Geometry of the Wiman–Edge pencil, I: algebrogeometric aspects
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Abstract
In 1981 William L. Edge discovered and studied a pencil \(\mathscr {C}\) of highly symmetric genus 6 projective curves with remarkable properties. Edge’s work was based on an 1895 paper of Anders Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel Geometry of the Wiman–Edge pencil, II: hyperbolic, conformal and modular aspects (in preparation), we consider \(\mathscr {C}\) from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.
Keywords
Wiman–Edge pencil Wiman curve Icosahedral symmetryMathematics Subject Classification
14H37 14H15 14J261 Introduction
The purpose of this paper is to explore what we call the Wiman–Edge pencil \(\mathscr {C}\), a pencil of highly symmetric genus 6 projective curves with remarkable properties. The smooth members of \(\mathscr {C}\) can be characterized (see Theorem 3.3) as those smooth genus 6 curves admitting a nontrivial action of alternating group \(\mathfrak {A}_5\). This mathematical gem was discovered in 1981 by William L. Edge [10, 11], who based his work on a discovery in 1895 by Anders Wiman of a nonhyperelliptic curve \(C_0\) of genus 6 (now called the Wiman curve) with automorphism group isomorphic to the symmetric group \(\mathfrak {S}_5\). The results of both Wiman and Edge are in the satisfying style of 19th century mathematics, with results in terms of explicit equations.
In this paper and its sequel [12], we consider \(\mathscr {C}\) from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects. In particular we will view \(\mathscr {C}\) through a variety of lenses: from algebraic geometry to representation theory to hyperbolic and conformal geometry. We will address both modular and arithmetic aspects of \(\mathscr {C}\), partly through Hodge theory. Given the richness and variety of structures supported by \(\mathscr {C}\), we can say in hindsight that the Wiman–Edge pencil deserved such a treatment, and indeed it seems odd that this had not happened previously.
In the introduction of his Lectures on the Icosahedron [18], Felix Klein writes: “A special difficulty, which presented itself in the execution of my plan, lay in the great variety of mathematical methods entering in the theory of the Icosahedron.” We believe that this is still very much true today, in ways Klein probably never anticipated. This, by the way, is followed by the sentence: “On this account it seemed advisable to take granted no specific knowledge in any direction, but rather to introduce, where necessary, such explanations \(\ldots \)” While we are writing only about one aspect of Klein’s book, in this paper we have tried to take Klein’s advice seriously.
1.1 The Wiman–Edge pencil
Edge showed that it is more natural to view the pencil as a pencil \(\mathscr {C}\) of curves on a quintic del Pezzo surface S obtained by blowing up the four base points of the pencil. In this picture, the lift \(C_0\) to S of Wiman’s curve W is, as Edge discovered, “a uniquely special canonical curve of genus 6” on S. That is, the standard action of \(\mathfrak {S}_5\) on the quintic del Pezzo surface S permutes the 1parameter family of smooth genus 6 curves on S but leaves invariant a unique such curve, namely the Wiman sextic \(C_0\).
1.2 This paper
The purpose of the present paper is twofold. First, we reprove all previously known facts about the Wiman–Edge pencil that can be found in Edge’s papers. Instead of computations based on the pencil’s equation (used also in a later paper [14] whose authors it seems were not aware of Edges’s work), our proofs rely on the representation theory of the symmetry group of the pencil, and also on the modulitheoretical interpretation of the quintic del Pezzo surface. Although our approach is not new, and has been used also in [4] or [16], respectively, we would like to believe that our methods are more conceptual and geometric.
The other goal of the paper is to answer some natural questions that arise while one gets familiar with the pencil. Thus we give a purely geometric proof of the uniqueness of the Wiman–Edge pencil as an \(\mathfrak {A}_5\)invariant family of stable curves of genus 6, and in particular, the uniqueness of the Wiman curve W as a nonhyperelliptic curve of genus 6 with the group of automorphisms isomorphic to \(\mathfrak {S}_5\). In fact, we give two different proofs of this result: an algebraic geometrical one given in Theorem 3.3 below, and a second one in the sequel [12] that is essentially grouptheoretical and topological.
Secondly, we give a lot of attention to the problem only barely mentioned by Edge and later addressed in the paper [20]: describe an \(\mathfrak {A}_5\)equivariant projection of the Wiman–Edge pencil to a Klein plane realizing an irreducible 2dimensional projective representation of \(\mathfrak {A}_5\). It reveals a natural relation between the Wiman–Edge pencil and the symmetry of the icosahedron along the lines of Klein’s book [18]. In particular, we relate the singular members of the pencil with some attributes of the geometry of the Clebsch diagonal cubic surface as well with some rational plane curves of degree 6 and 10 invariant with respect to a projective group of automorphisms isomorphic to \(\mathfrak {A}_5\) which were discovered by Roy Winger [26].
In the sequel [12] we will discuss some other aspects of the Wiman paper related to hyperbolic geometry, the moduli space of curves and Shimura curves.
1.3 Sectionbysection outline of this paper
Section 2 collects a number of mostly known facts regarding quintic del Pezzo surfaces, but with emphasis on naturality, so that it is straightforward to keep track of how the automorphism group of such a surface acts on the vector spaces associated to it. We also recall the incarnation of such a surface as the Deligne–Knudsen–Mumford compactification \(\overline{\mathscr {M}}_{0,5}\) and mention how some its features can be recognized in either description. Perhaps new is Lemma 2.1 and its use to obtain Proposition 2.3.
Section 3 introduces the principal object of this paper, the Wiman–Edge pencil \(\mathscr {C}\). Our main result here is that its smooth fibres define the universal family of genus 6 curves endowed with a faithful \(\mathfrak {A}_5\)action. We also determine the singular fibers of \(\mathscr {C}\). Each of these turns out to be a stable curve. Indeed, the whole pencil consists of all the stable genus 6 curves endowed with a faithful \(\mathfrak {A}_5\)action that can be smoothed as an \(\mathfrak {A}_5\)curve. We further prepare for the next two sections by describing \(\mathfrak {S}_5\)orbits in \(\overline{\mathscr {M}}_{0,5}\), thus recovering a list due to Coble. This simplifies considerably when we restrict to the \(\mathfrak {A}_5\)action and this what we will only need here.
The following two sections concern the projection to a Klein plane. These are \(\mathfrak {A}_5\)equivariant projections as mentioned above. Section 4 concentrates on the global properties of this projection, and proves among other things that the ramification curve of such a projection is in fact a singular member of the Wiman–Edge pencil. It is perhaps worthwhile to point out that we prove this using the Thom–Boardman formula for cusp singularities. This is the only instance we are aware of where such a formula for a second order Thom–Boardman symbol is used to prove an algebrogeometric property. We also show that the image of the Wiman–Edge pencil in a Klein plane under the projection is no longer a pencil, but a curve of degree 5 with two singular points.
As its title makes clear, Sect. 5 focusses on the images in the Klein plane of special members of the Wiman–Edge pencil. We thus find ourselves suddenly staring at a gallery of planar representations of degree 10 of genus 6 curves with \(\mathfrak {A}_5\)action, probably all known to our predecessors in the early 20th century if not earlier. Among them stand out (what we have called) the Klein decimic and the Winger decimic. Less exciting perhaps at first is the case of a conic with multiplicity 5; but this turns out to be the image of a member of the Wiman–Edge pencil which together with its \(\mathfrak {S}_5\)conjugate is characterized by possessing a pencil of even theta characteristics.
In the final section, Sect. 6, we look at the \(\mathfrak {S}_5\)orbit space of \(\overline{\mathscr {M}}_{0,5}\) (which is just the Hilbert–Mumford compactification of the space of binary quintics given up to projective equivalence) and make the connection with the associated invariant theory.
1.4 Conventions
Throughout this paper the base field is \(\mathbb {C}\) and S stands for a quintic del Pezzo surface (the definition is recalled in Sect. 2). The canonical line bundle \(\Omega _M^n\) of a complex nmanifold M will be often denoted by \(\omega _M\). As this is also the dualizing sheaf for M, we use the same notation if M is possibly singular, but has such a sheaf (we only use this for curves with nodes).
For a vector space V, let \(\mathbb {P}(V)\) denote the projective space of 1dimensional subspaces of V and \(\check{\mathbb {P}} (V)=\mathbb {P}(V^\vee )\) denotes the projective space of hyperplanes of V. We write Open image in new window for dth symmetric power of the vector space V. For a space or variety X we denote by Open image in new window the quotient of \(X^d\) by the permutation group \(\mathfrak {S}_d\).
2 Del Pezzo surfaces of degree 5
2.1 A brief review
Here we recall some known facts about quintic del Pezzo surfaces, i.e., del Pezzo surfaces of degree 5, that one can find in many sources (such as [8]). By definition a del Pezzo surface is a smooth projective algebraic surface S with ample anticanonical bundle \(\omega _S^{1}\). For such a surface S, the first Chern class map Open image in new window is an isomorphism. Denoting by Open image in new window the canonical class of S, i.e., the first Chern class of \(\omega _S\) (and hence minus the first Chern class of the tangent bundle of S), the selfintersection number \(d = (K_S)^2 = K_S^2\) is called the degree of S. With the exception of surfaces isomorphic to Open image in new window (which are del Pezzo surfaces of degree 8), a del Pezzo surface of degree d admits a birational morphism \(\pi :S\rightarrow \mathbb {P}^2\) whose inverse is the blowup of \(9d\) distinct points (so we always have \(d\leqslant 9\)) satisfying some genericity conditions. But beware that when \(d\leqslant 6\), there is more that one way to contract curves in S that produce a copy of \(\mathbb {P}^2\).
The image of a nonsingular rational curve on S with selfintersection number \(1\) (in other words, an exceptional curve of the first kind) is a line on the anticanonical model, and any such line is so obtained. Any line on the quintic del Pezzo surface S is either the strict transform of a line through \(p_i\) and Open image in new window , \(1\leqslant i<j\leqslant 4\), or the preimage of \(p_i\), \(i=1,\dots , 4\), so that there are \(\left( {\begin{array}{c}4\\ 2\end{array}}\right) +4=10\) lines on S.
The strict transform \(\mathscr {P}_i\) of the pencil of lines through \(p_i\) makes a pencil \(\mathscr {P}_i\) of rational curves on S whose members are pairwise disjoint: they are the fibers of a morphism from S to a projective line. The strict transform of the conics through \(p_1,\dots , p_4\) makes up a pencil \(\mathscr {P}_5\) with the same property. The intersection number of a member of \(\mathscr {P}_i\) with \(K_S\) is equal to the intersection number of a cubic with a line in \(\mathbb {P}^2\) minus 1 if \(i\ne 5\), resp. cubic with a conic minus 4 if \(i = 5\), hence is equal to 2. This is therefore also a conic in the anticanonical model, and that is why we refer to \(\mathscr {P}_i\) as a pencil of conics and call the corresponding morphism from S to a projective line a conic bundle.
Every pencil of conics has exactly three singular fibers; these are unions of two different lines. For example, the pencil \(\mathscr {P}_i, i\ne 5\), has singular members equal to the preimages on S of the lines Open image in new window joining the point \(p_i\) with the point Open image in new window . The singular members of pencil \(\mathscr {P}_5\) are proper transforms of the pairs of lines Open image in new window with all indices distinct.
The edges of the Petersen graph represent reducible conics on S, and there are indeed Open image in new window of them. The three reducible conics of a conic bundle are then represented by three disjoint edges whose indices are three 2element subsets of a subset of Open image in new window of cardinality 4. If we regard the missing item as a label for the conic pencil, we see that Open image in new window is realized as the full permutation group of the set of conic pencils.
2.2 The modular incarnation
A quintic del Pezzo surface has the—for us very useful—incarnation as the Deligne–Mumford moduli space of stable 5pointed rational curves, \(\overline{\mathscr {M}}_{0,5}\), see [17]. In the present case, this is also the Hilbert–Mumford (or GIT) compactification of \(\mathscr {M}_{0,5}\), which means that a point of \(\overline{\mathscr {M}}_{0,5}\) is uniquely represented up to automorphism by a smooth rational curve C and a 5tuple \((x_1,\dots , x_5)\in C^5\) for which the divisor Open image in new window has all its multiplicities \(\leqslant 2\). One of the advantages of either model is that the \(\mathfrak {S}_5\)action is evident. The 10 lines appear as the irreducible components of the boundary Open image in new window of the compactification: these are the loci defined by Open image in new window , where Open image in new window is a 2element subset of Open image in new window . We thus recover the bijection between the set of 10 lines and the collection of 2element subsets of Open image in new window .
The conic bundles also have a modular interpretation, namely as forgetful maps: if a point of \(\overline{\mathscr {M}}_{0,5}\) is represented by a Deligne–Mumford stable curve \((C; x_1, \dots , x_5)\), then forgetting \(x_i\), \(i=1,\dots , 5\), followed by contraction unstable components (and renumbering the points by Open image in new window in an order preserving manner) yields an element of \(\overline{\mathscr {M}}_{0,4}\). There is a similar definition in case a point of \(\overline{\mathscr {M}}_{0,5}\) is represented in a Hilbert–Mumford stable manner, but beware that a Hilbert–Mumford stable representative of \(\overline{\mathscr {M}}_{0,4}\) is given uniquely up to isomorphism by a 4tuple \((x_1, \dots , x_4)\in C^4\) for which the divisor Open image in new window is either reduced or twice a reduced divisor; if for example \(x_1=x_2\), then we also require that \(x_3=x_4\) (but \(x_2\not = x_3\)). The moduli space \(\overline{\mathscr {M}}_{0,4}\) is a smooth rational curve and Open image in new window consists of three points indexed by the three ways we can partition Open image in new window into two 2element subsets. The resulting morphism \(f_i:\overline{\mathscr {M}}_{0,5}\rightarrow \overline{\mathscr {M}}_{0,4}\) represents \(\mathscr {P}_i\) and over the three points of Open image in new window lie the three singular fibres.
Let us focus for a moment on \(\mathscr {P}_5\), in other words, on \(f_5:\overline{\mathscr {M}}_{0,5}\rightarrow \overline{\mathscr {M}}_{0,4}\). Each \(x_i\), \(i=1, \dots , 4\), then defines a section \(\overline{\mathscr {M}}_{0,4}\rightarrow \overline{\mathscr {M}}_{0,5}\) of \(f_5\), and \(f_5\) endowed with these four sections can be understood as the “universal stable 4pointed rational curve”. Note that the images of these sections are irreducible components of Open image in new window and hence lines in the anticanonical model. They are indexed by the unordered pairs Open image in new window , \(i=1,\dots , 4\). The singular fibers of \(f_5\) are those over the 3element set Open image in new window ; each such fiber is a union of two intersection lines, so that in this way all 10 lines are accounted for.
The \(\mathfrak {S}_5\)stabilizer of the conic bundle defined by \(f_5\) is clearly \(\mathfrak {S}_4\). Let us take a closer look at how \(\mathfrak {S}_4\) acts on \(f_5\). First observe that every smooth fiber of \(f_5\) (i.e., a strict transform of a smooth conic) meets every fiber of every \(f_i\), \(i\not =5\), with multiplicity one. We also note that \(\mathfrak {S}_4\) acts on the 3element set Open image in new window as its full permutation group, the kernel being the Klein Vierergruppe. So the Vierergruppe acts trivially on \(\overline{\mathscr {M}}_{0,4}\), but its action on the universal pointed rational curve \(\overline{\mathscr {M}}_{0,5}\) is of course faithful. The homomorphism \(\mathfrak {S}_4\rightarrow \mathfrak {S}_3\) can also be understood as follows. If we are given a smooth rational curve C endowed and a 4element subset \(X\subset C\), then the double cover \(\widetilde{C}\) of C ramified at X is a smooth genus 1 curve. If we enumerate the points of X by Open image in new window , then we can choose \(x_1\) as origin, so that \((\widetilde{C}, x_1)\) becomes an elliptic curve. This then makes Open image in new window the set of elements of \((\widetilde{C},x_1)\) of order 2. Thus \(\mathscr {M}_{0,4}\) can be regarded as the moduli space of elliptic curves endowed with a principal level 2 structure. This also suggests that we should think of X as an affine plane over \(\mathbb {F}_2\); for this interpretation the three boundary points of \(\overline{\mathscr {M}}_{0,4}\) are cusps, and correspond to the three directions in this plane. We now can think of \(\mathfrak {S}_4\) as the affine group of X, the Vierergruppe as its translation subgroup, and the quotient \(\mathfrak {S}_3\) as the projective linear group Open image in new window .
The character table of \(\mathfrak {S}_5\)
type  1  (12)  (12)(34)  (123)  (123)(45)  (1234)  (12345) 

