Cubic curves and totally geodesic subvarieties of moduli space

In this paper we present the ﬁrst example of a primitive, totally geodesic subvariety F ⊂ M g,n with dim( F ) > 1. The variety we consider is a surface F ⊂ M 1 , 3 deﬁned using the projective geometry of plane cubic curves. We also obtain a new series of Teichm¨uller curves in M 4 , and new SL 2 ( R )-invariant varieties in the moduli spaces of quadratic diﬀerentials and holomorphic 1-forms.


Introduction
Let M g,n denote the moduli space of compact Riemann surfaces of genus g with n marked points.A complex geodesic is a holomorphic immersion f : H → M g,n that is a local isometry for the Kobayashi metrics on its domain and range.It is known that M g,n contains a complex geodesic through every point and in every possible direction.
We say a subvariety V ⊂ M g,n is totally geodesic if every complex geodesic tangent to V is contained in V .It is primitive if it does not arise from a simpler moduli space via a covering construction.
A Teichmüller curve is a totally geodesic subvariety of M g,n of dimension one.These rare and remarkable objects are closely related to billiards in polygons, Jacobians with real multiplication, and dynamical rigidity.They are uniformized by Fuchsian groups defined over number fields, but they are generally not arithmetic.
This paper gives the first example of a primitive, totally geodesic Teichmüller surface in moduli space.We also obtain a new infinite series of Teichmüller curves in M 4 , and new SL 2 (R)-invariant subvarieties in moduli spaces of quadratic differentials and holomorphic 1-forms.
Our constructions depend in a fundamental way on the classical subject of cubic curves in the plane (Section 2) and space curves of genus four (Section 4), giving an unexpected connection between algebraic geometry and Teichmüller theory.
The flex locus F ⊂ M 1,3 .A point in M 1,3 is specified by a pair (A, P ) consisting of a compact Riemann surface A of genus one, and an unordered set P ⊂ A with |P | = 3.We can also regard P as an effective divisor of degree three.
The flex locus F ⊂ M 1,3 is defined by: (A, P ) : ∃ a degree three rational map π : A → P 1 such that (i) the fibers of π are linearly equivalent to P , and (ii) P is a subset of the cocritical points of π.
Here x ∈ A is a cocritical point of π if (1.1) {x, x } = (a fiber of π) for some critical point x of π. (We allow x = x.) We refer to F as the flex locus because, when A is defined by a homogeneous cubic polynomial f , π is given by projection from a point where det D 2 f = 0. See Section 3.
The main result of this paper, proved in Section 5, is: Strata.The surface F is closely related to an algebraic threefold G ⊂ M 4 , which is abundantly populated by new Teichmüller curves.Our proof of Theorem 1.1 depends on this relation.
Within a given stratum, we can impose the additional condition that there exists an involution J : X → X, with fixed points (p 1 , . . ., p n ), such that J * (ω) = −ω.The resulting locus is a Prym stratum ΩM g (a 1 , . . ., a n ) − ⊂ ΩM g .Note that J is uniquely determined by ω, provided g > 1.We allow a i = 0, to account for fixed points that are not zeros.Since the number of fixed points of J is preserved under limits, each Prym stratum is a closed subvariety of ΩM g .
There is a natural action of SL 2 (R) on ΩM g that preserves both types of strata, and whose orbits project to complex geodesics in M g .
The gothic locus G ⊂ M 4 .Given a Riemann surface X with a distinguished involution J, we say a holomorphic map p : X → B is odd if there exists an involution j ∈ Aut(B) such that p(J(x)) = j(p(x)) for all x ∈ X.
Here Z(ω) denotes the zero set of ω.The condition that p sends the three zeros of ω to a single point implies that p * (ω) = 0.
We refer to the variety G obtained by projecting ΩG to M 4 as the gothic locus.(The terminology is inspired by Figure 1.) The relationship between F and G can be summarized as follows: given any form (X, ω) ∈ ΩG with involution J, we obtain a point (A, P ) ∈ F by setting (A, q) = (X, ω 2 )/J, and marking the poles of q.
Using this natural map ΩG → F , in Section 4 and Section 5 we will show: Theorem 1.4.The space ΩG ⊂ ΩM 4 is a closed, irreducible variety of dimension four, locally defined by real linear equations in period coordinates.
In particular, the variety ΩG is locally isomorphic to a finite union of four-dimensional subspaces of C 10 .
Corollary 1.5.The locus ΩG is invariant under the natural action of SL 2 (R).
The crux of the proof of Theorem 1.4 is a lower bound on dim ΩG coming from our study of F .The surprise is that a small number of conditions on the periods of ω produce an elliptic curve B and a map p : X → B.
The fact that F is totally geodesic follows readily from Corollary 1.5, by transporting SL 2 (R)-orbits in ΩG to complex geodesics in F .

