Non-varying sums of Lyapunov exponents of abelian differentials in low genus
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmüller geodesic flow is the same for all Teichmüller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmüller curves with various geometrically defined divisors on moduli spaces of curves.
Lyapunov exponents of dynamical systems are often hard to calculate explicitly. For the Teichmüller geodesic flow on the moduli space of Abelian differentials at least the sum of the positive Lyapunov exponents is accessible for two cases. The moduli space decomposes into various strata, each of which carries a finite invariant measure with full support. For these measures the sum of Lyapunov exponents can be calculated using [EMZ03] together with [EKZ]. On the other hand, the strata contain many Teichmüller curves, e.g. those generated by square-tiled surfaces. For Teichmüller curves an algorithm in [EKZ] calculates the sum of Lyapunov exponents, of course only one Teichmüller curve at a time.
On several occasions, one likes to have estimates, or even the precise values of Lyapunov exponents for all Teichmüller curves in the same stratum simultaneously. For example, it is shown in [DHL] that Lyapunov exponents are responsible for the rate of diffusion in the wind-tree model, where the parameters of the obstacle correspond to picking a flat surface in a fixed stratum. One would like to know this escape rate not only for a specific choice of parameters nor for the generic value of parameters but for all parameters.
Zorich communicated to the authors, that, based on a limited number of computer experiments about a decade ago, Kontsevich and Zorich observed that the sum of Lyapunov exponents is non-varying among all the Teichmüller curves in a stratum roughly if the genus plus the number of zeros is less than seven, while the sum varies if this sum is greater than seven.
In this paper we show that a more precise version of this numerical observation indeed is true. More precisely, we treat the moduli space of genera less than or equal to five. For each of its strata – with three spin-related exceptions – we either exhibit an example showing that the sum is varying – the easy part – or prove that the sum is non-varying. The latter will be achieved by showing empty intersection of Teichmüller curves with various geometrically defined divisors on moduli spaces of curves. We remark that each stratum requires its own choice of divisor and its individual proof of disjointness, with varying complexity of the argument. In complement to our low genus results we mention a theorem of [EKZ] that shows that for all hyperelliptic loci the sum of Lyapunov exponents is non-varying.
We now give the precise statement of what emerged out of the observation by Kontsevich and Zorich. Let be a partition of . Denote by the stratum parameterizing genus Riemann surfaces with Abelian differentials that have distinct zeros of order . We say that the sum of Lyapunov exponents is non-varying in (a connected component of) a stratum , if for all Teichmüller curves generated by a flat surface in its sum of Lyapunov exponents equals the sum for the finite invariant measure supported on (the area one hypersurface of) the whole stratum.
For all strata in genus but the principal stratum the sum of Lyapunov exponents is non-varying.
For the principal stratum, the sum of Lyapunov exponents is bounded above by . This bound can be attained for Teichmüller curves in the hyperelliptic locus, e.g., for Teichmüller curves that are unramified double covers of genus two curves and also for Teichmüller curves that do not lie in the hyperelliptic locus.
For the strata with signature , , , , and as well as for the hyperelliptic strata in genus the sum of Lyapunov exponents is non-varying.
For all the remaining strata, except maybe and , the sum of Lyapunov exponents is varying and bounded above by .
We give more precise upper bounds for the sum stratum by stratum in the text. We remark that e.g. for the sharp upper bound is , which is attained for hyperelliptic curves, whereas for all non-hyperelliptic curves in this stratum the sum of Lyapunov exponents is bounded above by . This special role of the hyperelliptic locus is visible throughout the paper.
For , since there are quite a lot of strata, we will not give a full discussion of upper bounds for varying sums, but restrict to the cases where the sum is non-varying.
For the strata with signature , and as well as for the hyperelliptic strata in genus the sum of Lyapunov exponents is non-varying.
For all the other strata, except maybe , the sum of Lyapunov exponents is varying.
We also expect the three unconfirmed cases , and to be non-varying111Recently Yu and Zuo [YZ] confirmed the three cases using filtration of the Hodge bundle. See also [CM, Theorem A.9] for a detailed explanation., but a proof most likely requires a good understanding of the moduli space of spin curves, on which much less is known than on the moduli space of curves.
