine Weyl group acts transitively on the connected components of V • , but the action need not be free. The action of P ∨ /Q ∨ on the fundamental alcove A can have fixed points. Also W a is in general not a Coxeter group.
Proof. Let t(Z)w ∈ Wa and γ ∈ P ∨ /Q ∨ be such that t(Z)wγ(x) = x. Define y := γ(x) ∈Ā. Now t(Z)w(y) = x with x, y ∈Ā, and becauseĀ is a strict fundamental domain for the action of Wa we can conclude x = y, so γ(x) = x. This also implies that t(Z)w(x) = x.

The centralizer of an involution in a reflection group
In this section we recall some known results on centralizers of involutions in reflection groups. Let (W, S) be a finite Coxeter group and let u ∈ W be an involution. We want to determine the centralizer CW (u) of u in W . By the classification of involutions in Coxeter groups there is a subset I ⊆ S such that u is conjugate in W to the involution wI : the unique longest element −1 in the parabolic subgroup WI . Felder en Veselov in [7] observe the following. Proof. The element u is the unique longest element of WI , so that wuw −1 = u for all w ∈ NW (WI ) and consequently: NW (WI ) < CW (u). For the other inclusion let w ∈ CW (u). We need to prove that wsα i w −1 ∈ WI , or equivalently: w · αi ∈ R ∩ VI for all i ∈ I. This holds since the element w preserves the eigenspace decomposition of the involution u.
Using this result we can use the classification of normalizers of parabolic subgroups of reflection groups by Howlett [8]. We want to mention here that a lot of the results of this section also appear in [11]. Let u = wI and decompose V into ±1-eigenspaces for u: This defines two orthogonal root systems and corresponding Coxeter groups . Observe that W − u = WI and that the eigenspace V − u is spanned by the roots of R − u . The group Wu = W − u × W + u is generated by all reflections that commute with u and is contained in the centralizer CW (u) of u. This centralizer also contains a non-reflection part Gu which now describe. Consider the two Weyl elements where the sum runs over all positive roots of R ± u , which we denote by R ± u (+). The set is a root system which can be written as an orthogonal disjoint union of subroot systems R c u = R1 ∪ R2. These factors are isomorphic root systems and are exchanged by the involution u. The Weyl group W (R c u ) is a product whose factors are exchanged by conjugacy with u. The group Gu consists of all elements of W (R c u ) that commute with u and is the diagonal of this product This group is generated by pairs of commuting reflections sαsu·α with α ∈ R1 and is isomorphic to W (R1). Now we can formulate the main theorem of this section.
Theorem 7.3. The centralizer of an involution u ∈ W splits as a semidirect product where G + u is the reflection group defined by G + u = {w ∈ W ; wI = I} which contains W + u as a normal subgroup.

Root tori and their invariants
If the type of a root system R occurs in Equation 5 we say it is of del Pezzo type. Such root systems are products of root systems of type ADE so that all roots have the same length. This implies that we can identify the root (resp weight) and the coroot (resp coweight) lattices R and R ∨ (resp P and P ∨ ). To simplify notation we use this identification from now on. To a root system R of del Pezzo type we associate the complex torus T = C * ⊗ P = Hom(Q, C * ). It has a natural action of the Weyl group W .
Theorem 7.4. The quotient W \T is an affine toric variety and the algebra of W -invariants of Z[Q] is the semi-group algebra given by where This theorem is a generalisation of classical exponential invariant theory for root systems as described in [1] VI §3. The main theorem of that section states that the algebra of Winvariants of Z[P ] is a polynomial algebra This is a toric analogue of a well known theorem of Chevalley. The algebra C[P ] is the coordinate ring of the algebraic torus T = C * ⊗Q. Since the W -invariants form a polynomial algebra we can rephrase the theorem as W \T ∼ = C n . The torus T is a finite cover of T where the group of deck transformations is isomorphic to P/Q and there is an isomorphism of orbifolds W \T ∼ = (P/Q)\C n .
The action of W on T on the complement of the mirrors T • is not free in general: the group P/Q can have fixed points in T • . The stabilizers are described by the following lemma. Proof. Consider the exponential sequence where exp : z → e 2πiz . By tensoring from the right with P we obtain another exact sequence From the sequence we read off that W \T ∼ = W a \V C where the extended affine group W a = P W acts on V C by the formula for Z ∈ Q and w ∈ W . Write z = log t, by Lemma 7.1 the group StabW (t) is isomorphic to The group StabW a (z) is a Weyl group generated by the reflection in the mirrors that contain z.