Teichmüller curves and real multiplication. Let
] denote the real quadratic order of discriminant D, where D > 0 and D = 0 or 1 mod 4.
Let (X, ω) be a form whose membership in ΩG is ratified by an involution J and a map p : X → B. Then we also have a natural map φ : X → A = X/J.Taking the quotient of Jac(X) by divisors pulled back from A and B, we obtain the polarized Abelian surface Let ΩG D ⊂ ΩG denote the locus where C admits real-multiplication by O D with ω as an eigenform.Its projection to M 4 will be denoted by G D .In Section 8 we will show: Theorem 1.6.For every discriminant D > 0, the locus G D ⊂ M 4 is a finite union of Teichmüller curves.
Theorem 1.7.If D is not a square, then every component of G D is geometrically primitive.
Theorem 1.8.If the stabilizer of a form in ΩG contains a hyperbolic element γ, then SL(X, ω) is a lattice and (X, ω) ∈ ΩG D for some D with Q( Polygon models for gothic forms.To conclude, we will describe an explicit construction of forms in the locus ΩG.In Section 9 we will show: Theorem 1.9.For any a, b > 0, the holomorphic 1-form (X, ω) = (P (a, b), dz)/ ∼ lies in ΩG.
Theorem 1.10.If, in addition, there are rational numbers x, y and d ≥ 0 such that Corollary 1.11.Every real quadratic field K arises as Outline of the paper.
1.In Section 2 and Section 3 we describe the surface F from the perspective of classical projective geometry.Every pair (A, P ) ∈ M 1,3 can be presented as a triple of collinear points on a smooth cubic curve in the plane, Similarly, every degree three rational map π S : A → P 1 , with fibers linearly equivalent to P , is obtained by projection from a point S ∈ P 2 .We find that π S has a triple of collinear cocritical points if and only if S itself lies on a related cubic curve, the Hessian HA ⊂ P 2 .
Since the moduli space M 1 of smooth cubics is one-dimensional, this shows that F itself is two-dimensional.In fact, F is naturally swept out by an open subset ‹ F of the universal Cayleyan, a smooth surface discussed in Section 3. 2. In Section 4 we use the fact that dim F = 2 to show dim ΩG = 4; while in Section 5, we show that in period coordinates, ΩG is contained in a finite union of four-dimensional linear spaces defined over R. It follows that ΩG is SL 2 (R)-invariant and that F is totally geodesic.
The algebraic formula for G given in equation (4.7) below also provides a direct proof that dim ΩG ≥ 4. 3.In Section 6 we review the theory of covering constructions, and exhibit another totally geodesic surface S 11 ⊂ M 1,3 which arises in this way.We then give a topological proof that F is primitive.4. In Section 7 we show, via an analysis of the Kobayashi metric, that T F is not isomorphic to any traditional Teichmüller space T g,n .This result gives a geometric proof that F is primitive, and suggest that one might regard T F itself as a new type of Teichmüller space, on an equal footing with T g,n . 5. In Section 8 we show that the loci G D ⊂ M 4 are finite unions of Teichmüller curves; the proof is similar to the case of the Weierstrass curves in M 2 [McM03a].Finally, in Section 9 we show that Figure 1 defines 1-forms in ΩG and, for suitable parameters, these forms generate Teichmüller curves in G D .
Notes and references.The components of G D with √ D irrational give a new, infinite series of geometrically primitive Teichmüller curves.
The previously known examples consist of four infinite series and two sporadic cases.The first three series come from the Weierstrass curves W D ⊂ M g , defined for g = 2, 3 and 4 [Cal04], [McM03a], [McM06].The fourth is the Bouw-Möller series, which gives finitely many more examples in M g for every g > 1 [BM10]; see also [Vee89], [Hoo13], [Wri13].Finally, there are two sporadic examples associated to the Coxeter diagrams E 7 and E 8 ; see [KS00], [Vor96] and [Lei04].
The locus ΩG itself has many interesting properties.For example, it is the first known primitive, SL 2 (R)-invariant subvariety of ΩM g defined over Q (in period coordinates), aside from the obvious examples like strata.It is also an example of an affine invariant manifold of rank 2. For more on general properties of affine invariant manifolds, see [Wri14] and [Wri15].
A program provided by A. Eskin led us to focus on the cathedral forms and provided evidence that they should lie in a new invariant subvariety of ΩM 4 .A special case of Theorem 1.10 was first proved using the algorithm described in [Muk17a], which showed directly that SL(X, ω) is a lattice for (x, y, d) = (0, 1/2, 2).
Further results and useful background can be found in the surveys [Mas06], [M 09] and [Zor06].

Cubic curves
In this section we recall some classical constructions from projective geometry.These constructions associate, to any smooth plane cubic curve A ⊂ P 2 , three other curves: the Hessian HA ⊂ P 2 , the Cayleyan CA, and the satellite Cayleyan SA.The last two reside in the dual projective plane P2 .In the next section, we will see that the points in the flex locus lying over A are naturally parametrized by SA.
Plane cubics.Let A be a plane cubic curve, given as the zero set of a homogeneous polynomial f : C 3 → C of degree three.We say A is a triangle if it is projectively equivalent to the cubic Z(XY Z), and a Fermat cubic if it is equivalent to Z(X 3 + Y 3 + Z 3 ).We will be mostly interested in the case where f is irreducible and A is smooth.
Polars, satellites and projections.Let S = [s] denote the point in P 2 determined by a nonzero vector s = (s 0 , s 1 , s 2 ) ∈ C 3 .The polar conic of A with respect to S is defined by The satellite conic of A (cf. [Sal79, p. 62]) is defined by Projection from S defines a rational map π S : A → P 1 .
(Intrinsically, the target is the linear system of hyperplanes through S.) If A is smooth and S ∈ A, then projection from S to P 1 is a rational map of degree three, and one can readily check that Moreover, the cocritical points of π S (defined by equation (1.1)) come from its satellite: we have Note.Relation (2.3) holds, more generally, for any smooth hypersurface A ⊂ P n , and relation (2.4) holds whenever A is cubic, as does the alternative formula: Lattès maps.For the remainder of this section, we assume that the cubic curve A = Z(f ) is smooth.The tangent line to A at x will be denoted by T x A ⊂ P 2 .The space of tangent lines forms the dual sextic in the dual projective plane.Let {x, x } denote the points where T x A meets A. If a line L meets A at (a, b, c), then the three points (a , b , c ) also lie on a line L = δ A (L).This construction defines the holomorphic Lattès map (2.6) δ A : P2 → P2 associated to A, of interest in complex dynamics (see, e.g., [DJ10], [Ron10], [BL01]).Its algebraic degree is four.The critical values of A coincide with the dual sextic Ǎ.
The Hessian.The Hessian of A is the cubic curve defined by HA = Z(det D 2 f ).The nine flexes of A are given by HA ∩ A.
The Hessian can be described geometrically in terms of the polars and satellites of A; namely, (2.7) HA = {S ∈ P 2 : the polar conic Pol(A, S) is singular}, and (2.8) A ∪ HA = {S ∈ P 2 : the satellite conic Sat(A, S) is singular}.
These statements follow directly from equations (2.5) and (2.2), using the fact that a conic Z( a ij x i x j ) is singular if and only if det(a ij ) = 0.When the conic is singular, it becomes a pair of lines; thus (2.8) implies: (2.9)For S ∈ A, the projection π S : A → P 1 has three collinear cocritical points ⇐⇒ S ∈ HA.
From cocritical to critical.Since it plays an important role in the sequel, we sketch a more geometric proof of (2.9).Suppose S ∈ A and π S : A → P 1 has three collinear cocritical points P i , i = 1, 2, 3, with corresponding critical points Q i ∈ A. Let L be the line through {P 1 , P 2 , P 3 }, and let R i be the line joining S to P i , i = 1, 2, 3. Then 3 1 P i + 2Q i forms the base locus of the pencil of cubics |A + λR 1 R 2 R 3 |.Within this pencil one can find a reducible cubic of the form L + C. Since the conic C has three tangent lines R i that pass through a single point, it is singular; and since We then have M ⊂ Pol(A, S), so S ∈ HA by (2.7).The converse is immediate, as we will see below.
The Cayleyan.For each S ∈ HA, we have a pair of distinct lines such that The Cayleyan CA ⊂ P2 is the set of all lines that arise in this way; that is, The point S is uniquely determined by L, since it lies on T x A for all x ∈ L ∩ A. Thus we have a natural degree two covering map, CA → HA.The curve CA is also cubic; see equation (2.14) below.
The satellite Cayleyan.Note that if S ∈ A and x ∈ A is a critical point of π S , then x is a cocritical point.As a consequence, if S ∈ HA and Pol(A, We refer to the set of lines that arise in this way as the satellite Cayleyan, Since CA is irreducible, so is SA.It is generically a curve of degree 12 and genus one, with interesting singularities.