The above theorems seem to be the end of this non-varying phenomenon. We cannot claim that there is not a single further stratum of genus greater than five and not hyperelliptic, where the sum is non-varying. But while the sum in strata with a single zero is always non-varying for , the sum does vary in both non-hyperelliptic components of the stratum , as we show in Proposition 7.4.
As mentioned above, by [EKZ] for hyperelliptic strata in any genus the sum of Lyapunov exponents is non-varying. This has significance not only in dynamics, but also in the study of birational geometry of moduli spaces. In Theorem 8.1 we mention one application to the extremality of certain divisor classes on the moduli space of pointed curves, which answers a question posed by Harris [Har84, p. 413], [HM98, Problem (6.34)].
We now describe our strategy. One can associate three quantities ’slope’, ’Siegel-Veech constant’ and ’the sum of Lyapunov exponents’ to a Teichmüller curve. Any one of the three determines the other two. Hence it suffices to verify the non-varying property for slopes. To do this, we exhibit a geometrically defined divisor on the moduli space of curves and show that Teichmüller curves in a stratum do not intersect this divisor. It implies that those Teichmüller curves have the same slope as that of the divisor.
The slope of the divisors, more generally their divisor classes in the Picard group of the moduli space, can be retrieved from the literature in most cases we need. In the remaining cases, we apply the standard procedure using test curves to calculate the divisor class.
Frequently, we also need to consider the moduli space of curves with marked points or spin structures, but the basic idea remains the same. For upper bounds of sums of Lyapunov exponents, they follow from the non-negative intersection property of Teichmüller curves with various divisors on moduli spaces.
Technically, some of the complications arise from the fact, that the disjointness of a Teichmüller curve with a divisor is relatively easy to check in the interior of the moduli space, but requires extra care when dealing with stable nodal curves in the boundary.
In a sequel paper [CM] we consider Teichmüller curves generated by quadratic differentials and verify many non-varying strata of quadratic differentials in low genus. These results immediately trigger a number of questions. Just to mention the most obvious ones. What about measures supported on manifolds of intermediate dimension? What about the value distribution for the sums in a stratum where the sum is varying? We hope to treat these questions in the future.
This paper is organized as follows. In Sections 2, 3 and 4 we give a background introduction to moduli spaces and their divisors, as well as to Teichmüller curves and Lyapunov exponents. In particular, in Section 3.3 we study the properties of Teichmüller curves that are needed in the proof and in Section 4.3 we describe the upshot of our strategy. Our main results for , and are proved in Sections 5, 6 and 7, respectively. Finally in Section 8 we discuss an application of the Teichmüller curves in the hyperelliptic strata to the geometry of moduli spaces of pointed curves.
Acknowledgements. This work was initiated when the first author visited the Hausdorff Research Institute for Mathematics in Summer 2010. Both authors thank HIM for hospitality. The evaluation of Lyapunov exponents were performed with the help of computer programs of Anton Zorich and Vincent Delecroix. The authors are grateful to Anton and Vincent for sharing the programs and the data. The first author also would like to thank Izzet Coskun for stimulating discussions on the geometry of canonical curves and thank Alex Eskin for leading him into the beautiful subject of Teichmüller curves. Finally the authors want to thank the anonymous referees for a number of suggestions which helped improve the exposition of the paper.
2. Background on moduli spaces
2.1. Strata of and hyperelliptic loci
Let denote the vector bundle of holomorphic one-forms over the moduli space of genus curves minus the zero section and let denote the associated projective bundle. The spaces and are stratified according to the zeros of one-forms. For and , let denote the stratum parameterizing one-forms that have distinct zeros of order .
Denote by the Deligne-Mumford compactification of . The boundary of parameterizes stable nodal curves, where the stability means the dualizing sheaf of the curve is ample, or equivalently, the normalization of any rational component needs to possess at least three special points coming from the inverse images of the nodes. The bundle of holomorphic one-forms extends over , parameterizing stable one-forms or equivalently sections of the dualizing sheaf. We denote the total space of this extension by .
Points in , called flat surfaces, are usually written as for a one-form on . For a stable curve , denote the dualizing sheaf by . We will stick to the notation that points in are given by a pair with .
For and , let denote the moduli space of quadratic differentials that have distinct zeros or poles of order . The condition ensures that the quadratic differentials in have at most simple poles. Namely, parameterizes pairs of a Riemann surface and a meromorphic section of with the prescribed type of zeros and poles. Pairs are called half-translation surfaces. They appear occasionally to provide examples via the following construction.