Real root tori and their connected components
Complex conjugacy on C * defines a real form on the complex torus T = C * ⊗ P . This in turn defines a real form on the quotient W \T. Let q : T → W \T be the quotient map. The real points of W \T are the points q(t) such that t andt are in the same W -orbit so that We will prove in Proposition 7.6 that we can assume that w is an involution in W . Every involution in W defines a real form on T by composing with complex conjugation. The real points of such a real form are given by The following proposition is a slight modification of a result due to Tits ( [11], Proposition 2.2) to the present situation. The proof is similar to the one given there. Proposition 7.6 (Tits).

Tu(R)
Proof. Let t ∈ T be such that w · t =t for some w ∈ W . We will prove that there is a w in the reflection part Stab r W (t) of the stabilizer StabW (t) such that u = ww is an involution in StabW (t). The reflection part of the stabilizer is a finite reflection group which acts on the tangent space TtT through its complexified reflection representation. Since StabW (t) is also the stabilizer oft we see that This implies that w permutes the chambers of Stab r W (t) so that we can find a w ∈ Stab r W (t) such that u = ww leaves a chamber invariant. Since Stab r W (t) acts simply transitively on its chambers it follows that u 2 = 1.
The group W permutes the real tori Tu according to w · Tu = T wuw −1 so W -equivalent real tori correspond to conjugate involutions. Furthermore the stabilizer of a real torus Tu(R) in W is precisely the centralizer CW (u) of u in W . We want to study the real tori Tu and especially their connected components in more detail. The involution u acts naturally on the weight lattice P and there exists a so called normal basis for P such that This is described in detail in [3]. The decomposition of Equation 19 is not unique but the triple (n1, n2, n3) which we call the type of the involution u ∈ W is an invariant of the involution. A choice of normal basis determines an isomorphism with n1, n2, n3 ∈ N and n1 + n2 + 2n3 = n. This product consists of factors of split (R * ), compact (S 1 ) and complex (C * ) type. To determine the numbers ni we have the following lemma from [3].
Lemma 7.7. There are isomorphisms of abelian groups where the first of these can be identified with the component group π0(Tu(R)).
Proof. We construct the first of these isomorphisms. After choosing a normal basis for P we can use the lattice decomposition (19) to see that the lattice ker(u − 1) is isomorphic to P1,u ⊕ P3,u. Similarly the lattice im(u + 1) is isomorphic to 2P1,u ⊕ P3,u and the quotient of these lattices is P1,u/2P1,u.
To determine the number of connected components of CW (u)\Tu(R) we need to compute the number of orbits under the action of CW (u) on the connected components of Tu(R).
The following lemma shows that in fact only W + u acts non-trivially on the components. Lemma 7.8. The group W − u Gu is contained in the kernel of the action of CW (u) on ker(u − 1)/ im(u + 1).

Connected components of W \T 1
For the trivial involution u = 1 and the real torus T1 we have The decomposition of this real torus into connected components is described by We can use the basis of fundamental weights { 1, . . . , n} of P to identify T with (C * ) n through the isomorphism ti ⊗ i → (t1, . . . , tn).
In this way we can also identify the component group P/2P of T1 with the subgroup {−1, 1} n ⊂ (R * ) n .
Remark 7.9. An element of P/2P can be represented by a colouring of the Coxeter diagram of W where the ith node is coloured white if the corresponding coefficient of i is 1 and coloured black if it is −1. To determine the action of W on two-coloured Coxeter diagrams first observe that a simple reflection for W acts on the fundamental weights as where the sum runs over the set Ij of neighbouring vertices of the jth vertex of the Coxeter diagram. Now the generator si only acts nontrivially if the ith node vi is black. In this case the action of si changes the colour of all neighbouring vertices of vi but leaves vi unchanged.
It is often convenient to use the group 1 2 P/P for representing the connected components of Tu(R) instead of the group P/2P . The reason for this is that there are bijections of orbit spaces We can count the points in the intersection 1 2 P ∩Ā and the group P/Q is typically small. Its action is easily determined for Weyl groups of type ADE. We do this for root systems of type An in Example 7.10.
Proof. For a root system of type An all the roots have coefficient 1 in the highest root so the fundamental alcove is the convex hull of the fundamental weights. From this we determinē The group P/Q is cyclic of order n + 1 and is generated by γ1 which acts as the permutation (01 . . . n) on the indices of the fundamental weights { i}. A small calculation shows that where we use the notation 0 = 0 and the indices are considered mod n + 1. A typical γ1-orbit (for which n + 1 = 2i) is of the form A representative is given by i/2. If n is even then all orbits are of this form and there are n/2 orbits. If n is odd then there is one additional orbit with n + 1 = 2i given by A representative is given by (n+1)/2 .