Normal form.
Here is an explicit description of the polar and satellite lines for an arbitrary smooth cubic A ⊂ P 2 as seen from a point S ∈ HA − A.
It is easy to see that S ∈ L 1 .Choose affine coordinates (x, y) on C 2 ⊂ P 2 so that S = [0 : 1 : 0] is the vertical point at infinity, and L 1 is the x-axis.Then, it is readily verified that A is defined by a cubic equation of the form where b, c ∈ C[x] are polynomials of degrees (at most) one and three respectively; and that the polar and satellite lines for any such cubic are defined by the vanishing of the linear forms: (2.11) In particular, all four lines pass through a single point in P 2 .Figure 2 shows the lines L 1 and L 1 for the cubic defined by equation (2.10) with b(x) = x − 1 and c(x) = x(x + 1)(x − 2).
The Hesse pencil.We conclude with a discussion of the Hesse family of cubics, which will be used in our description of the flex locus.A useful reference for this topic is [AD09].
The Hesse pencil is the family of cubic curves A t ⊂ P 2 , defined for t ∈ P 1 by (2.12) The curve A 0 is a Fermat cubic, while A ∞ is the triangle defined by XY Z = 0.The cubic A t is smooth over the points (2.13) otherwise it is a triangle.The base locus E = A 0 ∩ A ∞ of the Hesse pencil coincides with the nine flexes of A 0 , as well as the flexes of every other smooth curve in the family.Thus, we can regard an element of the Hesse pencil as an elliptic curve with marked 3-torsion.More precisely, we have a natural isomorphism and j defines a covering map of orbifolds with deck group PSL 2 (F 3 ) ∼ = A 4 .Note that A t is a Fermat cubic if and only if j(t) = 0, which gives t = 0, −2 or 1 ± √ −3.When A belongs to the Hesse pencil, so do the cubics HA and CA (using the natural dual basis to identify P 2 and P2 ).In fact, HA t = A h and CA t = A c for the values . Using these formulas one can verify that, for a smooth cubic curve A: Either A is a Fermat cubic and HA and CA are triangles, or HA and CA are also smooth cubics.