\(\mathbf {1}\)  1  1  1  1  1  1  1 
\(\text {sgn}\)  1  \(\) 1  1  1  \(\) 1  \(\) 1  1 
V  4  2  0  1  \(\) 1  0  \(\) 1 
4  \(\) 2  0  1  1  0  \(\) 1  
W  5  1  1  \(\) 1  1  \(\) 1  0 
5  \(\) 1  1  \(\) 1  \(\) 1  1  0  
 6  0  \(\) 2  0  0  0  1 
The character table of \(\mathfrak {A}_5\)
type  (1)  (12)(34)  (123)  (12345)  (12354) 

\(\mathbf {1}\)  1  1  1  1  1 
V  4  0  1  \(\) 1  \(\) 1 
W  5  1  \(\) 1  0  0 
I  3  \(\) 1  0  \((1+\sqrt{5})/2\)  \((1\sqrt{5})/2\) 
\(I'\)  3  \(\) 1  0  \((1\sqrt{5})/2\)  \((1+\sqrt{5})/2\) 
We will be more concerned with the last orbit that gives rise to two special conics in the pencil. So every conic bundle has two special conics as fibers and \(\mathfrak {S}_5\) permutes these 10 special conics transitively. Two distinct special conics intersect unless they are in the same pencil. The two points \(0,\infty \) have the same \(\mathfrak {S}_3\)stabilizer in \(\mathbb {P}^1\) of order 3. In the action of \(\mathfrak {S}_4\) on the total space of the pencil, the two special conics are fixed by the subgroup \(\mathfrak {A}_4\). It follows from the preceding that these two fibers define the locus represented by \((x_1, \dots , x_5)\in C^5\) (with C a smooth rational curve) for which there exists an affine coordinate z on C such that Open image in new window is the union of \(\{\infty \}\) and the root set of Open image in new window . Our discussion also shows that the subgroup \(\mathfrak {A}_5\) has two orbits in that set, with each orbit having exactly one special conic in each conic bundle.
2.3 Representation spaces of \(\mathfrak {S}_5\)
In what follows we make repeated use of the representation theory of \(\mathfrak {S}_5\) and \(\mathfrak {A}_5\), and so let us agree on its notation. In Tables 1 and 2, the columns are the conjugacy classes of the group, indicated by the choice of a representative. We partially followed Fulton–Harris [13] for the notation of the types of the irreducible representations of \(\mathfrak {S}_5\). Here \(\mathbf {1}\) denotes the trivial representation, \(\text {sgn}\) denotes the sign representation, V denotes the Coxeter representation, that is, the standard 4dimensional irreducible representation of \(\mathfrak {S}_5\) and E stands for Open image in new window . Note that each of these representations is real and hence admits a nondegenerate \(\mathfrak {S}_5\)invariant quadratic form, making it selfdual.
Our labeling of the irreducible representations of \(\mathfrak {A}_5\) overlaps with that of \(\mathfrak {S}_5\), and this is deliberately so: the \(\mathfrak {S}_5\)representations V and Open image in new window become isomorphic when restricted to \(\mathfrak {A}_5\), but remain irreducible and that is why we still denote them by V. The same applies to W and Open image in new window (and of course to \(\mathbf {1}\) and \(\text {sgn}\)). On the other hand, the restriction to \(\mathfrak {A}_5\) of \(\mathfrak {S}_5\)representation Open image in new window is no longer irreducible, but is isomorphic as an \(\mathfrak {A}_5\)representation Open image in new window (cf. Table 2). The representations I and \(I'\) differ by the outer automorphism of \(\mathfrak {A}_5\) induced by conjugation with an element of Open image in new window . Both I and \(I'\) are realized as the group of isometries of Euclidean 3space that preserve an icosahedron. In particular, they are real (and hence orthogonal).
We will often use the fact that the natural map Open image in new window (given by conjugation) is an isomorphism of groups. So the outer automorphism group of \(\mathfrak {A}_5\) is of order 2 and representable by conjugation with an odd permutation.
The isomorphism Open image in new window depends of course on the model of S as a blowup \(\mathbb {P}^2\), but since Open image in new window permutes these models transitively, this isomorphism is unique up to an inner automorphism. This implies that the characters of Open image in new window , or equivalently, the isomorphism types of the finite dimensional irreducible representations of this group, are naturally identified with those of \(\mathfrak {S}_5\).
Another example is \(H^0(S,\omega _S^{1})\), a 6dimensional representation of \(\mathfrak {S}_5\). Using the explicit action of \(\mathfrak {S}_5\) on S and hence on the space of cubic polynomials representing elements of \(H^0(S,\omega _S^{1})\), we find that it has the same character as Open image in new window . This is why we shall write \(E_S\) for \(H^0(S,\omega _S^{1})\), so that Open image in new window and Open image in new window .
2.4 The Plücker embedding
We here show how S can be obtained in an intrinsic manner as a linear section of the Grassmannian of lines in projective 4space. The naturality will make this automatically Open image in new window equivariant. Let \(C_\infty \) denote the union of the 10 lines on S. It is clear that \(C_\infty \) is a normal crossing divisor in S (under the modular interpretation of \(S\cong \overline{\mathscr {M}}_{0,5}\) it is the Deligne–Mumford boundary).
Lemma 2.1
Proof
Remark 2.2
Proposition 2.3
Proof
A locally free sheaf \(\mathscr {E}\) of rank r on a compact variety X that is globally generated determines a morphism Open image in new window whose composite with the Plücker embedding Open image in new window is given by the invertible sheaf \(\det \mathscr {E}\). Applying this to our situation, we find that the composite of f with the Plücker embedding is given by the complete linear system \(K_S\). We know that it defines a closed embedding, from this the first claim follows.
To see the second claim, we compute the character of the representation of \(\mathfrak {S}_5\) in Open image in new window by means of the formula \(\chi _{{\scriptscriptstyle \bigwedge ^2}W}(g) = (\chi _W(g)^2\chi _W(g^2))/2\). The standard character theory gives us the decomposition (1). \(\square \)
Remark 2.4
The moduli space \(\overline{\mathscr {M}}_{0,6}\) of stable 6pointed genus zero curves is isomorphic to the blowup of \(\mathbb {P}^3\) at the five vertices of its coordinate simplex followed by the blowup of the proper transforms of the lines of this simplex (see [17]). There is a natural map from \(\overline{\mathscr {M}}_{0,6}\) to the Hilbert–Mumford compactification of \(\mathscr {M}_{0,6}\). The latter appears as the image an \(\overline{\mathscr {M}}_{0,6}\) in a 4dimensional projective space via the linear system of quadrics in \(\mathbb {P}^3\) through the five coordinate vertices so that this reproduces a copy of our map \(\Phi \) above. The image of this map is a cubic hypersurface, called the Segre cubic \(\mathscr {S}_3\).
The Segre cubic has 10 nodal points, each of which is the image of an exceptional divisor over a line of the coordinate simplex. Observe that the modular interpretation of the morphism \(\overline{\mathscr {M}}_{0,6}\rightarrow \mathscr {S}_3\) makes evident an action of \(\mathfrak {S}_6\), although in the above model, only its restriction to the subgroup \(\mathfrak {S}_5\) is manifest. The ambient 4dimensional projective space of the Segre cubic \(\mathscr {S}_3\) is the projectivization of an irreducible 5dimensional representation of \(\mathfrak {S}_6\) corresponding to the partition (3, 3) (see [8, p. 470]). \(\square \)
2.5 The anticanonical model
Some of what follows can be found in ShepherdBarron [23]; see also Mukai [19].
Corollary 2.5
The image of \(QQ'\) in \(H^0(S, \omega _S^{2})\), which spans a copy of the sign representation, has divisor \(C_\infty \).
Proof
Remark 2.6
It is a priori clear that \(\mathfrak {S}_5\) permutes the pentagons of lines listed above, but we see that it in fact preserves the 12element set and the preceding discussion makes explicit how.
3 The Wiman–Edge pencil and its modular interpretation
3.1 The Wiman–Edge pencil and its base locus
We found in Sect. 2.5 that there are exactly two \(\mathfrak {S}_5\)invariant quadrics on Open image in new window , one defined by \(Q+Q'\) and spanning the \(\mathbf {1}\)summand, the other by \(QQ'\) and spanning the \(\text {sgn}\)summand and (by Corollary 2.5) cutting out on S the 10line union \({C_\infty }\). The trivial summand spanned by \(Q+Q'\) cuts out a curve that we shall call the Wiman curve and denote by \({C_0}\). We shall find it to be smooth of genus 6. We observe that the plane spanned by Q and \(Q'\) is the fixed point set of \(\mathfrak {A}_5\) in Open image in new window and hence defines a pencil \(\mathscr {C}\) of curves on S whose members come with a (faithful) \(\mathfrak {A}_5\)action. This pencil is of course spanned by \(C_0\) and \(C_\infty \) and these are the only members that are in fact \(\mathfrak {S}_5\)invariant. We refer to \(\mathscr {C}\) as the Wiman–Edge pencil; we sometimes also use this term for its image in \(\mathbb {P}^2\) under the natural map \(\pi :S\rightarrow \mathbb {P}^2\).
Lemma 3.1
(Base locus) The base locus Open image in new window of \(\mathscr {C}\) is the unique 20element \(\mathfrak {S}_5\)orbit in \(C_\infty \). The curves \(C_0\) and \(C_\infty \) intersect transversally so that each member of the Wiman–Edge pencil is smooth at \(\Delta \).
Proof
Since this Open image in new window is \(\mathfrak {S}_5\)invariant, it suffices to determine how a line L on S meets \(C_0\). We first note that the intersection number Open image in new window (taken on S) is Open image in new window . When we regard L as an irreducible component of \(C_\infty \), or rather, as defining a vertex of the Petersen graph, then we see that the other lines meet L in three distinct points and that the \(\mathfrak {S}_5\)stabilizer of L acts on L as the full permutation group of these three points. So if we choose an affine coordinate z on L such that the three points in question are the third roots of 1, then we find that the \(\mathfrak {S}_5\)stabilizer of L acts on L with three irregular orbits: two of size 3 (the roots of \(z^31\) and the roots of \(z^3+1\)) and one of size 2 ( Open image in new window ). It follows that the \(C_0\) meets L in the size 2 orbit. In particular, the intersection of \(C_0\) with \(C_\infty \) is transversal and contained in the smooth part of \(C_\infty \). \(\square \)
3.2 Genus 6 curves with \(\mathfrak {A}_5\)symmetry
Proposition 3.2
(\(\mathfrak {A}_5\) and \(\mathfrak {S}_5\) orbit spaces) Let C be a smooth projective curve genus 6 endowed with a faithful action of \(\mathfrak {A}_5\). Then \(\mathfrak {A}_5\backslash C\) is of genus zero and \(\mathfrak {A}_5\) has four irregular orbits with isotropy orders 3, 2, 2 and 2.
If the \(\mathfrak {A}_5\)action extends to a faithful \(\mathfrak {S}_5\)action, then \(\mathfrak {S}_5\backslash C\) is of genus zero and \(\mathfrak {S}_5\) has three irregular orbits with isotropy orders 6, 4 and 2. The union of these irregular \(\mathfrak {S}_5\)orbits is also the union of the irregular \(\mathfrak {A}_5\)orbits: the \(\mathfrak {S}_5\)orbit with isotropy order 6, resp. 4, is an \(\mathfrak {A}_5\)orbit with isotropy order 3, resp. 2, and the \(\mathfrak {S}_5\)orbit with isotropy order 2 decomposes into two \(\mathfrak {A}_5\)orbits with the same isotropy groups. (In other words, the double cover \(\mathfrak {A}_5\backslash C\rightarrow \mathfrak {S}_5\backslash C\) only ramifies over the points of ramification of order 6 and 4.) Such a curve exists.
See also [4, Theorem 5.1.5].
Proof