If the quadratic differential is not a global square of a one-form, there is a natural double covering such that . This covering is ramified precisely at the zeros of odd order of and at its poles. It gives a map
where the signature is determined by the ramification type. Indeed is an immersion (see [KZ03, Lemma 1]).
There are two cases where the domain and the range of the map have the same dimension:
see [KZ03, p. 637]. In both cases we call the image a component of hyperelliptic flat surfaces of the corresponding stratum of Abelian differentials. Note that for both cases the domain of parameterizes genus zero curves. More generally, if the domain of parameterizes genus zero curves, we call the image a locus of hyperelliptic flat surfaces in the corresponding stratum. These loci are often called hyperelliptic loci, e.g. in [KZ03] and [EKZ]. We prefer to reserve hyperelliptic locus for the subset of (or its closure in , see also Section 2.5) parameterizing hyperelliptic curves and thus specify with ’flat surfaces’ if we speak of subsets of .
2.2. Spin structures and connected components of strata
A spin structure (or theta characteristic) on a smooth curve is a line bundle whose square is the canonical bundle, i.e. . The parity of a spin structure is given by . This parity is well-known to be deformation invariant. There is a notion of spin structure on a stable curve, extending the smooth case (see [Cor89], also recalled in [FV, Section 1]). We only need the following consequence. The moduli space of spin curves parameterizes pairs , where is a theta characteristic of . It has two components and distinguished by the parity of the spin structure. The spin structures on stable curves are defined such that the morphisms and are finite of degree and , respectively, cf. loc. cit.
Recall the classification of connected components of strata in by Kontsevich and Zorich [KZ03, Theorem 1 on p. 639].
Theorem 2.1 ([Kz03]).
The strata of have up to three connected components, distinguished by the parity of the spin structure and by being hyperelliptic or not. For , the strata and with an integer have three components, the component of hyperelliptic flat surfaces and two components with odd or even parity of the spin structure but not consisting exclusively of hyperelliptic curves.
The stratum has two components, and The stratum also has two components, and
Each stratum for or and has two components determined by even and odd spin structures.
Each stratum for has two components, the component of hyperelliptic flat surfaces and the other component .
In all the other cases, the stratum is connected.
Consider the partition . For with div(, the line bundle is an odd theta characteristic. Therefore, we have a natural morphism
Note that contracts the locus where . Similarly one can define such a morphism for even spin structures.
2.3. Picard groups of moduli spaces
Let be the moduli space (treated as a stack instead of the course moduli scheme) of genus curves with ordered marked points and let be the moduli space of genus curves with unordered marked points. We write for the rational Picard group of a moduli stack (see e.g. [HM98, Section 3.D] for more details).
We fix some standard notation for elements in the Picard group. Let denote the first Chern class of the Hodge bundle. Let , be the boundary divisor of whose generic element is a smooth curve of genus joined at a node to a smooth curve of genus . The generic element of the boundary divisor is an irreducible nodal curve of geometric genus . In the literature sometimes is denoted by . We write for the total boundary class.
For moduli spaces with marked points we denote by the relative dualizing sheaf of and its pullback to via the map forgetting all but the -th marked point. For a set we let denote the boundary divisor whose generic element is a smooth curve of genus joined at a node to a smooth curve of genus and the sections in lying on the first component.
The rational Picard group of is generated by and the boundary classes , .
More generally, the rational Picard group of is generated by , , , by and by , , where if and if .
The above theorem essentially follows from Harer’s result [Har83]. The reader may also refer to [Mum77] for a comparison between the rational Picard group of the coarse moduli scheme and of the moduli stack, as well as [AC87] for the Picard group with integral coefficients.
Alternatively, define to be the class with value on any family of stable genus curves with section corresponding to the -th marked point. By induction on , we have the relation (see e.g. [AC87, p. 161] and [Log03, p. 108])
Consequently, a generating set of can also be formed by the , and boundary classes.
For a divisor class in , define its slope to be
For our purpose the higher boundary divisors need not to be considered, as Teichmüller curves generated by Abelian differentials do not intersect for (see Corollary 3.2).