The flex locus
In this section we discuss the flex locus from the perspective of plane cubics.We begin by establishing an alternative definition of F , using the language of Section 2.
Theorem 3.1.A point (A, P ) ∈ M 1,3 lies in F if and only if there is a plane cubic model for A and a line L in the satellite Cayleyan SA such that Using the universal Cayleyan, we then show: Theorem 3.2.The flex locus F ⊂ M 1,3 is the image of a smooth, irreducible surface under a proper immersion.
Finally we define a four-dimensional bundle of quadratic differentials QF → F , analogous to ΩG → G, and show: Theorem 3.3.The locus QF ⊂ QM 1,3 is a closed, irreducible subvariety of dimension four.Markings.To prove Theorems 3.2 and 3.3 we will explicitly construct a smooth, irreducible surface ‹ F , a finite manifold cover of moduli space and a proper immersion F is of interest in its own right.A point in ‹ F corresponds to triple (A, P, π S ), with S ∈ HA, satisfying the definition of F in Section 1, together with a marking of the 3-torsion of A. The surface F itself is not smooth, and these choices serve to separate its sheets and resolve its orbifold points.
Cubic models.To make the connection to Section 2, recall that any Riemann surface A ∈ M 1 can be presented as a smooth cubic curve A ⊂ P 2 .This plane cubic model for A is unique up to automorphisms of A and P 2 .
Proof of Theorem 3.1.Suppose (A, P ) ∈ F .Let A ⊂ P 2 be the plane cubic model determined by the complete linear system |P |.Then P = A ∩ L for some line L ∈ P2 .Since (A, P ) ∈ F , there is a degree three rational map π : A → P 1 such that (i) the fibers of π are linearly equivalent to P , and (ii) P is contained in the cocritical points of π.Property (i) implies that π is given by projection from some point S ∈ P 2 − A; and (ii) implies that P is contained in the satellite conic Sat(A, S) (see assertion (2.4)).Since P consists of three distinct points, this implies we have L ⊂ Sat(A, S); hence the satellite conic is singular, and we have L ∈ SA.The converse is similar.
Cubics and lines.We now turn to the proof of Theorem 3.2.For concreteness, we will work with the family of Hesse cubics A t ⊂ P 2 defined by (2.12).
Consider the Zariski open subset of P 1 × P2 defined by Since every cubic occurs, up to isomorphism, in the Hesse family, we have a natural covering map of orbifolds u : › M 1,3 → M 1,3 given by We remark that the deck group Γ of › M 1,3 /M 1,3 has order 216; it satisfies where Aut(P 2 ) E denotes the group of projective transformations stabilizing nine basepoints E = A 0 ∩ A ∞ of the Hesse pencil [AD09].The fibers of u correspond to different markings of the flexes of A.
The universal Cayleyan.Next we define the universal Cayleyan over the Hesse family by The locus CA P 1 → P 1 is a smooth elliptic surface, even though some of its fibers, such as CA 0 , are triangles.In particular, CA P 1 is irreducible.Smoothness follows from the fact that c(t) has a simple pole in equation (2.14), so that near a triangular fiber CA P 1 is locally isomorphic to the surface xy = t.
Normalization of a cover of F .Finally we define a Zariski open subset of the universal Cayleyan by It is easy to see, e.g., from the example in Figure 2, that ‹ F is nonempty; and since CA P 1 is a smooth, irreducible surface, so is ‹ F .
Proof of Theorem 3.2.Since the map δ in equation (3.2) is proper, so is its restriction δ| ‹ F .It is also an immersion, since the critical values of δ A correspond to lines with |L ∩ A| < 3, and these configurations are excluded from › M 1,3 .Since u is a covering map of orbifolds, the composition u • δ is a proper immersion; and since SA = δ A (CA), its image is F by Theorem 3.1.
Corollary 3.4.The surface F is birational to P 2 .
Proof.The hyperelliptic involution −I ∈ SL 2 (F 3 ) belongs to the group Γ in equation (3.1), so the map u • δ : ‹ F → F factors through a rational quotient of the elliptic surface CA P 1 .Thus F itself is rational.
One can also check that in the example of Figure 2, S is uniquely determined by L 1 ∈ SA, and hence: The space of quadratic differentials QF → F .We conclude by defining a bundle of quadratic differentials to record the directions of Teichmüller geodesics in F .
Recall that the cotangent space to a point (Y, P ) ∈ M g,n is naturally identified with the vector space Q(Y, P ) of meromorphic quadratic differentials q on Y with (q)+P ≥ 0. A point in the moduli space of quadratic differentials, QM g,n → M g,n , is specified by a triple (Y, P, q) as above with q = 0. Now let (A, P ) ∈ M 1,3 be an elliptic curve with marked points whose membership in F is ratified by a rational map π : A → P 1 of degree three.Let QF (A, P, π) = {q ∈ Q(A, P ) : (q) = Z − P for some fiber Z of π}.
To take into account of the possibility that π is not unique, let QF (A, P ) = π Q(A, P, π).Finally, let QF → F denote the subspace of QM 1,3 → M 1,3 whose fiber over (A, P ) ∈ F is QF (A, P ).
Proof of Theorem 3.3.Recall that for (t, L) ∈ ‹ F , there is a unique S ∈ P 2 such that L ⊂ Pol(A t , S), and a unique L ⊂ Sat(A t , S) We then have a natural proper map immersion Qδ : Sheets of F .The proof shows that, given (A, P ) ∈ F , there are only finitely many possibilities for the associated map π : A → P 1 , and the different choices of π index the different sheets of the immersed surface F passing through (A, P ).