For \(G=\mathfrak {A}_5\), we have \(G =60\) and hence we then also have Open image in new window . The formula now becomes Open image in new window , which has as only solution Open image in new window .
The assertion concerning the map \(\mathfrak {A}_5\backslash C\rightarrow \mathfrak {S}_5\backslash C\) formally follows from the above computations.
The last assertion will follow from the Riemann existence theorem, once we find a regular \(\mathfrak {S}_5\)covering of Open image in new window with the simple loops yielding monodromies \(\alpha ,\beta ,\gamma \in \mathfrak {S}_5\) of order 6, 4, 2 respectively, which generate \(\mathfrak {S}_5\) and for which \(\alpha \beta \gamma =1\). This can be arranged: take \(\alpha =(123)(45)\) and \(\beta =(1245)\) and \(\gamma =(14)(23)\). \(\square \)
We next show that any smooth projective curve genus 6 endowed with a faithful action of \(\mathfrak {A}_5\) appears in the Wiman–Edge pencil. For this we shall invoke a theorem of Mukai [19], which states that a canonical smooth projective curve C of genus 6 lies on a quintic del Pezzo surface if and only if it is neither bielliptic (i.e., it does not double cover a genus 1 curve), nor trigonal (it does not triple cover a genus zero curve), nor isomorphic to a plane quintic.
Theorem 3.3
Every smooth projective curve of genus 6 endowed with a faithful \(\mathfrak {A}_5\)action is \(\mathfrak {A}_5\)equivariantly isomorphic to a member of the Wiman–Edge pencil. This member is unique up to the natural action of the involution \(\mathfrak {S}_5/\mathfrak {A}_5\).
Proof
Let C be such a curve. We first show that C is not hyperelliptic, so that we have a canonical model. If it were, then it is so in a unique manner so that the set of its 14 Weierstraß points is in invariant with respect to \(\mathfrak {A}_5\). But we found in Proposition 3.2 that \(\mathfrak {A}_5\) has in C one irregular orbit of size 20, three of size 30, and no others, and thus such an invariant subset cannot exist.
From now on we assume that C is canonical. We first show that it is neither trigonal, nor bielliptic, nor isomorphic to a plane quintic.
C is trigonal: This means that C admits a base point free pencil of degree 3. This pencil is then unique [2, p. 209] so that the \(\mathfrak {A}_5\)action on C permutes the fibers. Consider the associated morphism \(C\rightarrow \mathbb {P}^1\). The Riemann–Hurwitz formula then tells us that the ramification divisor of this morphism on C has degree 16. It must be \(\mathfrak {A}_5\)invariant. But our list of orbit sizes precludes this possibility and so such a divisor cannot exist.
C is isomorphic to a plane quintic: It is then so in a unique manner [2, p. 209] and hence the \(\mathfrak {A}_5\)action on C will extend as a projective representation to the ambient \(\mathbb {P}^2\).
The resulting projective representation cannot be reducible, for then \(\mathfrak {A}_5\) has a fixed point, \(p\in \mathbb {P}^2\) say, and the action of \(\mathfrak {A}_5\) on the tangent space Open image in new window will be faithful. But as Table 2 shows, \(\mathfrak {A}_5\) has no faithful representation. So the projective representation is irreducible and hence the projectivization of copy of I or \(I'\). Either representation is orthogonal and so the ambient projective plane contains an \(\mathfrak {A}_5\)invariant conic. The quintic defines on this conic an effective divisor of degree 10. Since \(\mathfrak {A}_5\) acts on the conic (a Riemann sphere) as the group of motions of a regular icosahedron, no \(\mathfrak {A}_5\)orbit on this conic has fewer than 12 points and so such a divisor cannot exist.
C is bielliptic: This means that C comes with an involution \(\iota \) whose orbit space is of genus 1. Let Open image in new window be the subgroup generated by \(\mathfrak {A}_5\) and \(\iota \). By a theorem of Hurwitz, Open image in new window and so Open image in new window can be 1, 2, 4 or 6.
If the index \(\leqslant 4\), then the argument above gives a map \(G\rightarrow \mathfrak {S}_4\). Its kernel is contained in \(\mathfrak {A}_5\), but cannot be trivial for reasons of cardinality. It follows that the kernel equals \(\mathfrak {A}_5\), so that \(\mathfrak {A}_5\) is normal in G. Since the image of \(\iota \) generates \(G/\mathfrak {A}_5\), it then follows that the index is 1 or 2.
In the last case, G is either isomorphic to Open image in new window or to \(\mathfrak {S}_5\), depending on whether or not conjugation by \(\iota \) induces in \(\mathfrak {A}_5\) an inner automorphism. In the first case, the action of \(\mathfrak {A}_5\) descends to a faithful action on the elliptic curve. This would make \(\mathfrak {A}_5\) an extension of finite cyclic group by an abelian group, which is evidently not the case. It follows that \(G\subset \mathfrak {S}_5\) and that \(\iota \) is conjugate to (12) or to (12)(34).
Denote by \(\chi \) the character of the \(\mathfrak {S}_5\)representation \(H^0(C, \Omega _C)\). Since the quotient of C by \(\iota \) has genus 1, we have Open image in new window . If \(\iota \) is conjugate to (12), then we then read off from Table 1 that \(\chi \) is the character of Open image in new window or Open image in new window . Its restriction to \(\mathfrak {A}_5\) is then the trivial character resp. Open image in new window . But this contradicts the fact that the \(\mathfrak {A}_5\)orbit space of C has genus zero. If \(\iota \) is conjugate to (12)(34), then we then read off from Table 2 that the \(\mathfrak {A}_5\)representation \(H^0(C, \Omega _C)\) takes on \(\iota \) a value \(\geqslant 2\), which contradicts the fact that this value equals Open image in new window .
According to Mukai [19], it now follows that C lies on a weak quintic del Pezzo surface \(S_C\) in Open image in new window . It may have singular points, and in fact quadric sections of a weak del Pezzo quintic form a divisor D in the moduli space \({\mathscr {M}}_6\). However, we claim that C must lie on a smooth del Pezzo surface.
To prove this claim, first note that if C lies on a singular surface then it has fewer than five \(g_4^1\)’s, where by a \(g_4^1\) we mean a linear series of degree 4 and dimension 1. In the plane model this is because three points are collinear or two points coincide. The five \(g_4^1\)’s are defined by four pencils through nodes of the sextic and the pencil of conics through four nodes. In the singular case, when, we choose three collinear points, there is no pencil of conics. The divisor D in \({\mathscr {M}}_6\) mentioned above is characterized by the fact that it has at most four \(g_4^1\)’s. Now, \(\mathfrak {A}_5\) acts on these \(g_4^1\)’s, and hence leaves them all invariant. Thus it preserves a map \(C\rightarrow \mathbb {P}^1\) of degree 4. This is impossible since there are no invariant subset of ramification points. This proves the claim.
It is also known [23] that \(S_C\) is unique. This uniqueness property implies that the faithful \(\mathfrak {A}_5\)action on C, which extends naturally to Open image in new window , will leave \(S_C\) invariant. A choice of an \(\mathfrak {A}_5\)equivariant isomorphism \(h:S_C\xrightarrow {\scriptscriptstyle \cong \;} S\) will then identify C in an \(\mathfrak {A}_5\)equivariant manner with a member of the Wiman–Edge pencil. Any two \(\mathfrak {A}_5\)equivariant isomorphisms \(h, h':S_C\xrightarrow {\scriptscriptstyle \cong \;} S\) differ by an automorphism of S, so by an element \(g\in \mathfrak {S}_5\). But the \(\mathfrak {A}_5\)equivariance then amounts to g centralizing \(\mathfrak {A}_5\). This can happen only when g is the identity. So h is unique. \(\square \)
Let \(\mathscr {B}\) denote the base of the Wiman–Edge pencil (a copy of \(\mathbb {P}^1\)) so that we have projective flat morphism \(\mathscr {C}\rightarrow \mathscr {B}\). Recall that \(\mathfrak {S}_5\) acts on the family in such a manner that the action on \(\mathscr {C}\rightarrow \mathscr {B}\) is through an involution \(\iota \) which has two fixed points. We denote by Open image in new window the locus over which this morphism is smooth. So the restriction over \(\mathscr {B}^\circ \) is a family of smooth projective genus 6 curves endowed with a faithful \(\mathfrak {A}_5\)action. It has the following modular interpretation.
Theorem 3.4
Proof
Theorem 3.3 (and its proof) has an obvious extension to families of genus 6curves with \(\mathfrak {A}_5\)action. This yields the first assertion. If Open image in new window are such that \(C_t\) and \(C_{t'}\) are isomorphic as projective curves, then as we have seen, an isomorphism Open image in new window is induced by an element of \(\mathfrak {S}_5\) and so Open image in new window . \(\square \)
We will find in Sect. 3.3 that the singular members of the Wiman–Edge pencil are all stable. We found already one such curve, namely the union of the 10 lines, and so this element will map in \(\mathscr {M}_6\) to the boundary.
From now on we identify the base of Wiman–Edge pencil with \(\mathscr {B}\).
Corollary 3.5
The Wiman curve \(C_0\) is smooth and is \(\mathfrak {S}_5\)isomorphic to the curve found in Proposition 3.2. It defines the unique \(\iota \)fixed point of \(\mathscr {B}^\circ \).
Proof
By Theorem 3.4, there is a unique member of the Wiman–Edge pencil whose base point maps the unique point of \(\mathscr {B}^\circ \) which supports a smooth genus 6 curve with \(\mathfrak {S}_5\)action. The Wiman–Edge pencil has two members with \(\mathfrak {S}_5\)action, one is the union of the 10 lines and the other is the Wiman curve. So it must be the Wiman curve. \(\square \)
Corollary 3.6
Let C be a smooth projective curve genus 6 endowed with a faithful action of \(\mathfrak {A}_5\). If the resulting map Open image in new window is not surjective, then it extends to an isomorphism Open image in new window , and \(C=C_0\) is the Wiman curve. The \(\mathfrak {A}_5\)representation, resp. \(\mathfrak {S}_5\)representation, \(H^0(C, \omega _C)\) is equivalent to Open image in new window , resp. E, and \(H^1(C; \mathbb {C})\) is equivalent to Open image in new window , resp. \(E^{ \oplus 2}\).
Proof
Since the \(\mathfrak {A}_5\)curve C is represented by a member of the Wiman–Edge pencil, we can assume it is a member of that pencil. Since C is canonically embedded, an automorphism of C extends naturally of the ambient projective space and hence also to S (as it is the unique quintic del Pezzo surface containing C). This implies that Open image in new window . This inclusion is an equality precisely when C is the Wiman curve.
As for the last assertion, we know that the representation space \(H^0(C, \omega _C)\) is as asserted, since it is a member of the Wiman curve. The representations Open image in new window , resp. E, are selfdual, and since \(H^1(C, \mathbb {C})\) contains \(H^0(C, \omega _C)\) as an invariant subspace with quotient the (Serre)dual of \(H^0(C, \omega _C)\), the last assertion follows also. \(\square \)
Remark 3.7
If an \(\mathfrak {A}_5\)curve C represents a point \(z\in \mathscr {B}^\circ \), then \(\mathfrak {A}_5\) acts nontrivially (and hence faithfully) on \(H^0(C,\omega _C^{2})\) and hence also on its Serre dual \(H^1(C,\omega _C^{1})\). Since \(\mathfrak {A}_5\) is not a complex reflection group, a theorem of Chevalley implies that the orbit space \(\mathfrak {A}_5\backslash H^1(C,\omega _C^{1})\) must be singular at the image of the origin. The local deformation theory of curves tells us that when \(\mathfrak {A}_5\) is the full automorphism group of C, the germ of this orbit space at the origin is isomorphic to the germ of \(\mathscr {M}_6\) at the image of z. Hence \(\mathscr {B}^\circ \) maps to the singular locus of \(\mathscr {M}_6\). (This is a special case of a theorem of Rauch–Popp–Oort which states that any curve of genus \(g\geqslant 4\) with nontrivial group of automorphisms defines a singular point of \(\mathscr {M}_g\).)