2.4. Linear series on curves
Many divisors on moduli spaces of curves are related to the geometry of linear series. Here we review some basic properties of linear series on curves (see [ACGH85] for a comprehensive introduction).
Let be a genus curve and a line bundle of degree on . Denote by the linear system parameterizing sections of mod scalars, i.e.
If , then . For a (projective) -dimensional linear subspace of , call a linear series . If for a divisor on , we also denote by or simply by the linear system.
If all divisors parameterized in a linear series contain a common point , then is called a base point. Otherwise, this linear series is called base-point-free. A base-point-free induces a morphism . The divisors in this correspond to (the pullback of) hyperplane sections of the image curve. For instance, a hyperelliptic curve admits a , i.e. a double cover of . The following fact will be used frequently when we prove the disjointness of Teichmüller curves with a geometrically defined divisor.
A point is not a base point of a linear system if and only if , where .
By the exact sequence
we know is either equal to or . The former happens if and only if every section of vanishes at , in other words, if and only if is a base point of . ∎
The canonical linear system is a , which induces an embedding to for a non-hyperelliptic curve. The image of this embedding is called a canonical curve. Let be an effective divisor of degree on . Denote by the linear subspace in spanned by the images of points in under the canonical map . The following geometric version of the Riemann-Roch theorem is useful for the study of canonical curves (see [ACGH85, p. 12] for more details).
Theorem 2.4 (Geometric Riemann-Roch).
In the above setting, we have
We will focus on the geometry of canonical curves of low genus. Curves of genus are always hyperelliptic. For non-hyperelliptic curves of genus , their canonical images correspond to plane quartics.
For , a non-hyperelliptic canonical curve in is a complete intersection cut out by a quadric and a cubic. Any divisor in a of spans a line in , by Geometric Riemann-Roch. This line intersects at , hence it is contained in the quadric by Bézout. If the quadric is smooth, it is isomorphic to . It has two families of lines, called two rulings. Any line in a ruling intersects at three points (with multiplicity), hence has two different linear systems corresponding to the two rulings. If the quadric is singular, then it is a quadric cone with a unique ruling, hence has a unique .
For , a general canonical curve is cut out by three quadric hypersurfaces in and it does not have any . On the other hand, a genus curve with a , i.e. a trigonal curve, has canonical image contained in a cubic scroll surface, which is a ruled surface of degree in . By Geometric Riemann-Roch, divisors in the span rulings that sweep out the surface (see e.g. [Rei97, Section 2.10]).
Recall that on a nodal curve , Serre duality and Riemann-Roch hold with the dualizing sheaf in place of the canonical bundle (see e.g. [HM98, Section 3.A] for more details). We also need the following generalized Clifford’s theorem for Deligne-Mumford stable curves (see e.g. [ACGH85, p. 107] for the case of smooth curves and [Cap10, Theorems 3.3, 4.11] for the remaining cases). Following [Cap10, Section 2.1] let be the number of intersection points of with its complement and . A divisor of degree on a stable curve is balanced, if for every irreducible component we have
Theorem 2.5 (Clifford’s theorem).
Let be a stable curve and an effective divisor on with . Then we have
if one of the following conditions holds: (i) is smooth; (ii) has at most two components and is balanced; (iii) does not have separating nodes, and is balanced.
Finally we need to consider the canonical system of a stable curve associated to its dualizing sheaf. This will help us discuss the boundary of Teichmüller curves. Recall the dual graph of a nodal curve whose vertices correspond to its irreducible components and edges correspond to intersections of these components. A graph is called -connected if one has to remove at least edges to disconnect the graph. The following fact characterizes canonical maps of stable curves based on the type of their dual graphs (see [Has99, Proposition 2.3]).
Let be a stable curve of genus . Then the canonical linear system is base point free (resp. very ample) if and only if the dual graph of is two-connected (resp. three-connected and is not in the closure of the locus of hyperelliptic curves).
2.5. Special divisors on moduli spaces
In the application for Teichmüller curves generated by flat surfaces we do not care about the coefficients of for in the divisor classes in , since Teichmüller curves do not intersect those components (see Corollary 3.2). As shorthand, we use to denote some linear combination of for . Similarly, in we use to denote some linear combination of all boundary divisors but . By the same reason we do not distinguish between and for a divisor class, since they only differ by boundary classes in .