The gothic locus
In this section we discuss the correspondence between quadratic differentials and Prym forms, and use it to relate the flex locus F to the threefold G ⊂ M 4 defined in Section 1.We will show: Theorem 4.1.The squaring map gives a natural algebraic isomorphism sq : ΩG → QF (−1 3 , 1 3 ).
Strata.We begin by reviewing notation for strata of quadratic differentials with marked points.Recall that a point of QM g,n is specified by a triple (Y, P, q).The stratum QM g,n (a 1 , . . ., a s ) ⊂ QM g,n is defined by the requirement that there exist distinct points (p 1 , . . ., p s ) in Y such that P = {p 1 , . . ., p n } and the divisor of q satisfies (q) = s 1 a i p i .
Here a i = 4g − 4, a i ≥ −1 for all i, and a i ≥ 1 if i > n.
From gothic to flex.First assume that (X, ω) ∈ ΩG.We then have a degree three map to an elliptic curve, p : X → B, and an involution j ∈ Aut(B), such that (4.3) p(J(x)) = j(p(x)).
By the definition of ΩG, Z is a fiber of p.
Choose the origin in B so that j(x) = −x, and let r : B → B/j ∼ = P 1 be the quotient map.Then r • p : X → P 1 is a J-invariant map of degree six.Consequently we have a unique degree three rational map π : A → P 1 making the diagram: commute.We will show that: (i) the fibers of π are linearly equivalent to P , and (ii) P is contained in the cocritical points of π.
To see (i), simply note that Z is a fiber of π since Z is a fiber of r • p, and equation (4.2) shows that Z is linearly equivalent to P because the canonical bundle of A is trivial.
To prove (ii), note that p maps Fix(J) into Fix(j) by (4.3), and that Fix(j) = B [2] coincides with the set of 2-torsion points in B. Let us denote the points of B [2] by {e 1 , e 2 , e 3 , e 4 }, with p(Z ) = (e 4 ).Since p −1 (e i ) is also J-invariant for i = 1, 2, 3, we can find P i , Q i ∈ X such that, as divisors, we have and P i = φ(P i ) ∈ P .Let Q i = φ(Q i ), and let e i = r(e i ).Then again as divisors, we have (4.5)π −1 (e i ) = P i + 2Q i for i = 1, 2, 3, and hence P i is a cocritical point of π.This proves (ii) and shows that (A, P, q) ∈ QF .
From flex to gothic.Now suppose (A, P, q) ∈ QF (−1 3 , 1 3 ).Let π : A → P 1 be a rational map of degree three verifying conditions (i) and (ii) above.Let P = {P 1 , P 2 , P 3 }, let π(P i ) = e i for i = 1, 2, 3, and let π(Z) = e 4 .Then we can find By the definition of the flex locus (Section 3), there is a plane cubic model for A and a pair of lines L ∈ CA and L ∈ SA such that P = A ∩ L and Let r : B → P 1 be an elliptic curve, presented as a 2-fold covering of P 1 branched over E = {e 1 , e 2 , e 3 , e 4 }.One can also regard B/P 1 as the Riemann surface of the function √ f , where (f ) = e 1 + e 2 + e 3 − 3e 4 .Now note that, because the divisors Z, P and Q are all linearly equivalent, is the divisor of the square of a rational function on A. Thus √ q and √ f • π define the same Riemann surface X/A, and hence the map π : A → P 1 lifts to a map p : X → B making diagram (4.4) commute.This lift intertwines the Z/2 Galois groups of X/A and B/P 1 , so p is odd.Then, since Z is a fiber of p, the form (X, ω) belongs to ΩG.
Proof of Corollary 4.2.By Theorem 3.3, QF is a closed, irreducible subvariety of QM 1,3 dimension four.Thus ΩG ⊂ ΩM 4 inherits these properties from QF , by Theorem 4.1 and the fact (see Section 1) that the Prym stratum ΩM 4 (2 3 , 0 3 ) − is closed in ΩM 4 .Remarks on the canonical model.The canonical embedding gives an illuminating geometric picture of the relationship between X and A.
Let (X, ω) be a 1-form in ΩM 4 (2 3 , 0 3 ) − with involution J and quotient curve A = X/J of genus one.The eigenspaces of J determine a splitting where we have identified Ω(A) = Cα with the span of a J-invariant form α ∈ Ω(X).Let Fix(J) = Z ∪ P , where Z = Z(ω).We have |Z| = |P | = 3.An involution of a hyperelliptic curve has at most four fixed points, provided it is not the hyperelliptic involution itself.Since | Fix(J)| = 6, the curve X is not hyperelliptic.Thus its canonical linear system provides an embedding X ⊂ PΩ(X) * ∼ = P 3 .
Every nonzero 1-form η on X determines a plane H η ⊂ P 3 .
It is classical that the space curve X, of degree six, is the transverse intersection Q ∩ C of an irreducible quadric and a cubic surface in P 3 [GH78, p. 258].The surface Q is uniquely determined by X, but C is not.
The automorphism J acts naturally on Ω(X), and hence on P 3 .Its fixedpoint set in P 3 is the union of the plane H α and the point H ⊥ α dual to Ω(A) ⊥ .Projection from this point yields a J-invariant map Φ : X → H α .The fibers of Φ consist of pairs of points that are interchanged by J. Hence A = Φ(X) ⊂ H α is a cubic plane curve, naturally isomorphic to X/J; and the fixed-points of J|X are given simply by Fix(J) = Z ∪ P = X ∩ H α = A ∩ Q.We can now make the cubic surface defining X canonical, by letting C be the cone over A with vertex H ⊥ α .The conic Q ∩ H α is singular, since its intersection with the line H ω ∩ H α consists of the three points Z. Since J(Q) = Q, this implies that Q itself is singular.Thus Q ∩ H α is a pair of lines through the singular point of Q -one of the form H ω ∩ H α , containing Z = Z(ω) -and the other containing the remaining fixed-points P .See Figure 3.
We remark that the fact that Q is singular implies: X lies on the θ-null divisor M 4 ⊂ M 4 .
Equations for X ∈ G.We can now give an explicit formula for the canonical model of any curve X in the gothic locus G ⊂ M 4 .Namely, we find that X = C ∩ Q can be defined in affine coordinates by the two equations To see this, recall that Theorem 4.1 associates to any form (X, ω) ∈ ΩG a point (A, P ) in the flex locus and a quadratic differential q ∈ Q(A, P ).As we saw in Section 2 (see equations (2.10) and (2.11)), we can choose affine coordinates on P 2 ∼ = H α so that A is given by the first equation in (4.7), and P = A ∩ L 1 , where L 1 is the satellite line defined by y + b(x) = 0.In these coordinates π S (x, y) = x, so the zeros Z(q) ⊂ A are cut out by the equation x + a = 0 for some a ∈ C. The zeros of q become the zeros of ω after passing to the double cover X → A. Since the quadric Q is characterized by the fact that it is symmetric under J and passes through P ∪ Z(ω), it is given by the second equation in (4.7) above.Moreover, the associated elliptic curve B is defined in coordinates (t, x) by t 2 + c(x)(x + a) = 0, and the degree three map p : X → B is given by the substitution t = yz.
Invariant varieties and period coordinates.We first remark that (I) implies (II) by general principles.
Recall that for any point (X 0 , ω 0 ) ∈ ΩM g , there exists a neighborhood U of (X 0 , ω 0 ) (or an orbifold chart) in its stratum and a natural open analytic embedding called the period map, that sends (X, ω) to the unique relative cohomology class [ω] such that [ω], γ = γ ω.Using this map, we implicitly identify U with an open subset of the vector space on the right.
A closed subset M ⊂ ΩM g (a 1 , . . ., a n ) is locally defined by real linear equations if, for each point (X 0 ω 0 ) ∈ M and U as above, there exists a finite set of complex subspaces