3.3 Singular members of the Wiman–Edge pencil
The singular members of the Wiman–Edge pencil were found by Edge [11]. A modern proof of his result can be found in [4, Theorem 6.2.9]. Here we obtain them in a different manner as part of a slightly stronger result that we obtain with a minimum of computation.
Let us begin with an a priori characterization of the reducible genus 5 curves with \(\mathfrak {A}_5\)action that occur in the Wiman–Edge pencil.
Lemma 3.8
There are precisely two reducible members \(C_\mathrm{c}\) and \(C'_\mathrm{c}\) of \(\mathscr {C}\) distinct from \(C_\infty \); each is is a stable union of five special conics whose intersection graph is \(\mathfrak {A}_5\)equivariantly isomorphic to the full graph on a 5element set. Any element of Open image in new window exchanges \(C_\mathrm{c}\) and \(C'_\mathrm{c}\).
Proof
Let C be such a member of \(\mathscr {C}\) and let Y be an irreducible component of C. Then Y cannot be a line and so its degree Open image in new window in the anticanonial model must be \(\geqslant 2\). Since \(\mathfrak {A}_5\) has no subgroup of index \(< 5\), the number r of irreducible components in the \(\mathfrak {A}_5\)orbit of Y, must be \(\geqslant 5\). But we also must have Open image in new window , and so the only possibility is that \((d, r)=(2,5)\).
An irreducible, degree 2 curve in a projective space is necessarily a smooth conic. Its \(\mathfrak {A}_5\)stabilizer has index 5 in \(\mathfrak {A}_5\), and so must be \(\mathfrak {S}_5\)conjugate to \(\mathfrak {A}_4\). This implies that Y is a special conic. The five irreducible components of its \(\mathfrak {A}_5\)orbit lie in distinct conic bundles \(\mathscr {P}_i\), and we number them accordingly \(Y_1, \dots , Y_5\). For \(1\leqslant i<j\leqslant 5\), any smooth member of \(\mathscr {P}_i\) meets any smooth member of Open image in new window with multiplicity one, and so this is in particular true for \(Y_i\) and Open image in new window . Hence the set of singular points of C is covered in an \(\mathfrak {A}_5\)equivariant manner by the set of 2element subsets of Open image in new window . The action of \(\mathfrak {A}_5\) on this last set is transitive, and so either the singular set of C is a singleton \(\{p\}\) (a point common to all the irreducible components) or the intersections Open image in new window , \(1\leqslant i<j\leqslant 5\), are pairwise distinct. The first case is easily excluded, for p must then be a fixed point of the \(\mathfrak {A}_5\)action and there is no such point. So C is as described by the lemma.
The sum of the classes of the five conic bundles is \(\mathfrak {S}_5\)invariant and hence proportional to \(K_S\). The intersection product with \(K_S\) is \(5.2=10\) and hence this class is equal to \(2K_S\), in other words, the class of the Wiman–Edge pencil. Since an \(\mathfrak {A}_5\)orbit of a special conic takes precisely one member from every conic bundle, its follows that the sum of such an orbit indeed gives a member of \(\mathscr {C}\). There two such orbits and so we get two such members. \(\square \)
There are precisely two faithful projective representations of \(\mathfrak {A}_5\) on \(\mathbb {P}^1\) up to equivalence, and they only differ by precomposition with an automorphism of \(\mathfrak {A}_5\) that is not inner. We will refer to theses two representations as the Schwarzian representations of \(\mathfrak {A}_5\). Both appear as the symmetry groups of a regular icosahedron drawn on the Riemann sphere. This action has three irregular orbits, corresponding to the vertices, the barycenters of the faces and the midpoints of the edges, and so are resp. 12, 20 and 30 in size. Their points come in (antipodal) pairs and the \(\mathfrak {A}_5\)action preserves these pairs. We give a fuller discussion in Sect. 4.1.
Lemma 3.9
There exists an irreducible stable curve C of genus 6 with six nodes endowed with a faithful \(\mathfrak {A}_5\)action. Such a C is unique up to an automorphism of \(\mathfrak {A}_5\).
Proof
Let C be such a curve with six nodes and denote by \(D\subset C\) its singular set. Then the normalization of \(\widetilde{C}\rightarrow C\) is of genus zero: \(\widetilde{C}\cong \mathbb {P}^1\). If \(\widetilde{D}\subset \widetilde{C}\) denotes the preimage of D, then \(\widetilde{D}\) consists of 12 points that come in six pairs. The \(\mathfrak {A}_5\)action on \(\widetilde{C}\) lifts to \(\widetilde{C}\) and will preserve \(\widetilde{D}\) and its decomposition into six pairs. From the above remarks it follows that \(\mathfrak {A}_5\) acts on \(\widetilde{C}\) as the symmetry group of an icosahedron drawn on \(\widetilde{C}\) which has \(\widetilde{D}\) as vertex set and such that antipodal pairs are the fibers of \(\widetilde{C}\rightarrow C\). This shows both existence and uniqueness up to an automorphism of \(\mathfrak {A}_5\). \(\square \)
Proposition 3.10
An irreducible singular member of the Wiman–Edge pencil is necessarily as in Lemma 3.9: a stable curve with six nodes and of geometric genus zero. It appears in the Wiman–Edge pencil together with its outer transform.
Proof
It is known that \(e(C_t)e(C)\) is equal to the sum of the Milnor numbers of singular points of \(C_t\). We know that \(\mathfrak {A}_5\) leaves invariant each fiber, but that no fiber other than \(C_\infty \) (which is reducible) or the Wiman curve \(C_0\) (which is smooth) is \(\mathfrak {S}_5\)invariant. In other words, the irreducible fibers come in pairs. Since \(\mathfrak {A}_5\) cannot fix a point on S (because it has no a nontrivial linear representations of dimension 2 and hence cannot act nontrivially on the tangent space at this point), and a proper subgroup of \(\mathfrak {A}_5\) has index \(\geqslant 5\), the irreducible fibers come as a single pair, with each member having exactly six singular points, all of Milnor number 1, that is, having six ordinary double points. Hence the normalization of such a fiber is a rational curve as in Lemma 3.9 \(\square \)
We sum up with the following.
Corollary 3.11
 (lines)
\({C_\infty }\), a union of 10 lines with intersection graph the Petersen graph.
 (conics)
a pair \(C_\mathrm{c},C'_\mathrm{c}\), each of which is a union of five conics whose intersection graph is the complete graph on five vertices.
 (irred)
a pair \(C_{\mathrm{ir}},C_{\mathrm{ir}}'\) of irreducible rational curves, each with six nodes.
Remark 3.12
Our discussion in Sect. 2.1 shows that Open image in new window , when regarded as a curve on \(\overline{\mathscr {M}}_{0,5}\), meets \(\mathscr {M}_{0,5}\) in the locus parameterizing 5pointed rational curves \((C; x_1, \dots , x_5)\) with the property that there exists an affine coordinate z for C such that Open image in new window contains the union of \(\{0\}\) and the roots of Open image in new window . So we can characterize the Wiman–Edge pencil on \(\overline{\mathscr {M}}_{0,5}\) as the pencil which contains in Open image in new window and these two loci. It is desirable to have a modular interpretation of this pencil.
3.4 Connection with the Dwork pencil
 (i)
an unramified T[5]covering \(\mathscr {T}_{0,5}\rightarrow \mathscr {M}_{0,5}\) endowed with an action Open image in new window which extends the T[5]action and is compatible with the \(\mathfrak {S}_5\)action on \(\mathscr {M}_{0,5}\),
 (ii)
a regular function \(\psi :\mathscr {T}_{0,5}\rightarrow \mathbb {C}\), equivariant with respect to the above character (so that \(\psi ^5\) is defined as an \(\mathfrak {S}_5\)invariant function on \(\mathscr {M}_{0,5}\)),
 (iii)
an Open image in new window equivariant lift of \(\psi \) from \(\mathscr {T}_{0,5}\) to the Fano variety Open image in new window of lines on the Dwork pencil.
The singular members of the Wiman–Edge pencil are accounted for as follows: the Fermat quintic (\(\psi =0\)) yields the sum of 10 lines with multiplicity 2 (\(2C_\infty \)), the sum of the five coordinate hyperplanes (\(\psi =\infty \)) yields \(C_\mathrm{c}+C'_\mathrm{c}\), and if \(\psi \) is a 5th root of unity, we get \(C_\mathrm{ir}+C'_\mathrm{ir}\).
3.5 Plane model of the Wiman–Edge pencil
Choose one of the branch tangents, say \(x+\epsilon y+\epsilon ^2 z = 0\). It intersects the Wiman sextic B at two nonsingular points Open image in new window and Open image in new window . We see that each double point of W is a biflecnode, i.e., the branch tangents at singular point intersect the curve at this point with multiplicity 4 instead of expected 3.
3.6 Irregular orbits in S
Recall that if a group acts on a set, an orbit is called regular if the stabilizer of one (and hence any) of its points is trivial; otherwise it is called an irregular orbit. For what follows it will helpful to have a catalogue of irregular an \(\mathfrak {S}_5\)orbits and \(\mathfrak {A}_5\)orbits in S. Here we can observe that an \(\mathfrak {S}_5\)orbit is an \(\mathfrak {A}_5\)orbit if and only if its \(\mathfrak {S}_5\)stabilizer is not contained in \(\mathfrak {A}_5\) (otherwise it splits into two \(\mathfrak {A}_5\)orbits). So a determination of the irregular \(\mathfrak {S}_5\)orbits determines one of the irregular \(\mathfrak {A}_5\)orbits. The \(\mathfrak {S}_5\)equivariant incarnation of S as the moduli space of stable, 5pointed genus zero curves makes this determination (which is in fact due to Coble) rather straightforward, as we will now explain.
A point of \(\overline{\mathscr {M}}_{0,5}\) is the same thing as a stable map Open image in new window (where ‘stable’ means here that every fiber has at most two elements), given up to a composition with a Möbius transformation. The \(\mathfrak {S}_5\)stabilizer of a stable map x consists of the set of \(\sigma \in \mathfrak {S}_5\) for which there exists a Open image in new window with the property that Open image in new window . Since x is stable, its image has at least three distinct points, and so \(\rho (\sigma )\) will be unique. It follows that \(\rho \) will be a group homomorphism. Its image will be a finite subgroup of Open image in new window with the property that it has in \(\mathbb {P}^1\) an orbit of size \(\leqslant 5\).
Klein determined the finite subgroups of Open image in new window up to conjugacy: they are the cyclic groups, represented by the group \(\mu _n\) of nth roots of unity acting in \(\mathbb {C}\subset \mathbb {P}^1\) as scalar multiplication; the dihedral groups, represented by the semidirect product of \(\mu _n\) and the order 2 group generated by the inversion \(z\mapsto z^{1}\); and the tetrahedral, octahedral and icosahedral groups, which are isomorphic to \(\mathfrak {A}_4,\mathfrak {S}_4,\mathfrak {A}_5\) respectively. The octahedral and icosahedral groups have no orbit of size \(\leqslant 5\) in \(\mathbb {P}^1\), and hence cannot occur here. The tetrahedral group has one such orbit: it is of size 4 (the vertices of a tetrahedron), but since we want a degree 5 divisor, it then must have a fixed point, and this is clearly not the case.
We then end up with the following list, due to Arthur Coble [7, pp. 400–401].
Theorem 3.13
 Open image in new window