For non-trivial involutions u it is more complicated to determine the connected components of CW (u)\Tu(R).
After choosing a normal basis for P in which u takes a normal form we can identify the component group π0(Tu(R)) with P1,u/2P1,u. However since there is no canonical choice for P1,u we have to compute W + u \ (P1,u/2P1,u) case by case. The real torus Tu(R) can be written as the disjoint union of its connected components in the following way (23) 7.7 Connected components of real tori of type E 7 In this section we determine all connected components of the space where u ∈ W is an involution in the Weyl group W of type E7 and Tu is the corresponding real torus. The results are listed in  If u is an involution with n1 = 0 then the real torus Tu(R) is connected. Consequently the quotient CW (u)\Tu(R) is also connected. Now suppose that n1 = 1 so that P1,u/2P1,u ∼ = Z/2Z. Since {0} is a single W + u -orbit, so is { } with ∈ P a generator for P1,u/2P1,u and there are two connected components. From Table 4 we see that this is the case for A 3 1 and D4 which we represent by I = {s2, s4, s7} and I = {s2, s3, s4, s7} respectively. In both cases the fundamental weight 6 is a generator for P1,u/2P1,u. For n1 > 2 the situation becomes more complicated and we have to determine the action of W + u on the generators of P1,u/2P1,u. These cases are u = 1, A1, A 2 1 or A 3 1 and we treat them below. 1 The involution u = 1 is of type (7, 0, 0) so that we can use Formula 22. The closure of the fundamental alcove A intersected with lattice of half weights consists of the six elements {0, 1/2, 5/2, 6, 6/2, 7/2}. The group P/Q is of order two and acts on this set by γ6 which interchanges 0 ↔ 6 and 1/2 ↔ 5/2. We conclude that there are four orbits in W \ (P/2P ) represented by {0, 5, 6, 7}.
u ∈ W n 1 n 2 n 3 #components representatives Table 4: The number of components of C W (u)\T u (R) for all conjugacy classes of involutions u ∈ W . We also list the corresponding representatives in W + u \ (P 1,u /2P 1,u ).

A1
The involution A1 is of type (5, 0, 1). As a representative we pick I = {s1}. Let Su be the matrix of u with respect to the basis of fundamental weights for P and let Bu be a matrix whose columns represent a normal basis in the sense of Equation 19. The normal basis for P is not uniquely determined but we fix the choice below.
A basis for P1,u and system of simple roots for W + u are then given by respectively We need to determine the action of W + u on P1,u. First we observe that the parabolic subgroup W (A5) of W + u generated by the reflections represented by grey nodes in the diagram is of type A5 and acts on P1,u in the usual way. We see from example 7.10 that there are four orbits for W (A5)\ (P1,u/2P1,u) represented by {0, 7, 3, 4}. The remaining generating reflection sI acts on the basis for P1,u as sα I ( 7, 3, 4, 5, 6) = ( 7 + 5, 3 + 2 5, 4 + 5, 5, 6).
The involution A 2 1 is of type (3, 0, 2). As a representative we choose I = {s1, s6}. As a basis for the lattice P1,u we can choose Z{ 3, 4, 7} which is of type A3. The group W + u is of type D4A1 with simple system The corresponding diagram is given below. The parabolic subgroup W (A3) of W + u of type A3 generated by the reflection represented by the grey nodes of the diagram acts on P1,u in the usual way. We can represent the orbits of W (A3)\ (P1,u/2P1,u) by {0, 3, 4}. The reflection sα II acts trivially on these orbits. The reflection sα I acts as sα I ( 7, 3, 4) = ( 7 + 3, 3, 4).

The complement of the mirrors
In this section we prove that for a root system of type ADE satisfying certain assumptions the connected components of the space (W \T • )(R) are of the form This implies that removing the mirrors from Tu(R) does not add new components to the quotient CW (u)\Tu(R). In particular the number of connected components of CW (u)\T • u (R) for involutions u in W (E7) are the same as the numbers in Table 4.