The conclusion of (I) means that ΩG has this form.
There is a natural action of the connected group GL + 2 (R) on any complex vector space V with a real structure, defined by The action of SL 2 (R) on ΩM g has this form in period coordinates (see, e.g., [McM03a,§3]).Evidently any subspace of V defined over R is SL 2 (R)-invariant, and so: Any closed set M ⊂ ΩM g locally defined by real linear equations is SL 2 (R) invariant.(5.1) (The assumption that M is closed is used to prove the group action is global.)Thus (I) implies (II) for ΩG, and invariance is inherited by QF via Theorem 4.1, using the fact that the squaring map in equation (4.1) respects the action of SL 2 (R).
A criterion for linearity.For the proof of (I) we will use the following general result.
Theorem 5.1.Let M ⊂ ΩM g (a 1 , . . ., a n ) be an algebraic variety whose irreducible components have dimension ≥ d.Suppose that for every (X, ω) ∈ M , there is a d-dimensional subspace S, defined over a real number field, such that Then M is locally defined by real linear equations in period coordinates, and dim(M ) = d.
Proof.Fix a point (X, ω) ∈ M .Let S be the collection of all d-dimensional subspaces S as above.Then there exists a ball U about (X, ω) in period coordinates such that M ∩ U is an analytic variety of dimension ≥ d, contained in S. Since each S ∈ S is defined over a number field, the collection S is countable.Consequently dim(M ∩ U ) = d.
By Noetherian properties of analytic sets, S ∩ M ∩ U has nonempty interior in S for only finitely many subspaces {S 1 , . . ., S n } ⊂ S. Since S i ∩ U is connected, it is contained in M for all i.By the Baire category theorem, The Abelian surface C(X|A, B).Recall from equation (1.2) that any (X, ω) ∈ ΩG, with associated maps X → A and X → B of degrees two and three respectively, determines a polarized Abelian surface The submersion Jac(X) → C determines an orthogonal splitting, with ω ∈ Ω(C).Similarly, on the level of relative cohomology, we obtain an exact sequence Since J|Z(ω) = Id and H 1 (X) J = H 1 (A), we have a natural isomorphism defined over Q.
The class [ω] lies in the H 1 (C) factor above, since J * ω = −ω and ω pushes forward to zero on B.
Proof of (I).By Theorems 3.3 and 4.1, ΩG is an irreducible four-dimensional algebraic subvariety of ΩM 4 (2, 2, 2).As we have just seen, the relative periods of any form (X, ω) ∈ ΩG lie in a four-dimensional subspace H 1 (C) ⊂ H 1 (X, Z(ω)) defined over Q.By Theorem 5.1, these two facts imply that ΩG is locally defined by real linear equations in period coordinates.
Proof of (III).Recall that the Teichmüller norm on the cotangent space Q(A, P ) is given by q = X |q|, and that the Beltrami coefficient q/|q| represents a unit vector tangent to the Teichmüller geodesic generated by q, which is itself the projection to M 1,3 of the orbit SL 2 (R) • q.
By Theorem 3.2, F is an immersed smooth surface.Let (A, P ) be a point on a sheet of F with associated degree three map π : A → P 1 .Let T 1 be the unit tangent space to this sheet at (A, P ), and let Q 1 be the unit ball in QF (A, P, π).Note that T 1 and Q 1 are both homeomorphic to S 3 .Lemma 6.1.Every form in QM 1,3 (−1 3 , 1 3 ) is either primitive or the pullback of a form in QM 0,4 via a covering construction.
Proof.Let (A, P, q) be a form in QM 1,3 (−1 3 , 1 3 ).Suppose q = p * (q ) with q ∈ Q(A , P ) and deg(p) > 1.If A has genus one, then p must be a covering map of degree three; but then q has just one pole, which is impossible.Thus A ∼ = P 1 .Any zero of q must be simple, and have deg(p) simple pre-images; thus in the presence of a zero, we must have deg(p) ≤ 3 and q must have at least five poles.This contradicts the fact that q only has three poles.Thus q has four poles and no zeros, i.e., q ∈ QM 0,4 ; and since q only has simple zeros, p can only be branched over the poles of q .
Proof of (II).Suppose that F ⊂ M 1,3 arises via a covering construction f π : M h,m → M 1,3 .Then every complex geodesic contained in F is generated by the lift of a quadratic differential in QM h,m .Thus QF contains no primitive form, contrary to statement (I) above.
Proof of (I) for ΩG.For later applications to Teichmüller curves, we conclude by proving a similar primitivity theorem for forms in ΩG.Lemma 6.2.Every form in ΩM 4 (2 3 , 0 3 ) − is either primitive or the pullback of a form in ΩM 1 .
Let J be the unique involution of X fixing Z(ω), and let j be the hyperelliptic involution of Y ; it is the unique involution fixing Z(η).Since J * (ω) = −ω, and j * (η) = −η, the map p is odd; that is, p(J(x)) = j(p(x)) for all x ∈ X.Thus (X, ω 2 )/J is the pullback of the quadratic differential (Y, η 2 )/j.Since the latter is not in QM 0,4 , this contradicts Lemma 6.1.