For example, \(\langle (12)\rangle \) is the stabilizer of \((0,0,\infty , 1,z)\) when z is generic. An orbit of this type has size 60. It is a regular \(\mathfrak {A}_5\)orbit.
 Open image in new window

For example, \(\langle (12)(34)\rangle \) is the stabilizer of Open image in new window when z is generic. An orbit of this type has size 60 and decomposes into two \(\mathfrak {A}_5\)orbits of size 30.
 Open image in new window

For example, \(\langle (1234)\rangle \) is the stabilizer of \((1,\sqrt{1}, 1,\sqrt{1},\infty )\). This is an \(\mathfrak {S}_5\)orbit of size 30. It is also an \(\mathfrak {A}_5\)orbit of type Open image in new window (take \(z=\sqrt{1}\)).
 \(\mathfrak {D}_4^{\mathrm{odd}}\)

For example \(\langle (12),(34)\rangle \) is the stabilizer of \((0,0, 1 , 1,\infty )\). This is an \(\mathfrak {S}_5\)orbit of size 30, which is also an \(\mathfrak {A}_5\)orbit of type Open image in new window (let \(z\rightarrow 0\)).
 \(\mathfrak {S}_3^{\mathrm{ev}}\)

For example \(\langle (23)(45),(123)\rangle \) is the stabilizer of \((1, \zeta _3, \zeta _3^2, 0,\infty )\). This is an \(\mathfrak {S}_5\)orbit of size 20 which splits into two \(\mathfrak {A}_5\)orbits of size 10.
 \(\mathfrak {D}_8\)

For example, \(\langle (12),(1324)\rangle \) is the stabilizer of \((0, 0,\infty ,\infty , 1)\). This is the unique \(\mathfrak {S}_5\)orbit of size 15. It is also an \(\mathfrak {A}_5\)orbit of type \(\mathfrak {D}_4^{\mathrm{ev}}\).
 \(\mathfrak {C}_6\)

For example, \(\langle (12)(345)\rangle \) is the stabilizer of \((\infty , \infty , 1, \zeta _3, \zeta _3^2)\). This is a single orbit of size 20 which is also an \(\mathfrak {A}_5\)orbit of type \(\mathfrak {C}_3\).
 \(\mathfrak {D}_{10}\)

For example, \(\langle (25)(34),(12345)\rangle \) is the stabilizer of \((1, \zeta _5, \zeta _5^2, \zeta _5^3, \zeta _5^4)\). The associated orbit is of size 12 and splits into two \(\mathfrak {A}_5\)orbits of size 6.
Remark 3.14
It is clear from Theorem 3.13 that we have a curve of irregular \(\mathfrak {A}_5\)orbits of size 30. However the locus of such points in S has 15 irreducible components. This is because the preimage of an \(\mathfrak {A}_5\)orbit under the map Open image in new window is generically of the form Open image in new window . An example of the closure such an irreducible component is the preimage of the line defined in \(\mathbb {P}^2\) by \(t_2=t_0+t_1\) under the blowup of the vertices of the coordinate vertex (it is pointwise fixed under an even linear permutation of these vertices). Since this line does not pass through any of the four vertices, this also shows that its preimage in S is a rational normal curve of degree 3. We thus obtain an \(\mathfrak {S}_5\)invariant curve on S of degree 45 (defined by a section of \(\omega _S^{9}\)) with 15 irreducible components. A priori this section is \(\mathfrak {A}_5\)invariant, but as Clebsch [6] showed, it is in fact \(\mathfrak {S}_5\)invariant (see also Remark 6.6).
Since we have already determined the size of some orbits in the anticanonical model, we can now interpret the orbits thus found. We do this only insofar it concerns \(\mathfrak {A}_5\)orbits, because that is all we need. Note that this is not a closed subset. Here is what remains, but stated in terms of the Wiman–Edge pencil:
Corollary 3.15
 \(\mathfrak {C}_3\)

This 20element orbit is the base point locus \(\Delta \) of \(\mathscr {C}\).
 \(\mathfrak {D}_4^{\mathrm{ev}}\)

This 15element orbit is the singular locus of \(C_\infty \).
 \(\mathfrak {S}^{\mathrm{ev}}_3\)

This consists of two 10element orbits in Open image in new window equal to Open image in new window and Open image in new window .
 \(\mathfrak {D}_{10}\)

This consists of two 6element orbits, namely the singular loci of the two irreducible members of the Wiman–Edge pencil, Open image in new window and Open image in new window .
Proof
Theorem 3.13 yields a complete list of the irregular \(\mathfrak {A}_5\)orbits in terms of \(\overline{\mathscr {M}}_{0,5}\) of size smaller than 30. We have already encountered some of these orbits as they appear in this corollary. All that is then left to do is to compare cardinalities. \(\square \)
4 Projection to a Klein plane
4.1 The Klein plane
The two representations I and \(I'\) of \(\mathfrak {A}_5\) are the complexification of two real representations that realize \(\mathfrak {A}_5\) as the group of motions of a regular icosahedron. They differ only in the way we have identified this group of motions with \(\mathfrak {A}_5\). The full group of isometries of the regular icosahedron (including reflections) is a direct product Open image in new window , and is in fact a Coxeter group of type \(H_3\), a property that will be quite helpful to us when we need to deal with the \(\mathfrak {A}_5\)invariants in the symmetric algebra of I. Both \(\mathfrak {A}_5\)actions give rise to \(\mathfrak {A}_5\)actions on the unit sphere in Euclidean 3space. Via the isomorphism Open image in new window , they can also be considered as actions of \(\mathfrak {A}_5\) on the Riemann sphere \(\mathbb {P}^1\). These projective representations are what we have called the Schwarzian representations and are the only two nontrivial projective representations of \(\mathfrak {A}_5\) on \(\mathbb {P}^1\) up to isomorphism.
We observed above that a Schwarzian representation has three irregular orbits of sizes 12, 20 and 30, corresponding respectively to the vertices, the barycenters and the midpoints of the edges of a spherical icosahedron. The antipodal map, when considered as an involution \(\mathbb {P}^1\), is antiholomorphic: it comes from assigning to a line in \(\mathbb {C}^2\) its orthogonal complement. We can think of this as defining an \(\mathfrak {A}_5\)invariant real structure on \(\mathbb {P}^1\) without real points. In particular, the involution is not in the image of \(\mathfrak {A}_5\). Yet it preserves the \(\mathfrak {A}_5\)orbits, so that each orbit decomposes into pairs. The preimages of Open image in new window under the degree 2 isogeny Open image in new window define two representations of degree 2 of an extension \(\widetilde{\mathfrak {A}}_5\) of \(\mathfrak {A}_5\) by a central subgroup of order 2, called binary icosahedral group. As above, these two representations of \(\widetilde{\mathfrak {A}}_5\) differ by an outer automorphism. If we take the symmetric square of such a representation, then the kernel \(\{ \pm 1\}\) of this isogeny acts trivially and hence factors through a linear representation of \(\mathfrak {A}_5\) of degree 3. This is an icosahedral representation of type I or \(I'\).
We can phrase this solely in terms of a given Schwarzian representation of \(\mathfrak {A}_5\) on a projective line K. For then the projective plane P underlying the associated icosahedral representation is the one of effective degree 2 divisors on K (the symmetric square of K) and K embeds \(\mathfrak {A}_5\)equivariantly in P as the locus defined by points with multiplicity 2. This is of course the image of K under the Veronese embedding and makes K appear as a conic.
We will identify K with its image in P, and following Klein [18] we refer to this image as the fundamental conic. It is also defined by a nondegenerate \(\mathfrak {A}_5\)invariant quadratic form on the icosahedral representation (which we know is selfdual). We call P a Klein plane. So an Open image in new window can be understood as a 2element subset of K. The latter spans a line in P and this is simply the polar that the conic K associates to x (when \(x\in K\), this will be the tangent line of X at x).
Following Winger, we can now identify all the irregular \(\mathfrak {A}_5\)orbits in P. As we have seen, K has exactly three irregular \(\mathfrak {A}_5\)orbits having sizes 30, 20 and 12, each of which being invariant under an antipodal map. This antipodal invariance implies that the antipodal pairs in the above orbits span a collection of resp. 15, 10 and 6 lines in P, each of which makes up an \(\mathfrak {A}_5\)orbit. When we regard these pairs as effective divisors of degree 2, they also yield \(\mathfrak {A}_5\)orbits in Open image in new window of the same size (K parameterizes the nonreduced divisors). The bijection between lines and points is induced by polarity with respect to K. This yields all the irregular orbits in the Klein plane:
Lemma 4.1
(Winger [26, Section 1]) There are unique irregular \(\mathfrak {A}_5\)orbits in P having size 12, 20 (both in K), 6, 10 or 15 (all three in Open image in new window ). The remaining irregular orbits in P have size 30 and are parametrized by a punctured rational curve. Further, the points with stabilizer a fixed \(\tau \in \mathfrak {A}_5\) of order 2 is open and dense in the image of the map \(K\rightarrow P\) given by \(z\in K\mapsto (z) +(\tau (z))\).
Proof
We can think of a point of K as a point on the icosahedron in Euclidean 3space. An element of Open image in new window is represented by an effective degree 2 divisor on K which has the same \(\mathfrak {A}_5\)stabilizer. If we identify K with its \(\mathfrak {A}_5\)action as a spherical icosahedron in Euclidean 3space with its group of motions, then such a divisor spans an affine line in Euclidean 3space with the same stabilizer. When the line passes through the origin, we get the three orbits of sizes 30, 20 and 12, otherwise the stabilizer is of order 2. \(\square \)
We call a member of the 6element \(\mathfrak {A}_5\)orbit in Open image in new window , a fundamental point and denote this orbit by F. We call the polar line of such a point a fundamental line.
4.2 Two projections
The irreducible \(\mathfrak {S}_5\)representation \(E_S\) splits into two 3dimensional irreducible \(\mathfrak {A}_5\)representations \(I_S\) and \(I'_S\). The two summands give a pair of disjoint planes Open image in new window and Open image in new window in Open image in new window , to which we shall refer as Klein planes. We abbreviate them by P resp. \(P'\), and denote by \(K\subset P\) and Open image in new window the fundamental (\(\mathfrak {A}_5\)invariant) conics. We have \(\mathfrak {A}_5\)equivariant (Klein) projections \(p:S\dasharrow P\) and \(p':S\dasharrow P'\) of the anticanonical model Open image in new window with center \(P'\), resp. P. Precomposition with an element of Open image in new window exchanges these projections.
Proposition 4.2
The first assertion of Proposition 4.2 is [20, Theorem 5].
Proof
To see that \(p:S\rightarrow P\) is surjective, suppose its image is a curve, say of degree m. The preimage of a general line in P in S is an anticanonical curve and hence connected. This implies that \(m=1\). But then S lies in hyperplane and this is a contradiction.
So p is a surjection of nonsingular surfaces. If some irreducible curve is contracted by p, then this curve will have negative selfintersection. The only curves on S with that property are the lines, and then a line is being contracted. Since \(\mathfrak {A}_5\) acts transitively on the lines, all of them are then contracted. In other words, the exceptional set is the union \(C_\infty \) of the 10 lines on S. But \(C_\infty \) has selfintersection \((2K_S)^2=20>0\) and hence cannot be contracted.
So the preimage of a point in P is finite. This is also the intersection of the quintic surface S with a codimension 2 linear subspace and so this fiber consists of five points, when counted with multiplicity.
For the last assertion, we notice that since one of the components of f has degree 5, the degree of \(S\xrightarrow {\scriptscriptstyle f}f(S)\) must divide 5. So it is either 1 or 5. If it is 5 then p and \(p'\) will have the same generic fiber so that \(p'\) factors through p via an isomorphism \(h:P\xrightarrow {\scriptscriptstyle \cong \,}P'\). But this would make the \(\mathfrak {A}_5\)representations I and \(I'\) projectively equivalent, which is not the case. (We could alternatively observe that then the elements of Open image in new window preserve the fibers of p so that we get in fact an action of \(\mathfrak {S}_5\) on P which makes p \(\mathfrak {S}_5\)equivariant. But there is no projective representation of \(\mathfrak {S}_5\) on \(\mathbb {P}^2\).) \(\square \)
The proof of the next proposition makes use of the Thom–Boardman polynomial for the \(A_2\)singularity locus. Let us first state the general result that we need. Let f be a morphism between two compact, nonsingular complex surfaces. Assume first that f has a smooth locus of critical points (\(=\) ramification divisor) \(\Sigma ^1(f)\). As Whitney and Thom observed, \(f\Sigma ^1(f)\) need not be a local immersion: generically it will have a finite set \(\Sigma ^{1,1}(f)\) of (Whitney) cusp singularities; at such a point f exhibits a stable mapgerm which in localanalytic coordinates can be given by Open image in new window .
In the more general case when \(\Sigma ^1(f)\) is a reduced divisor (when defined by the Jacobian determinant), \(\Sigma ^{1,1}(f)\) is defined as a 0cycle on the source manifold. While is not so hard to prove that the degree of \(\Sigma ^{1,1}(f)\) is a characteristic number of the virtual normal bundle Open image in new window of f, it is another matter to obtain a closed formula for it. In the present case it is equal to Open image in new window (see for instance [21, Theorem 5.1], where this is listed as the case \(A_2\)).
Proposition 4.3
(The ramification curve) The ramification curve R of the finite morphism \(p:S\rightarrow P\) is a singular irreducible member of the Wiman–Edge pencil, and hence obtained as an \(\mathfrak {A}_5\)curve by means of the procedure of Lemma 3.9.
Proof
The divisor class [R] is given by the wellknown formula Open image in new window . Since Open image in new window is \(3\) times the class of a line, and \(p^*\) takes the class of a line to the class of hyperplane section (i.e., \(K_S\)), we have Open image in new window . It follows that \([R]=2K_S\). Since R is \(\mathfrak {A}_5\)invariant, it must be a member of the Wiman–Edge pencil. In particular, R is reduced.
Remark 4.4
Proposition 4.5
The preimage Open image in new window of the fundamental conic in P is a nonsingular member of the Wiman–Edge pencil. The ordered 4tuple \((C_0, C_\infty , C_K, C_{K'})\) consists of four distinct members of the Wiman–Edge pencil which lie in harmonic position: there is a unique affine coordinate for \(\mathscr {C}\) that identifies this 4tuple with \((0, \infty , 1, 1)\).
Proof
It is clear that \(C_K\) is an \(\mathfrak {A}_5\)invariant member of \(2K_S\) and therefore a member of the Wiman–Edge pencil. It is defined by the quadric Q and similarly \(C_{K'}\) is defined by the quadric \(Q'\). The last clause of the proposition then follows as \(C_0\), resp. \(C_\infty \), is defined by \(Q+Q'\), resp. \(QQ'\).
If \(C_K\) is singular, then it must be one of \(C_\mathrm{c}, C'_\mathrm{c}, R, R'\). We shall exclude each of these possibilities.
Suppose that \(C_K=C_\mathrm{c}\) or \(C'_\mathrm{c}\). Then let Y be an irreducible component of \(C_K\). Now both Y and K are conics and since p is a linear projection, \(Y\xrightarrow {\scriptscriptstyle p\;} K\) must be of degree 1. On the other hand, R meets \(C_K\) transversally in \(\Delta \), and so Y meets R transversally in four distinct points. This implies that \(Y\xrightarrow {\scriptscriptstyle p\;} K\) must have degree 3 and we arrive at a contradiction.
We cannot have \(C_K=R\), for then p would ramify along Open image in new window and so Open image in new window would be 2divisible: this would make it twice an anticanonical divisor (a hyperplane section) and R is clearly not of that type. If \(C_K=R'\), then Open image in new window must ramify in the singular part Open image in new window of \(R'\) and so R contains Open image in new window . This contradicts the fact that the two members R and \(R'\) of the Wiman–Edge pencil intersect transversally. \(\square \)
We can also improve the statement about the birationality of the map \((p,p')\).
Theorem 4.6
Proof
We have already proved that the map is of degree 1 onto its image and hence coincides with the normalization of the image.
It remains to show that \((p,p')\) is a local isomorphism. Suppose it is not. Since the projection p, resp. \(p'\), is a local isomorphism on Open image in new window , resp. Open image in new window , the map could only fail to be local isomorphism at some \(x\in R\cap R'\). Let L be the kernel of the derivative at x. Then L is mapped to 0 under the composition with the both projections. This implies that L coincides with the tangent line of R and \(R'\) at x. But we know that two curves in the Wiman–Edge pencil intersect transversally at a base point. This contradiction proves the assertion.
Next represent h, resp. \(h'\), by general lines \(\ell \subset P\), resp. Open image in new window . Then \(f^*(h.h')\) is represented by Open image in new window . This is evidently a plane section of S and hence is a class of degree 5. Since p and \(p'\) are also of degree 5, it follows that \(f_*(1)\) is as asserted. \(\square \)
Remark 4.7
4.3 The projection of irregular orbits
Let us describe the images of the irregular \(\mathfrak {A}_5\)orbits in S under the projection map \(p:S\rightarrow P\). Since the projection is \(\mathfrak {A}_5\)equivariant, the image of an irregular \(\mathfrak {A}_5\)orbit in S is an irregular orbit in the Klein plane. According to Lemma 4.1, there are \(\mathfrak {A}_5\)orbits in P of size 6, 10, 15 (all outside the fundamental conic) and of 12 and 20 (all on the fundamental conic), of which those of size 6 and 10 come in pairs, the others being unique. The other irregular orbits in P are of size 30 and are parametrized by a rational curve. On the other hand, by Corollary 3.15, in S there are two \(\mathfrak {A}_5\)orbits in S of size 6 and 10, one of size 15 and 20, and an irreducible curve of orbits of size 30.
Irregular \(\mathfrak {A}_5\)orbits in P of cardinality \(\not =30\) explained by \(\mathfrak {A}_5\)orbits in S
special \(\mathfrak {A}_5\)orbit \(\mathscr {O}\) in S  \(p(\mathscr {O})\subset K\)?  