The hyperelliptic locus in
Denote by the closure of locus of genus hyperelliptic curves. We call the hyperelliptic locus in . Note that is a divisor if and only if . A stable curve lies in the boundary of if there is an admissible cover of degree two , for some nodal curve whose stabilization is . We refer to [HM98, Section 3.G] for an excellent introduction to admissible covers.
The class of the hyperelliptic locus calculated e.g. in [HM98, p. 188] is given as follows:
hence it has slope .
Divisors of Weierstrass points
Let be the divisor parameterizing a curve with a Weierstrass point. In [Cuk89, (2.0.12) on p. 328], the class of was calculated for all , which specializes as follows:
The theta-null divisor
Consider the divisor parameterizing such that admits an odd theta characteristic whose support contains . The class of was calculated in [Far10, Theorem 0.2], which specializes as follows:
The Brill-Noether divisors
The Brill-Noether locus in parameterizes curves that possesses a . If the Brill-Noether number
then is indeed a divisor. We remark that nowadays is more commonly used to denote the Brill-Noether divisors, but we decide to reserve for the moduli space only.
There are pointed versions of this divisor. Let be a tuple of integers. Let be the locus in of pointed curves with a line bundle of degree such that admits a and . This Brill-Noether locus is a divisor, if the generalized Brill-Noether number
The hyperelliptic divisor and the Weierstrass divisor could also be interpreted as Brill-Noether divisors, but we stick to the traditional notation for them.
The class of the classical Brill-Noether divisor for was calculated in [HM82, p. 24], in particular
If and the class of the Brill-Noether divisor was calculated in [Log03, Theorem 5.4]. It has class
In particular for , it specializes as follows:
For , it specializes as follows:
If and , the class of the divisor was also calculated in [Log03]. It specializes to
Generalizing the calculation of Logan for and to arbitrary weight , one obtains the divisor called in the proof of [Far09, Theorem 4.9]. From the proof one deduces
is a degeneration of the divisor in [Far09]. A partial degeneration is in . It has class
Since this divisor class was not explicitly written out in [Far09], below we give a proof.
Proof of Equation (10).
Using the same logic in the proof of [Far09, Theorems 4.6, 4.9], have non-varying coefficients in , which is in our notation, and in . Hence we have
We have to take into account, because the test curves used below intersect . Let be a general curve of genus five. Take a fixed general point on and move another point along . Call this family . We have
The intersection number can be calculated using [Log03, Proposition 3.4] by setting , and it equals . Note that Logan counts the number of pairs (), which equals , but for our purpose can be either or , so we double the counting. We thus obtain a relation
Now fix a general point and move another point along . Call this family . We have
The intersection number can also be calculated using [Log03, Proposition 3.4] by setting , and it equals . This equals Logan’s counting, since in the pair now has weight , which distinguishes it from . We then obtain another relation
Combining the two relations we conclude that , which completes the proof. ∎
Consider a linear series for a linear subspace of dimension , and the multiplication map
Define the Gieseker-Petri locus
The divisor class of the Gieseker-Petri locus in the case was calculated in [EH87, Theorem 2]. It specializes to
Alternatively, one can describe in as follows. The canonical image of a genus non-hyperelliptic curve is contained in a quadric surface in . Then is the closure of the locus where this quadric is singular (see e.g. [ACGH85, p. 196]).
3. Teichmüller curves and their boundary points
We quickly recall the definition of Teichmüller curves and of square-tiled surfaces which serve as main examples. New results on the boundary behavior of Teichmüller curves needed later are collected in Section 3.3.
3.1. Teichmüller curves as fibered surfaces
A Teichmüller curve is an algebraic curve in the moduli space of curves that is totally geodesic with respect to the Teichmüller metric. There exists a finite unramified cover such that the monodromies around the ’punctures’ are unipotent and such that the universal family over some level covering of pulls back to a family of curves . We denote by a relatively minimal semistable model of a fibered surface of fiber genus with smooth total space. Let be the set of points with singular fibers, hence . See e.g. [Möl06] for more on this setup. By a further finite unramified covering (outside ) we may suppose that the zeros of on extend to sections of . We denote by the images of these sections.
Teichmüller curves arise as the -orbit of special flat surfaces or half-translation surfaces, called Veech surfaces. We deal here with the first case only and denote by a generating flat surface, if its orbit gives rise to a Teichmüller curve. Teichmüller curves come with a uniformization , where is the affine group (or Veech group) of the flat surface . Let denote the trace field of the affine group and let denote the Galois closure of .