The Kobayashi metric on F
In this section we give a more geometric proof that F is primitive, by showing that T F is not isomorphic to any traditional Teichmüller space.The proof is a variation on a theme of Royden; it is based on an analysis of the Kobayashi metric.
The Kobayashi metric.Recall that T F denotes an irreducible component of the preimage of F in T 1,3 .Since T F is totally geodesic, it is a smooth, contractible complex manifold; in fact T F is homeomorphic, via the exponential map, to its tangent space at any point.We can regard T F as the universal cover of the surface ‹ F introduced in Section 3. It is well known that the Kobayashi and Teichmüller metrics agree on T g,n [Gar87,§7].Similarly: The Kobayashi and Teichmüller metrics agree on T F , and the inclusion T F ⊂ T 1,3 is an isometry.
In fact, any two distinct points x, y ∈ T F lie on a unique complex geodesic D ⊂ T 1,3 ; since T F is totally geodesic, we have D ⊂ T F ⊂ T 1,3 ; and since inclusions are contractions, D is also a Kobayashi geodesic in T F .
The vertices of a norm.Let V be a two-dimensional normed complex vector space.We say V has n vertices if the set of v ∈ V where the function v is real-analytic is the complement of n distinct lines through the origin.
To distinguish T F from T g,n , we will show: Theorem 7.1.The cotangent space at any point in T 0,5 has five vertices in the Teichmüller norm, while the cotangent space at a typical point in T F has six.
Pairs of quadratic differentials.The following general result describes points where the Teichmüller norm is not smooth.Consider quadratic differentials q 0 , q 1 ∈ Q(X, P ), where (X, P ) ∈ M g,n .Fix x ∈ X, and let m i = ord x (q i ) denote the order of vanishing of q i at x (or −1 if q i has a simple pole there).
Proof.Note that |q 0 + tq 1 | is a convex function of t ∈ R, and if a sum of convex functions is C 2 , then each function has a bounded (distributional) second derivative.Thus it suffices to show that U |q 0 + tq 1 | has an unbounded second derivative in some neighborhood U of x; this is done in [Roy71, §2] and, allowing simple poles, in [Gar87, §9.4].
Proof of Theorem 7.1.First note that the function q is real-analytic on any stratum of QM g,n , since it can be expressed as a polynomial in the absolute periods of √ q and their complex conjugates.In particular, q is real-analytic on the generic stratum QM 0,5 (−1 5 , 1).Thus for any (X, P ) ∈ M 0,5 , the norm is real-analytic except possibly on the five lines V i ⊂ Q(X, P ) where q = q 0 has only four poles.For any such q 0 , we can find an x ∈ P and a q 1 ∈ Q(X, P ) such that q 1 has a simple pole at x, but q 0 does not.Applying Lemma 7.2 with (m 0 , m 1 ) = (0, −1), we find that q is not C 2 at q 0 and hence Q(X, P ) has five vertices.Now consider the cotangent space QF (A, P, π) to a point (A, P ) ∈ T F .Note that for S ∈ HA, the projection π S : A → P 1 has six simple critical values except in the finitely many cases where S lies on the tangent line to a flex of A. Thus π also has six critical values for a typical point in T F .
Since T F is totally geodesic, the Teichmüller norm on its cotangent space at (A, P ) is the restriction of the Teichmüller norm on Q(A, P ).Recall that for any q = 0 in Q(A, P, π), we have (q) = Z − P where Z is a fiber of π, and that the generic stratum here is QM 1,3 (−1 3 , 1 3 ).Thus the norm is real-analytic except possibly along the six lines V i ⊂ QF (A, P, π) where (q 0 ) = Z − P and the support of Z contains a simple critical point x of π.In this case x ∈ P , and hence m 0 = ord x (q 0 ) = 2, while m 1 = ord x (q 1 ) = 0 for most other q 1 ∈ QF (A, P, π); thus q fails to be C 2 at q 0 by Lemma 7.2.Therefore the normed cotangent space to T F typically has six vertices.
Proof of Theorem 1.2.Suppose T F is isomorphic to a traditional Teichmüller space.Then we have T F ∼ = T 0,5 ∼ = T 1,2 as a complex manifold, since there is only one two-dimensional Teichmüller space up to isomorphism.Since the Kobayashi metric on a space depends only on its complex structure, and agrees with the Teichmüller metric in these cases, this implies that the cotangent bundles of T 0,5 and T F are isomorphic as bundles of normed vector spaces, contradicting Theorem 7.1.
Proof of Corollary 1.3.Since F has codimension one, if it arises from a totally geodesic surface V 0 ⊂ M h,m via a covering construction, then we must have V 0 = M h,m and hence T F ∼ = T h,m , contrary to Theorem 1.2.
Remark.For a visualization of the Teichmüller norm on the cotangent space to M 0,5 , see [Muk17b].

Teichmüller curves in M 4
A Teichmüller curve is geometrically primitive if it is generated by a primitive 1-form (in the sense of Section 6).Every Teichmüller curve arises from a unique geometrically primitive Teichmüller curve via a covering construction [M 06, Th. 2.5].It is useful to work with 1-forms, rather than quadratic differentials, so the class [ω] ∈ H 1,0 (X) can be discussed from the point of view of Hodge theory.
In this section we will show: (I) the locus G D ⊂ M 4 is a finite union of Teichmüller curves; (II) if D is not a square, then every component of G D is geometrically primitive; (III) if the stabilizer of a form in ΩG contains a hyperbolic element γ, then SL(X, ω) is a lattice and (X, ω) ∈ ΩG D for some D with Q(tr γ) = Q( √ D).
These assertions are restatements of Theorems 1.6, 1.7 and 1.8.Aside from (II), the proofs follow the same lines as the proofs in genus two given in [McM03a].
After replacing (X, ω) by another point in its SL 2 (R) orbit, we can assume that Then ψ is a pseudo-Anosov mapping, with invariant foliations defined by the harmonic forms Re(ω) and Im(ω).These forms give a ψ * -eigenbasis for a two-dimensional subspace V ⊂ H 1 (C), with eigenvalues λ and λ −1 .By the theory of pseudo-Anosov mappings, these eigenvalues are simple and all other eigenvalues of ψ * |H 1 (C) have modulus less than λ.
In particular, the eigenvalues of T |H 1 (C) are given by t and t , where t = λ + λ −1 and |t | < t; The corresponding eigenspaces V and V satisfy V = V ⊥ .
The ring Z[T ] is quadratic since T has exactly two eigenvalues.Let D be the discriminant of the maximal quadratic order in End(C) containing Z[T ].Then (X, ω) ∈ ΩG D , and hence SL(X, ω) is a lattice by (I).The field K = Q(tr γ) is the same as the field generated by the eigenvalues of T , so K = Q( √ D).