Base locus \(\Delta \) of \(\mathscr {C}\)  20  Yes  20 
A regular orbit in \(C_K\)  60  Yes  12 
Singular part of \(C_\infty \)  15  No  15 
Singular part of \(C_\mathrm{c}\) or \(C'_\mathrm{c}\)  10 (2)  No  10 
Singular part of R or \(R'\)  6 (2)  No  6 
4.4 The projection of the Wiman–Edge pencil
We will later investigate the pimages of the special members of the Wiman–Edge pencil, but at this point it is convenient to already make the following observation.
Proposition 4.8
The divisor \(p_*C_\infty \) is the sum of the 10 lines that are spanned by the antipodal pairs in the 20element orbit \(p(\Delta )\) on K. Each singular point of \(p_*C_\infty \) lies on exactly two lines and the resulting \(\left( {\begin{array}{c}10\\ 2\end{array}}\right) =45\) double points make up two irregular \(\mathfrak {A}_5\)orbits, one of which is the unique 15element orbit defined by pairs of lines which meet in S (and so the other has size 30).
Proof
The image of a line on S is a line in P and so \(p_*C_\infty \) is a sum of 10 lines. The polars of these lines make up an \(\mathfrak {A}_5\)orbit in P of size \(\leqslant 10\). There is only one such orbit and it has exactly 10 elements.
The singular locus of \(C_\infty \) is a 15element orbit and we observed that this orbit maps bijectively onto the unique 15element orbit in P. It follows from the discussion in Sect. 3.6, that the stabilizer of each singular point of \(C_\infty \) is the group \(\mathfrak {D}_4^{\mathrm{ev}}\). Its projection has the same stabilizer group. Thus the image of Open image in new window consists of 15 points. Since there is only one orbit in P of cardinality 15, the remaining \(4515 = 30\) points form an orbit of \(\mathfrak {A}_5\) in P. \(\square \)
This has implications for a generic member C of \(\mathscr {C}\), as follows. The curve \(p_*C_\infty \) being reduced and of geometric genus 6, it follows that \(p_*C\) has the same property. As p is linear, \(p_*C\) is a plane curve of the same degree as C, namely 10. So the arithmetic genus of \(p_*C\) is \((101)(102)/2=36\), and hence its genus defect is 30. Since the singular set of C specializes to a subset of the singular set of \(C_\infty \), it follows that this singular set consists of 30 nodes and makes up an \(\mathfrak {A}_5\)orbit (but remember that such orbits move in a curve and can degenerate into an orbit of smaller size).
So \(C\in \mathscr {C}\mapsto p_*C\) defines a morphism from the base \(\mathscr {B}\) of the Wiman–Edge pencil (a copy of \(\mathbb {P}^1\)) to \(\mathscr {O}_P(10)\). We denote its image by \(p_*\mathscr {B}\). It is clear that every point of \(p_*\mathscr {B}\) will be an \(\mathfrak {A}_5\)invariant curve. An \(\mathfrak {A}_5\)invariant curve in P admits an \(\mathfrak {A}_5\)invariant equation (because every homomorphism \(\mathfrak {A}_5\rightarrow \mathbb {C}^\times \) is trivial), and so every member of \(p_*\mathscr {C}\) lands in the projectivization of Open image in new window (recall that P is the projectivization of the dual of I).
The formula (3) also proves the following
Proposition 4.9
(The net of \(\mathfrak {A}_5\)decimics) All members of the net of \(\mathfrak {A}_5\)decimics intersect the fundamental conic transversally at 20 points and they are all tangent at these points to one of the 10 lines that joins two antipodal points.
Of course, the 20 points in the statement of Proposition 4.9 are the projection of the set \(\Delta \) of base points of the Wiman–Edge pencil.
Proposition 4.10
The members of \(p_*\mathscr {C}\) distinct from 5K are reduced and intersect K transversally in \(p_*\Delta \). The map \(\mathscr {B}\rightarrow p_*\mathscr {B}\) defined by \(C\mapsto p_*C\) is injective, and \(p_*\mathscr {B}\) is a curve of degree 5 in \(\mathscr {O}_P(10)\).
Proof
Let \(C_1, C_2\) be members of the Wiman–Edge pencil distinct from \(C_K\). Since \(p_*C_i\) is reduced, the map \(p:C_i\rightarrow p_*C_i\) is a normalization and so the equality \(p_*C_1=p_*C_2\) lifts to an \(\mathfrak {A}_5\)equivariant isomorphism \(C_1\cong C_2\). As \(\mathscr {C}\) is the universal family, it follows that \(C_1=C_2\). So the map \(\mathscr {B}\rightarrow p_*\mathscr {B}\) is injective.
It also follows that for \(z\in P\) generic, then through each of the five points of \(p^{1}(z)\) passes exactly one member of \(\mathscr {C}\) and these members are distinct and smooth. This means that the hyperplane in \(\mathscr {O}_P(10)\) of decimics passing through z meets \(p_*\mathscr {B}\) transversally in five points and so the curve in question has degree 5. \(\square \)
Proposition 4.11
The plane curve \(p_*\mathscr {B}\) has two singular points, namely the points represented by \(p_*R\) and by 5K, where it has a singularity of type \(A_4\), resp. \(E_8\) (having localanalytic parameterizations Open image in new window , resp. Open image in new window ).
Proof
Let \(x\in \Delta \). Since p has simple ramification at x, we can find localanalytic coordinates \((z_1,z_2)\) at x and \((w_1,w_2)\) at p(x) such that \(p^*w_1=z_1^2\) and \(p^*w_2=z_2\), and such that K is at p(x) given by \(w_2=0\). So the ramification locus R is given at x by \(z_1=0\) and \(C_K\) by \(z_2=0\).
A tangent direction at x not tangent to \(C_K\) has in the \((z_1,z_2)\)coordinates a unique generator of the form \((\lambda , 1)\). We therefore can regard \(\lambda \) as a coordinate for the complement in Open image in new window of the point defined by \(T_xC_K\) and hence as a coordinate for the complement Open image in new window of the point representing \(C_K\). This means that the member of Open image in new window corresponding to \(\lambda \) has a local parametrization at x given by Open image in new window . Its image under p has then the local parametrization Open image in new window , which shows that \(\lambda ^2\), when regarded as a regular function on \(\mathscr {B}^+\), is in fact a regular function on its image \(p_*\mathscr {B}^+\).
5 Images of some members of the Wiman–Edge pencil in the Klein plane
5.1 The preimage of a fundamental conic
We will characterize \(C_K\) and \(C_{K'}\) as members of \(\mathscr {C}\) by the fact that they support an exceptional even theta characteristic. Recall that a theta characteristic of a projective smooth curve C is a line bundle \(\kappa \) over \(C_K\) endowed with an isomorphism \(\phi :\kappa ^{\otimes 2}\cong \omega _{C}\). It is called even or odd according to the parity of the dimension of \(H^0(C, \kappa )\).
Proposition 5.1
The morphism \(C_K\rightarrow K\) is obtained as the complete linear system of an even theta characteristic \(\kappa \) on \(C_K\) followed by the (Veronese) embedding \(K\subset P\). The \(\mathfrak {A}_5\)action on \(C_K\) lifts to an action of the binary icosahedral group \(\widetilde{\mathfrak {A}}_5\) on \(\kappa \) in such a way that \(H^0(C_K, \kappa )\) is an irreducible \(\widetilde{\mathfrak {A}}_5\)representation of degree 2. Similarly for Open image in new window , albeit that \(H^0(C_{K'}, \kappa ')\) will be the other irreducible \(\widetilde{\mathfrak {A}}_5\)representation of degree 2.
There are no other pairs \((C, \theta )\), where C is a member of Open image in new window and \(\theta \) is an \(\mathfrak {A}_5\)invariant theta characteristic with \(\dim \theta =1\).
Proof
Observe that the preimage of a line in P meets \(C_K\) in a canonical divisor and that any effective degree 2 divisor on K spans a line in P. This implies that the fibers of \(C_K\rightarrow K\) belong to the divisor class of a theta characteristic Open image in new window . The \(\mathfrak {A}_5\)action on \(C_K\) need not lift to such an action on \(\kappa \), but its central extension, the binary icosahedral group \(\overline{\mathfrak {A}}_5\), will (in a way that makes \(\phi \) equivariant). Thus \(H^0(C_K, \kappa )\) becomes an \(\widetilde{\mathfrak {A}}_5\)representation. It contains a 2dimensional (base point free) subrepresentation which accounts for the morphism \(C_K\rightarrow K\). To see that this inclusion is an equality, we note that by Clifford’s theorem as cited above, \(\dim H^0(C_K, \kappa )\leqslant 3\). If it were equal to 3, then \(\widetilde{\mathfrak {A}}_5\) would have a trivial summand in \(H^0(C_K, \kappa )\) and hence so would \(H^0(C_K, \kappa ^{\otimes 2})= H^0(C_K, \omega _{C_K})\). This contradicts the fact that the latter is of type Open image in new window as an \(\mathfrak {A}_5\)representation.
It is clear that we obtain \((C_{K'}, \kappa ')\) as the transform/pullback of \((C_K, \kappa )\) under an element of Open image in new window .
If \((C, \theta )\) is as in the proposition, then C lies on an \(\mathfrak {A}_5\)invariant quadric of rank 3. This quadric will be defined by \(\lambda Q+\lambda ' Q'\) for some Open image in new window (with Q and \(Q'\) as in Sect. 2.5), and so the rank condition implies \(\lambda =0\) or \(\lambda '=0\). In other words, \((C, \theta )\) equals \((C_K,\kappa )\) or \((C_{K'}, \kappa ')\). \(\square \)
5.2 The image of reducible singular members
Recall that \(C_\mathrm{c}\) and \(C'_\mathrm{c}\) are the \(\mathfrak {A}_5\)orbit of a special conic. So the following proposition tells us what their pimages are like.
Proposition 5.2
The projection p maps each special conic isomorphically onto a conic in P that is tangent to four lines that are the projections of lines on S, and passes through four points of the 10element \(\mathfrak {A}_5\)orbit.
Proof
Let Y be a special conic. Since Y is part of \(C_\mathrm{c}\) or \(C'_\mathrm{c}\), Y meets \(\Delta \) in four distinct points and \(p_*Y\) is a degree 2 curve which meets \(p(\Delta )\) in four points. In particular p maps Y isomorphically onto its image (not on a double line). For each \(a\in Y\cap \Delta \), p(a) and its antipode span a line in P and according to Proposition 4.10, p(Y) is tangent to the line at p(a). \(\square \)
5.3 The decimics of Klein and Winger
Let us call a projective plane endowed with a group G of automorphisms isomorphic to \(\mathfrak {A}_5\) a Klein plane (but without specifying an isomorphism \(G\cong \mathfrak {A}_5\)). Such a plane is unique up to isomorphism (it is isomorphic to both P and \(P'\)). For what follows it is convenient to make the following definition.
Definition 5.3
(Special decimics) We call a reduced, Ginvariant curve in a Klein plane of degree 10 a special decimic if it is singular at each fundamental point and its normalization is rational. If the singularity at a fundamental point is an ordinary node, we call it a Winger decimic; otherwise (so when it is worse than that) we call it a Klein decimic.
We will see that each of these decimics is unique up to isomorphism, and that p(R) is a Klein decimic and \(p(R')\) a Winger decimic. We will also show that the singularity of a Klein decimic at a fundamental point must be a double cusp (i.e., with local analytic equation \((x^3y^2)(y^3x^3)\)). As to our naming: a Klein curve appears in [18, Chapter 4, Section 3] (p. 218 in the cited edition), and a Winger curve appears in [26, Section 9].
Let us first establish the relation between special decimics and the Wiman–Edge pencil. Let K be a copy of a \(\mathbb {P}^1\) and let Open image in new window be a subgroup isomorphic to \(\mathfrak {A}_5\). Recall that K has a unique Gorbit \(F^\#\) of size 12 (think of this as the vertex set of a regular icosahedron) which comes in six antipodal pairs. Let us denote by Open image in new window the antipodal involution, and let \(\overline{K}\) be obtained from \(\widetilde{Y}\) by identifying every \(z\in F^\#\) with Open image in new window as to produce an ordinary node. This is just the curve of arithmetic genus 6 that we constructed in Lemma 3.9.