The variation of Hodge structure (VHS) over a Teichmüller curve decomposes into sub-VHS
where is the VHS with the standard ’affine group’ representation, are the Galois conjugates and is just some representation ([Möl06, Proposition 2.4]). One of the purposes of our work is to shed some light on what possibilities for the numerical data of can occur.
3.2. Square-tiled surfaces
A square-tiled surface is a flat surface , where is obtained as a covering of a torus ramified over one point only and is the pullback of a holomorphic one-form on the torus. It is well-known that in this case is commensurable to , hence has no Galois conjugates or equivalently, the rank of is .
In order to specify a square-tiled surface covered by squares, it suffices to specify the monodromy of the covering. Take a standard torus by identifying via affine translation the two pairs of parallel edges of the unit square . Consider the closed, oriented paths on the horizontal axis and on the vertical axis. The indices and correspond to ’up’ and ’right’, respectively. Note that and form a basis of , where is a base point in . Going along and induces two permutations on the sheets of a degree cover of . Hence can be regarded as elements in the symmetric group . Conversely, given such a pair , one can construct a degree cover of (possibly disconnected) ramified over one point only. The domain of the covering is connected if and only if the subgroup in generated by acts transitively on the letters. Moreover, the ramification profile over is determined by the commutator .
The surface in Figure 1 corresponds to a degree 5, genus 2, connected cover of the standard torus:
It is easy to see that the monodromy permutations for this square-tiled surface are given by . Here a cycle means the permutation sends to for and sends back to . One can check that . Therefore, the corresponding covering has a unique ramification point marked by with ramification order arising from the length- cycle, since locally the three sheets labeled by get permuted at that point. By Riemann-Hurwitz, the domain of the covering has genus equal to . The pullback of from is a one-form in the stratum .
Based on the monodromy data, one can directly calculate the Siegel-Veech constant as well as the sum of Lyapunov exponents (introduced in the next section) for a Teichmüller curve generated by a square-tiled surface (see [EKZ]). Later on we will use square-tiled surfaces to produce examples of Teichmüller curves that have varying sums of Lyapunov exponents.
3.3. Properties of Teichmüller curves
Here we collect the properties of the boundary points of Teichmüller curves that are needed in the proofs in the subsequent sections. We will use to denote the closure of a Teichmüller curve in the compactified moduli space.
Let be a partition of . If is another partition and if it can be obtained from by successively combining two entries into one, we say that is a degeneration of . For instance, is a degeneration of . Geometrically speaking, combining two entries corresponds to merging two zeros of order into a single zero of order .
Suppose is a Teichmüller curve generated by a flat surface in and let be a degeneration of . Then in is disjoint from .
The claim is obvious over the interior of the moduli space. We only need to check the disjointness over the boundary. The cusps of Teichmüller curves are obtained by applying the Teichmüller geodesic flow to the direction of the flat surface in which decomposes completely into cylinders. The stable surface at the cusp is obtained by ’squeezing’ the core curves of these cylinders. This follows from the explicit description in [Mas75]. Since the zeros of are located away from the core curves of the cylinders, the claim follows. ∎
For a nodal curve, a node is called separating if removing it disconnects the curve.
The section of the canonical bundle of each smooth fiber over a Teichmüller curve extends to a section of the dualizing sheaf for each singular fiber over the closure of a Teichmüller curve. The signature of zeros of is the same as that of . Moreover, does not have separating nodes. In particular, does not intersect for .
The first statement follows from the description in the preceding proof. The fact that does not have separating nodes is a consequence of the topological fact that a core curve of a cylinder can never disconnect a flat surface. It implies that does not intersect the boundary divisors for on , because by definition a curve parameterized in for possesses at least one separating node. ∎
For Teichmüller curves generated by a flat surface in the degenerate fibers are irreducible.
For Teichmüller curves generated by a flat surface in , with both odd, the degenerate fibers are irreducible or consist of two components of genus for joined at nodes for an odd number such that .