Explicit polygonal constructions
In this section we will prove Theorems 1.9 and 1.10.The latter gives new examples of Teichmüller curves in M 4 .In the course of the proof we will give explicit equations for ΩG in period coordinates.
Cylinder deformations.We begin with some notation.We write the usual action Let C ⊂ X be a collection of parallel cylinders, and let φ : (X, ω) → (X , ω ) be a PL (piecewise linear) map for the flat metrics |ω| and |ω |.We say (X , ω ) is a cylinder deformation of (X, ω), and write (9.1) (X , ω ) = γ C • (X, ω), if Dφ = γ on C and Dφ = Id on the rest of X (cf.[Wri15]).Note that we must have γ(v) = v on vectors v parallel to ∂C; in particular, if det(γ) = 1, then γ is a shear.Conversely, any linear map γ that fixes vectors parallel to ∂C determines a new form γ C • (X, ω) ∈ ΩM g .A symmetric form in ΩG.Now consider the holomorphic 1-form (X 0 , ω 0 ) ∈ ΩM 4 (2, 2, 2) obtained by gluing together three regular hexagons along parallel edges as indicated in Figure 4.The three zeros of ω are indicated by white, black and gray dots.The edges are oriented to run counter-clockwise around the right and left hexagons, in the complex directions v i , w i ∈ C. We claim: The form (X 0 , ω 0 ) lies in the gothic locus ΩG.
In fact, rotation of each hexagon by 60 • preserves the gluing pattern, and thus descends to an automorphism T : X 0 → X 0 that cyclically permutes the three zeros of ω 0 .If we let J = T 3 and define an elliptic curve by B = X 0 / T 2 , then J * ω 0 = −ω 0 and the degree three quotient map π : X 0 → B sends Z(ω 0 ) to a single point.Thus (X 0 , ω 0 ) belongs to ΩG by the definition in Section 1.
Period coordinates.We can regard v i and w i as linear coordinates on the cohomology group H 1 (X 0 , Z(ω 0 )), since each oriented edge connects two zeros of ω 0 .
In these coordinates, a sheet of ΩG is locally defined by the linear relations: (9.2) v i+3 = −v i and w i+3 = −w i , i = 1, 2, 3; and The first set of equations insures that J * (ω) = ω, while the second insures that ω ∈ H 1 (B) ⊥ .To see that the second condition is correct, just note that v 1 + v 3 + v 5 and w 1 + w 3 + w 5 span the T 2 -invariant subspace of H 1 (X 0 ), and hence represent a basis for H 1 (B).These equations define an open subset of a sheet of ΩG by Theorem 5.2.
Cylinders and H 1 (B).The shaded region in Figure 4 covers a collection of three parallel cylinders, C = C 1 ∪ C 2 ∪ C 3 ⊂ X.We claim: Any cylinder deformation of the form (X, ω) = γ C • (X 0 , ω 0 ) also lies in ΩG.
To see this, it suffices to show that shearing C preserves the period conditions in (9.2).The first set is preserved because J(C) = C.As for the second set, by examining the left hexagon in Figure 4, we see that the relative cycle v = 2v 1 − (v 1 + v 2 + v 6 ) can be represented by two arcs lying entirely in C.But modulo the first set of equations, we have v = v 1 + v 3 + v 5 ; thus v(ω 0 ) = 0, and hence v(ω) = 0. Similarly, the cycle w 1 + w 3 + w 5 can be represented by arcs outside of C, so its vanishing is also preserved by γ C .
Step 2 Step 3 PL maps. Figure 5 gives another presentation of the surface (X 0 , ω 0 ), in which the dark shaded regions cover two more parallel cylinders D = D 1 ∪D 2 ⊂ X 0 .Since J(D) = D, and the cycles v 1 + v 3 + v 5 and w 1 + w 3 + w 5 can be represented by arcs outside of D, deformations of the form γ D • (X 0 , ω 0 ) also lie in ΩG.In fact, by combining γ C , γ D and the action of GL + 2 (R), we can conclude: (9.3) If there exists a PL map φ : (X 0 , ω 0 ) → (X, ω) such that Dφ is constant on C, D and their complement, then (X, ω) ∈ ΩG.
Remark: An open subset of ΩG.Since ΩG is linear in period coordinates, Theorem 1.9 also holds for the open set of complex parameters (a, b) that determine embedded polygons.In fact, if we apply the action of GL + 2 (R) to the resulting forms, then we obtain a dense open subset of ΩG.Figures 6 and 7 show cathedral polygons of this more general type.

Figure 1
Figure1illustrates two copies of a polygon P (a, b) ⊂ C.This polygon is symmetric about the x-axis, and each of its edges has slope 0, ∞ or ±1.Pairs of parallel edges are glued together by translation to form a compact Riemann surface X = P (a, b)/ ∼ of genus four.The edge pairings for P (a, b) can be read off from the condition that regions with the same shade on the right or the left cover cylinders on X.The form dz|P (a, b) descends to a form ω ∈ Ω(X) with three double zeros, coming from the vertices of P (a, b).In Section 9 we will show:

Figure 3 .
Figure 3.The canonical model for X ⊂ P 3 gives six distinguished points Z ∪ P on the cubic curve A.

Figure 5 .
Figure 5. Preparation for cut and paste.

Figure 6 .
Figure 6.Result of cut and paste is a new polygon P .