Notice that the normalization of a special decimic factors through this quotient of K: it is the image of \(\overline{K}\) under a 2dimensional linear system of degree 10 divisors on \(\overline{K}\) that comes from a 3dimensional irreducible subrepresentation of \(H^0(\overline{K}, \mathscr {O}_{\overline{K}}(10))\). In order to identify these subrepresentations, we focus our attention on the dualizing sheaf \(\omega _{\overline{K}}\) of \(\overline{K}\). This is the subsheaf of the direct image of \(\omega _K(F^\#)\) characterized by the property that the sum of the residues in a fiber add up to zero. So \(H^0(\overline{K}, \omega _{\overline{K}})\) is the subspace of \(H^0(K,\omega _K(F^\#))\) consisting of differentials whose residues at the two points of any antipodal pair add up to zero.
Lemma 5.4
Proof
For every \(x\in F\) there is a linear form on \(H^0(K, \omega _K(F^\#))\) that assigns to \(\alpha \in H^0(K, \omega _K(F^\#))\) the sum of the residues at the associated antipodal pair on K. By the residue formula, these six linear forms add up to zero. Apart from that, residues can be arbitrarily described and we thus obtain the exact sequence. The character of the permutation representation on F (which can be thought of as the set of six lines through opposite vertices of the icosahedron) is computed to be that of trivial representation plus that of W. This implies that \(\widetilde{H}^0(F; \mathbb {C})\cong W\) as Grepresentations. \(\square \)
We know by Propositions 3.10 and 4.3 that the dualizing sheaf of \(\overline{K}\) defines the canonical embedding for \(\overline{K}\) and that the 6dimensional Grepresentation \(H^0(\overline{K}, \omega _{\overline{K}})\) decomposes into two irreducible subrepresentations of dimension 3 that are not of the same type. This enables us to prove that there are only two isomorphism types of special decimics.
Corollary 5.5
(Classification of special decimics) Let \(Y\subset P\) be a special decimic and let \(\overline{K}\rightarrow Y\) define a partial Gequivariant normalization (so that the six nodes of \(\overline{K}\) lie over the six fundamental points of P). Then Y can be identified in a Gequivariant manner with the image of \(\overline{K}\) under the linear system that comes from one of the two irreducible 3dimensional Gsubrepresentations of \(H^0(\overline{K}, \omega _{\overline{K}})\). In fact, \(p_*R\) and \(p'_*R\) are special decimics and every special decimic is isomorphic to one of them.
Proof
The line bundle \(\omega _K(F^\#)\) is of degree 10. The fact that \(\omega _{\overline{K}}\) is of degree Open image in new window implies that Open image in new window is the complete linear system of degree 10. As the Gembedding \(Y\subset P\) is of degree 10, the Gembedding \(Y\subset P\) is definable by a 3dimensional Ginvariant subspace of \(H^0(K, \omega _K(F^\#))\). It follows from Lemma 5.4 that this subrepresentation must be contained in \(H^0(\overline{K}, \omega _{\overline{K}})\), and hence is given by one of its 3dimensional summands.
Let us write \(E_K\) for \(H^0(\overline{K}, \omega _{\overline{K}})\) and \(\mathbb {P}_K\) for Open image in new window . The canonical map \(\overline{K}\rightarrow \mathbb {P}_K\) is an embedding and realizes \(\overline{K}\) as a member of the Wiman–Edge pencil: if we regard this embedding as an inclusion, then the \(\mathfrak {A}_5\)embedding \(R\subset \mathbb {P}_S\) is obtained from the Gembedding \(\overline{K}\subset \mathbb {P}_K\) via a compatible pair of isomorphisms \((G, E_K)\cong (\mathfrak {A}_5, E_S)\). The preimage Open image in new window of \(S\subset \mathbb {P}_S\) is a Ginvariant quintic del Pezzo surface which contains \(\overline{K}\). We have a decomposition Open image in new window into two irreducible subrepresentations such that the two associated linear systems of dimension 2 reproduce the projections Open image in new window and \(p'R\). So \(p_*R\) and \(p'_*R\) represent the two types of special decimics. \(\square \)
Theorem 5.6
The preimage Open image in new window of the set of fundamental points is the disjoint union of Open image in new window and Open image in new window . Moreover, \(p_*R'\) is a Winger decimic and \(p_*R\) is a Klein decimic.
Proof
We know that the singular set of R makes up an \(\mathfrak {A}_5\)orbit and that such an \(\mathfrak {A}_5\)orbit is mapped to F. The same is true for \(R'\) and so each fundamental point is a singular point of both \(p_*R\) and \(p_*R'\). For every singular point x of R, Open image in new window consists of \(k\leqslant 52= 3\) points. So Open image in new window is an \(\mathfrak {A}_5\)invariant set of size Open image in new window .
Our irregular orbit catalogue for the \(\mathfrak {A}_5\)action on S (see Sect. 3.6) shows that only \(k = 1\) is possible, so that this must be the 6element orbit in S different from Open image in new window , i.e., Open image in new window . This proves that the (disjoint) union of Open image in new window and Open image in new window make up a fiber of p. In particular, \(p_*R'\) has an ordinary double point at p(x) and hence is a Winger decimic. On the other hand, \(p_*R\) has multiplicity at least 4 at p(x): this is because the restriction of p to a local branch of R has a singularity at x (because of the presence of the other branch). So \(p_*R\) must be a Klein decimic. \(\square \)
Remark 5.7
We can now also say a bit more about the special decimics.
Proposition 5.8
The singular set of a Winger decimic consists of 36 ordinary double points, and a local branch at each fundamental point has that node as a hyperflex. The singular points that are not fundamental make up a 30element orbit.
Proof
For the first assertion, we essentially follow Winger’s argument. The normalization \(q:\widetilde{Y}\rightarrow Y\subset P\) is a rational curve that comes with an automorphism group \(G\cong \mathfrak {A}_5\). The points of \(\widetilde{Y}\) mapping to a flex point or worse make up a divisor D of degree Open image in new window (if t is an affine coordinate of \(\widetilde{Y}\), and Open image in new window is a coordinate system for P such that \(q^*z_0\), resp. \(q^*z_i\), is a polynomial of degree \(10i\), resp. \(\leqslant 10i\), then D is the defined by the Wronskian determinant \((z_0,z_1,z_2)\) and is viewed as having degree \((10+9+8)3=24\)). If a point of \(\widetilde{Y}\) maps to cusp or a hyperflex (or worse) then it appears with multiplicity \(\geqslant 2\). We prove that each of the two points of \(\widetilde{Y}\) lying over a fundamental point of P has multiplicity \(\geqslant 2\) in D. This implies that D is \(\geqslant \) twice the 12element Gorbit in \(\widetilde{Y}\) and hence, in view of its degree, must be equal to this. It will then follow that Y has only nodal singularities and that these will be 36 in number by the genus formula.
Let us first note that since G has no nontrivial 1dimensional character, Y admits a Ginvariant equation. Let \(x\in P\) be a fundamental point. Let \(U\subset P\) be the affine plane complementary to the polar of x and make it a vector space by choosing x as its origin. Then \(G_x\) acts linearly on U. The stabilizer \(G_x\) is a dihedral group of order 10, and we can choose coordinates \((w_1,w_2)\) such that \((w_1, w_2)\mapsto (w_2,w_1)\) and \((w_1, w_2)\mapsto (\zeta _5w_1,\zeta _5^{1}w_2)\) define generators of \(G_x\). The algebra of \(G_x\)invariant polynomials is then generated by \(w_1w_2\) and \(w_1{}^5+w_2{}^5\).
So Open image in new window is an \(\mathfrak {A}_5\)invariant 30element subset of P and it suffices to prove that this is an \(\mathfrak {A}_5\)orbit. This set does not meet the fundamental conic K (for \(p_*R'\) is transversal to that conic). There are unique \(\mathfrak {A}_5\)orbits in Open image in new window of size 6 and 10; the others have size 30 and 60. Hence Open image in new window is a 30element orbit in Open image in new window . \(\square \)
5.4 Construction of the Klein decimic
We now give a construction of the Klein decimic, which will at the same time show that it will have double cusp singularities at the fundamental points.
Let P be a Klein plane. We first observe that its fundamental set F does not lie on a conic. For if it did, then this conic would be unique, hence \(\mathfrak {A}_5\)invariant and so its intersection with K would then produce an orbit with \(\leqslant 4\) elements, which we know does not exist. This implies that for each \(x\in F\) there is a conic \(K_x\) which meets F in Open image in new window .
Lemma 5.9
If \(\ell _x\) is the polar line of x, then Open image in new window and hence \(K_x\) is tangent to K at these two points.
Proof
The divisor Open image in new window in K is \(\mathfrak {A}_5\)invariant and of degree Open image in new window . Given the \(\mathfrak {A}_5\)orbit sizes in K (12, 20, 30, 60), this implies that that this divisor is twice the 12element orbit. Hence \(K_x\) meets K in two distinct points of this orbit and is there tangent to K. Assigning to x the polar of \(K\cap K_x\) then gives us a map from F onto a 6element orbit of P. This orbit can only be F itself. We thus obtain a permutation of F which commutes with the \(\mathfrak {A}_5\)action. Since every point of F is characterized by its \(\mathfrak {A}_5\)stabilizer, this permutation must be the identity. \(\square \)
Corollary 5.10
The Cremona transformation \(q^\dagger q^{1}:P\dashrightarrow P^\dagger \) takes \(K\subset P\) to a Klein decimic in \(P^\dagger \). This Klein decimic has a double cusp (with localanalytic equation \((x^2y^3)(x^3y^2)=0\)) at each fundamental point, and is smooth elsewhere.
Proof
Since q is an isomorphism over K, we can identify K with \(q^{1}K\). Any singular point of \(q^\dagger q^{1}K\) will of course be the image of some \(K_x\). Since we can regard \(K_x\) as the projectivized tangent space of the fundamental point in \(P^\dagger \) to which it maps, Lemma 5.9 shows that the image of K at this fundamental point is as asserted: we have two cusps meeting there with different tangent lines. \(\square \)
Note that by the involutive nature of this construction, \(q(q^\dagger )^{1}\) will take \(K^\dagger \) to the Klein decimic in P.
Remark 5.11
Remark 5.12
We have seen that the curve of Wiman decimics in P contains both 5K and the \(\mathfrak {A}_5\)invariant sum of 10 lines. It has as a remarkable counterpart: the pencil of sextics spanned by 3K and the \(\mathfrak {A}_5\)invariant sum of six lines. This pencil was studied in detail by Winger [26] and is discussed in [8, 9.5.11]. The six lines meet K in the 12element orbit and this intersection is evidently transversal. It follows that this orbit is the base locus of the Winger pencil (as we will call it) and that all members of the pencil except 3K have a base point as a flex point (with tangent line the fundamental line passing through it). Note that a general member W of the pencil is smooth of genus 10.