Definition 7.11. Let q : T → W \T be the quotient map. The discriminant D T is the set of critical values of q. It consists of union of the q-images of the toric mirrors and the q-image of the set where we denote by V γ i C the fixed points in V C of the generator γi for P/Q. Lemma 7.12. The q-images of the real tori Tu(R) are disjoint in Proof. Suppose that t ∈ Tu 1 ∩ Tu 2 for involutions u1, u2 ∈ W . This implies that u1 · t = u2 · t =t so that in particular u1u2 · t = t. But then q(t) ∈ D T (R).
Since we are interested in connected components it suffices to consider the part of D T (R) of codimension one in (W \T)(R). This motivates the following definition.
Definition 7.13. The real discriminant D T,R of (W \T)(R) is the closure of the nonsingular part of D T (R). The difference is a D T (R) − D T,R has codimension ≥ 2 in (W \T)(R).
Proposition 7.14. If we assume that T P/Q ∩ Tu has codimension ≥ 2 for all involutions u ∈ W then q −1 D T,R = (Tu ∩ Hs) where the union runs over all involutions u ∈ W and reflections s ∈ W that commute with u.
Proof. Under the assumption of the proposition the set T P/Q does not contribute to the real discriminant. In this case an element t ∈ Tu is mapped to a nonsingular point of D T (R) by q if and only if there is a unique reflection s ∈ W that fixes t. Since the reflection usu also fixes t, we must have that s commutes with u.
The assumption is satisfied for type E7. In that case P/Q is generated by the involution γ6. The locus of fixed points V γ 6 has dimension four so that the codimension of T P/Q ∩Tu ≥ 3. In order to prove that the space of equation (24) is connected it is sufficient to prove that the space is connected. We prove the slightly stronger result that the quotient by the smaller group StabW u (T u ) is connected. We start with a lemma on the decomposition V = V + u ⊕ V − u into ±1-eigenspaces for u on the weight lattice P . Denote by P + u and P − u the orthogonal projections of P into V + u and V − u respectively.
Lemma 7.15. The lattice P − u is equal to the weight lattice P (W − u ) of W − u . If we also assume that −1 ∈ W then P + u = P (W + u ) and R + u spans V + u .
Proof. The simple system {αi}i∈I for the root system R − u is a basis for V − u . The dual basis is given by If −1 ∈ W then for every involution u ∈ W its opposite −u is also an involution in W . Furthermore V ± u = V ∓ −u and W ± u = W ∓ −u so that we have equalities .
Proof. Similar to the decomposition where [ ] ∈ P1,u/2P1,u (so that in particular ∈ V + u ). The stabilizer of T u in Wu also splits into a product StabW u (T u ) ∼ = Stab W + u exp The result now follows from applying Lemma 7.1 to these factors and taking the quotient.
Remark 7.17. While the roots of the Weyl group W + u span the vector space V + u the same need not be true for the Weyl group Stab W + u ( 1 2 ), even if the group W contains −1. An example is given by the Weyl group of type E7 and the trivial involution u = 1. In that case the Weyl group StabW ( 1 2 6) is of type E6 and has rank six while V is of dimension seven. A fundamental domain for this action is the product of a Weyl chamber of type E6 and its orthogonal complement in V which is a copy of R. From this discussion we see that the chamber C + u is not a Weyl chamber in the traditional sense but the product of a Weyl chamber and an affine space. In particular it is connected, as is the alcove A − u , which implies the following corollary.

The Geometry of the 20 components
In this section we complete the correspondence between the 20 connected components of the space (W \T • )(R) for a root system of type E7 and the components of the moduli space (Q • 1 ) R . For each of these components we find a representative pair (C, p) with p ∈ C(R). The results are listed in the tables of Section 9.
Theorem 8.1. The curves in the tables of Section 9 represent the 20 different components of (Q • 1 ) R .
Proof. It is clear that for the curves in the left columns the associated del Pezzo pair (Y, Z) satisfies Z ns (R) ∼ = R * so that they belong to the space StabW (Tu)\T • u for u of type 1, A1, A 2 1 , A 3 1 or D4. Similarly the curves in the right column satisfy Z ns (R) ∼ = S 1 and belong to StabW (T−u)\T • −u (so −u is of type E7, D6, D4A1, A 4 1 or A 3 1 ). Just check from the pictures whether Z ns (R) has one or two components. With the exception of the two M -curves labeled 6 and 7 for all of the curves in the table the topological types of the pairs (C(R), TpC(R)) are distinct. It is not possible to deform one of them into the other without passing through one of the strata (Q flex 1 ) R , (Q bit 1 ) R or (Q hflex 1 ) R so that they lie indeed in different components of (Q • 1 ) R . We need to prove that the M -curves labeled 6 and 7 are not in the same component. For this consider the affine quartics obtained by placing the tangent line TpC at infinity for these two curves. They are shown in Figure  10. 6 7 Figure 10: The triangle forms an obstruction to deforming these two curves into each other by Bezout's Theorem.