Winger shows that this pencil has, besides its two generators, which are evidently singular, two other singular members. Both are irreducible; one of them has as its singular locus a node at each fundamental point (the 6element orbit of \(\mathfrak {A}_5\)) and the other, which we shall call the Winger sextic, has as its singular locus a node at each point of the 10element orbit of \(\mathfrak {A}_5\), and each local branch at such a point has this point as a flex point. So their normalizations have genus 4 and 0 respectively. The former turns out be isomorphic to the Bring curve (whose automorphism group is known to be isomorphic to \(\mathfrak {S}_5\)) and the latter will be isomorphic as an \(\mathfrak {A}_5\)curve to K.
In fact, as an abstract curve with \(\mathfrak {A}_5\)action, the Winger sextic can be obtained in much the same way as the curve constructed in Lemma 3.9, simply by identifying the antipodal pairs of its 20element orbit so as to produce a curve with 10 nodes. This plane sextic is also discussed in [9]. With the help of the Plücker formula (e.g., [8, formula 1.50]), it then follows that the dual curve of the Winger sextic is a Klein decimic. (A priori this curve lies in the dual of P, but in the presence of the fundamental conic \(K\subset P\) we can regard it as a curve in P: just assign to each point of the Winger curve the Kpolar of its tangent line.)
Other remarkable members of this pencil include the two nonsingular Valentiner sextics with automorphism group isomorphic to \(\mathfrak {A}_6\) (^{4}).
6 Passage to \(\mathfrak {S}_5\)orbit spaces
The goal of this section to characterize the Wiman curve as a curve on \(\overline{\mathscr {M}}_{0,5}\), or rather its \(\mathfrak {S}_5\)orbit space in \(\mathfrak {S}_5\backslash \overline{\mathscr {M}}_{0,5}\). The latter is simply the moduli space of stable effective divisors of degree 5 on \(\mathbb {P}^1\).
6.1 Conical structure of \(\mathfrak {S}_5 \backslash \overline{\mathscr {M}}_{0,5}\)
We first want to understand how the \(\mathfrak {S}_5\)stabilizer \(\mathfrak {S}_{5, z}\) of a point \(z\in \overline{\mathscr {M}}_{0,5}\) acts near z. The finite group action on the complexanalytic germ of \(\overline{\mathscr {M}}_{0,5}\) at z can be linearized in the sense that it is complexanalytically equivalent to the action on the tangent space. It then follows from a theorem of Chevalley that the local orbit space at \(\mathfrak {S}_5\backslash \overline{\mathscr {M}}_{0,5}\) is nonsingular at the image of z if and only if \(\mathfrak {S}_{5, z}\) acts on \(T_z\overline{\mathscr {M}}_{0,5}\) as a complex reflection group.
Lemma 6.1
The \(\mathfrak {S}_5\)stabilizer of any point not in \(\Delta \) is a Coxeter group (and hence a complex reflection group) so that \(\mathfrak {S}_5\backslash \overline{\mathscr {M}}_{0,5}\) is nonsingular away from the point \(\delta \) that represents the orbit \(\Delta \).
Proof
Case 2: \(z=(0,0,1,\infty , \infty )\), and \(\mathfrak {S}_{5,z}\cong \mathfrak {D}_8\) acts in the obvious manner. We again find that Open image in new window is an isomorphism. The group \(\mathfrak {S}_{5,z}\) now acts as the Coxeter group \(B_2\): It consists of the signed permutations of \((v_0, v_\infty )\).
Case 3: \(z=(1,\zeta _5, \zeta _5^2, \zeta _5^3, \zeta _5^4)\), and \(\mathfrak {S}_{5,z}\cong \mathfrak {D}_{10}\) is generated by \(t\mapsto \zeta _5t\) and the involution \(t\mapsto t^{1}\) (which gives the exchanges \(\zeta _5^i\leftrightarrow \zeta _5^{i}\)). For \(i\geqslant 1\) let Open image in new window be the restriction of the vector field \(t^{i+1}{d}/{dt}\). Then scalar multiplication by \(\zeta _5\) multiplies \(v^{(i)}\) by \(\zeta _5^{i}\), so that this is a basis of eigenvectors for this action. The vectors Open image in new window span the image of Open image in new window , and hence \(v^{(2)}\) and \(v^{(3)}\) span \(T_z\overline{\mathscr {M}}_{0,5}\). Multiplication by \(\zeta _5\) sends Open image in new window to Open image in new window and the involution sends it to Open image in new window . Thus it acts through the Coxeter group \(I_2(5)\). \(\square \)
It is clear from Proposition 3.2 that the \(\mathfrak {A}_5\)orbit space of every smooth member of \(\mathscr {C}\) is a smooth rational curve. This is also true for a singular member, for we have seen that \(\mathfrak {A}_5\) is transitive on its irreducible components and that an irreducible component is rational (a quotient curve of a rational curve is rational). It is clear that \(\mathfrak {S}_5\backslash \overline{\mathscr {M}}_{0,5}\) will inherit this structure. Since the base locus \(\Delta \) is a single \(\mathfrak {A}_5\)orbit and all members of the Wiman–Edge pencil are nonsingular at \(\Delta \), it then follows that the image of the Wiman–Edge pencil gives \(\mathfrak {A}_5\backslash \overline{\mathscr {M}}_{0,5}\) the structure of a quasicone with vertex \(\delta \).
Proposition 6.2
Proof
The diagonal form of \(\sigma \) shows that it leaves invariant only two lines through the origin: the \(u_2\)axis defined by \(u_1=0\), on which it acts faithfully with order 6, and the \(u_1\)axis defined by \(u_2=0\) on which it acts with order 3. Since \(C_0\) and \(C_\infty \) are the only members of \(\mathscr {C}\) that are \(\mathfrak {S}_5\)invariant, these axes must define their germs at z. The \(\mathfrak {S}_5\)stabilizer of z (which we have identified with \(\mathfrak {C}_6\)) acts on the germ of \(C_\infty \) at 0 through a subgroup of \(\mathfrak {S}_3\) (for it preserves a line on the quintic del Pezzo surface) and so \(\mathfrak {C}_6\) cannot act faithfully on it. It follows that \(C_0\) is defined by \(u_1=0\).
Since \(\sigma ^3\) sends \((u_1,u_2)\) to Open image in new window , its algebra of invariants is Open image in new window . Now \(\sigma \) sends \((u_1, u_2^2)\) to \((\zeta _3 u_1, \zeta _3^{1}u_2^2)\), so that its orbit space produces a Kleinian singularity of type \(A_2\). Following Klein, its algebra of invariants is generated by \(u_1^3, u_1u_2^2, u_2^6\). So the degree 6 part of Open image in new window has basis \(u_0^6, u_0^3u_1^3, u_0^3u_1u_2^2, u_2^6, u_1^6, u_1^2u_2^4\), \(u_1^4u_2^2\), and this \(\mathfrak {C}_6\backslash \mathbb {P}^2\) in \(\mathbb {P}^5\). Its image is clearly of degree 6. We have seen that the image of \(C_0\) is given by \(u_1=0\) and the above basis restricted to nonzero monomials are \(u_0^6\) and \(u_2^6\) and so its image is a line. If we substitute \(u_2=\lambda u_1\) (with \(\lambda \) fixed), then the monomials in question span \(u_0^6,u_0^3u_1^3, u_1^6\) and so the image of this line is indeed a conic. \(\square \)
Remark 6.3
The conclusion of the previous lemma and proposition agrees with the explicit isomorphism Open image in new window from Remark 6.6 below. The only singular point of the quotient is the point Open image in new window , which is a rational double point of type \(A_2\). The points projecting to the singular point satisfy \(I_4=I_8 = 0\). We know that \(I_4 = 0\) defines the Wiman curve on S and \(I_8 = 0\) defines the union \(C_K+C_K'\). They intersect at 20 base points with the subgroup \(\mathfrak {S}_3^{\mathrm{odd}}\) as the stabilizer subgroup.
6.2 The cross ratio map
The embedding of \(\mathfrak {S}_5\backslash \overline{\mathscr {M}}_{0,5}\) in projective 5space that we found in Proposition 6.2 has a modular interpretation, which we now give.
Lemma 6.4
The map Open image in new window is injective.
Proof
On \(\mathscr {M}_{0,5}\) this is clear, since given an ordered 5element subset \((x_1, \dots , x_5)\) of a smooth rational curve C, there is a unique affine coordinate z on C such that \((z(x_1), z(x_2), z(x_3))=(0, \infty , 1)\), and then \(z(x_i)=f_i(C;x_1,x_2, x_3, x_4)\) for \(i=4,5\). Since this even allows \(x_4=x_5\), we have in fact injectivity away from the zero dimensional strata. A typical zero dimensional stratum is represented in a Hilbert–Mumford stable manner by a 5term sequence in \(\{1,2,3\}\) with at exactly two repetitions, e.g., (1, 2, 3, 1, 2) and then the value of \(f_i\) on it is computed by removing the ith term and stipulating that of the four remaining two items two that are not repeated are made equal (and are then renumbered in an order preserving manner such that all terms lie in Open image in new window ). (In our example the value of \(f_1\) is (2, 1, 1, 2), the value of \(f_2,f_3, f_4\) is (1, 2, 1, 2), and the value of \(f_5\) is (1, 2, 2, 1).) Elementary combinatorics shows that f is here injective, too. \(\square \)
Theorem 6.5
The morphism \(\widetilde{\Phi }^*\) takes the hyperplane class of Open image in new window to Open image in new window , and \(\Phi \) is projectively equivalent to the projective embedding found in Proposition 6.2. In particular, \(\widetilde{\Phi }(C_\infty )\) is a line in Open image in new window .
Proof
Remark 6.6
The discriminant \(\Delta \) of a binary quintic is a polynomial in coefficients of degree 8, and as such it must be a linear combination of \(I_4^2\) and \(I_8\). In fact, it is known that Open image in new window . It is also a square of an element Open image in new window and so D must correspond to the curve \(C_\infty \). It is clear that any \(\text {SL}_2\)invariant in Open image in new window is a linear combination of \(\Delta =D^2\) and \(I_4^2\), in other words, has the form \((aD+bI_4)(aDbI_4)\) for some \((a, b)\not =(0,0)\). So this defines on S a reducible \(\mathfrak {S}_5\)invariant divisor which is the sum of two \(\mathfrak {A}_5\)invariant divisors. In other words, it represents the union of two \(\mathfrak {S}_5\)conjugate members of the Wiman–Edge pencil.
The invariant \(I_{18}\), being the, up to proportionality unique, invariant of degree 18 must represent a unique curve on S defined by an \(\mathfrak {S}_5\)invariant section of \(\omega _S^{9}\). According to Clebsch [6, p. 298], the invariant \(I_{18}\) vanishes on a binary form when its four zeros are invariant with respect to an involution of \(\mathbb {P}^1\) and the remaining zero is a fixed point of this involution. This is precisely the locus parametrized by Open image in new window and according to Remark 3.14 indeed defined by a section of \(\omega _S^{9}\).
Footnotes
 1.
Edge calls them Hesse duads for the following reason. It follows from observing the Petersen graph the stabilizer subgroup of each line acts on the line by permuting three intersection points with other lines. If one considers these points as the zero set of a binary form \(\phi \) of degree 3, then its Hessian binary form of degree 2 given as the determinant of the matrix of second partial derivatives of \(\phi \) has zeros at two points, the Hessian duad.
 2.
We follow the now standard ATLAS notation for finite groups.
 3.
This surface is isomorphic to the Clebsch diagonal cubic surface in \(\mathbb {P}^4\) defined by Open image in new window , Open image in new window ; the evident \(\mathfrak {S}_5\)symmetry accounts for its full automorphism group and its isomorphism type is characterized by that property. The intersection of this cubic surface with the quadric defined by Open image in new window is the Bring curve mentioned in Remark 5.12; its automorphism group is also \(\mathfrak {S}_5\).
 4.
The first author uses the opportunity to correct the statement in [8, Remark 9.5.11] where the Valentiner curve was incorrectly identified with the Wiman sextic.
Notes
Acknowledgements
We thank Shigeru Mukai for helpful information regarding genus 6 curves.
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