The triangle drawn in the picture forms an obstruction to deforming one into the other: it is not possible to move the central oval of the curve 6 out of the triangle without contradicting Bezout's theorem (a line intersects C in four points). This is in agreement with Table 15 in Appendix 1 of [4] where certain affine M -quartics are classified.
To determine which of the curves in Section 9 corresponds to which component of (Q • 1 ) R it is convenient to have an alternative description of the components of (W \T • ) (R). For this we explicitly use the construction from the proof of Theorem 6.1 to associate to χ ∈ T u (R) seven points in general position on the nonsingular locus of an irreducible real plane nodal cubic Z ⊆ P 2 . As before we identify Z ns (C) ∼ = C * so that the points are defined by the formula χ → (P1, . . . , P7) with Pi = χ ei − e0 3 up to addition of an inflection point of Z. We start with the components corresponding to M -quartics.
If χ is an element of the compact torus T−1 ∼ = Hom(Q, S 1 ) then this construction determines seven points on the real point set Z(R) of a real plane nodal cubic with Z ns (R) ∼ = S 1 .
If χ is an element of the split torus T1 ∼ = Hom(Q, R * ) then the construction determines seven points on the real point set of a real plane nodal cubic with Z ns (R) ∼ = R * . If we choose the unique real inflection point of Z as the unit element for the group law on Z ns then these seven points are real. The Weyl group W acts on (R * ) 7 by permuting the coordinates and by triangular Cremona transformations in triples of points. For a seventuple t = (t1, . . . , t7) ∈ (R * ) 7 let m+ denote the number of positive coordinates and m− the number of negative coordinates. The permutation orbit of t is uniquely determined by the pair (m+, m−). From Formula 12 we see that if we perform a triangular Cremona transformation in ti, tj, t k the sign of these points remains unchanged and the remaining points change sign if and only if one or three of the three points are negative. This describes the action of W on the pairs (m+, m−) and there are four orbits. These correspond to the four components of where ∈ {0, 5, 6, 7}. The precise correspondence is shown in W (A 7 ) Z/2Z {(4, 3), (0, 7)} Table 5: Connected components for W \T 1 (R). The first column lists the representatives in W \ (P/2P ).
To determine which picture from the tables in Section 9 for u = 1 belongs to which of the components from Table 5 we determine the adjacency relations between the five components corresponding to pointed M -quartics in (Q • 1 ) R . Two components are adjacent if their corresponding pointed quartics (C, p) can be deformed into each other by moving through the stata (Q bit 1 ) R or (Q flex 1 ) R of codimension one. The effect of these two deformations is shown in Figure 11.  where we label components of W \T • 1 by a representative for the corresponding orbit W · (m+, m−) and the component W \T • −1 by (7).
This proves the proposition and shows that the pictures corresponding to the components are indeed the ones shown in the table for u = 1.
For χ ∈ Tu(R) with u of type A i 1 with i = 1, 2, 3 we can do a similar analysis. In this case the construction associates to χ: 7 − 2i real points and i pairs of complex conjugate points for a suitable representative u (not involving the reflection s7). For example the involution u = s6s4 of type A 2 1 acts as s6s4 · (P1, P2, P3, P4, P5, P6, P7) = (P1, P2, P3, P5, P4, P7, P6) on the Pi so that χ ∈ Ts 6 s 4 produces seven points in Z ns (C) ∼ = C * with P1, P2, P3 real points and (P4, P5) and (P6, P7) complex conjugate pairs. The centralizer CW (u) is more complicated in this case. It acts on the points by permutations preserving the real points and conjugate pairs and triangular Cremona transformations centered in a triples of real points or a real point and a pair of conjugate points. The orbits are calculated in Table 6 and confirm the numbers we computed earlier in Table 4. For χ ∈ Tu with u of type D4 or A 3 the situation is different. In this case u acts as a nonstandard Cremona transformation on the points. In fact it acts as a de Jonquiéres involution of order three centered in 5 of the points (for the definition we refer to [17]). The {(0, 1)} Table 6: Connected components of C W (u)\T u (R) for u of type A i 1 .
The second column shows the representatives for C W (u)\ (P 1,u /2P 1,u ) from Table 4.