Let be a degenerate fiber and a component of . The dualizing sheaf of restricted to has positive degree equal to , where is the intersection number of with its complement in . For the case , by Corollary 3.2 it implies that only has one component, hence it is irreducible. For the case , it implies that is either irreducible or has two components . For the latter suppose contains the -fold zero. By assumption is odd, hence is also odd. ∎
Let be a Teichmüller curve generated by a flat surface in . Suppose an irreducible degenerate fiber over a cusp of is hyperelliptic. Then is hyperelliptic, hence the whole Teichmüller curve lies in the locus of hyperelliptic flat surfaces.
Moreover, if and is not hyperelliptic, then no degenerate fiber of the Teichmüller curve is hyperelliptic.
The last conclusion does not hold for all strata. For instance, Teichmüller curves generated by a non-hyperelliptic flat surface in the stratum always intersect the hyperelliptic locus at the boundary, as we will see later in the discussion for that stratum.
As motivation for the proof, recall why a Teichmüller curve generated by with hyperelliptic stays within the corresponding locus of hyperelliptic flat surfaces. The hyperelliptic involution acts as on all one-forms, hence on . In the flat coordinates of given by Re and , the hyperelliptic involution acts by the matrix . The Teichmüller curve is the -orbit of and is in the center of . So if admits a hyperelliptic involution, so does for any .
Suppose the stable model of the degenerate fiber is irreducible of geometric genus with pairs of points identified. This stable curve being hyperelliptic means that there exists a semi-stable curve birational to that admits a degree two admissible cover of the projective line. In terms of admissible covers, this is yet equivalent to require that the normalization of is branched at branch points over a main component (i.e. the image of the unique component not contracted under that passage to the stable model) with covering group generated by an involution and, moreover, for each of the nodes there is a projective line intersecting in and with two branch points.
In the flat coordinates of given by , the surface consists of a compact surface with boundary of genus and half-infinite cylinders (corresponding to the nodes) attached to the boundary of . We may define canonically, by sweeping out the half-infinite cylinder at (or ) with lines of slope equal to the residue (considered as element in ) of at until such a line hits a zero of , i.e. a singularity of the flat structure.
With this normalization, the above discussion shows that for irreducible stable curves the hyperelliptic involution exchanges the half-infinite cylinders corresponding to and and it defines an involution of . As in the smooth case, acts as on .
To obtain smooth fibers over the Teichmüller curve (in a neighborhood of ) one has to glue cylinders of finite (large) height in place of the half-infinite cylinders of appropriate ratios of moduli. The hypothesis on acting on and on the half-infinite cylinders implies that is a well-defined involution on the smooth curves. Moreover, has two fixed points in each of the finite cylinders and fixed points on , making fixed points in total. This shows that the smooth fibers of the Teichmüller curve are hyperelliptic.
To complete the proof we have to consider the two-component degenerations for by Corollary 3.3. In all these cases, the hyperelliptic involutions can neither exchange the components (since the zeros are of different order) nor fix the components (since the zeros are of odd order).
For a hyperelliptic involution cannot fix the component, since is odd. It cannot exchange the two components and exchange a pair of half-infinite cylinders that belong to different nodes, since could then be used to define a non-trivial involution for each component. This involution fixes the zeros and this contradicts that is odd. If exchanges all pairs of half-infinite cylinders that belong to the same node, has two fixed points in each cylinder on the smooth ’opened up’ surface. Now we can apply the same argument as in the irreducible case to conclude that the ’opened up’ flat surfaces are hyperelliptic as well.
For a hyperelliptic involution can neither fix the component with the (unique) zero of order three, since 3 is odd, nor map it elsewhere, since the zeros are of different order. ∎
4. Lyapunov exponents, Siegel-Veech constants and slopes
4.1. Lyapunov exponents
Fix an -invariant, ergodic measure on . The Lyapunov exponents for the Teichmüller geodesic flow on measure the logarithm of the growth rate of the Hodge norm of cohomology classes during parallel transport along the geodesic flow. More precisely, let be the restriction of the real Hodge bundle (i.e. the bundle with fibers ) to the support of . Let be the lift of the geodesic flow to via the Gauss-Manin connection. Then Oseledec’s theorem shows the existence of a filtration
by measurable vector subbundles with the property that, for almost all and all , one has
where is the maximum value such that is in the fiber of over , i.e. . The numbers for are called the Lyapunov exponents of . Note that these exponents are unchanged if we replace the support of by a finite unramified covering with a lift of the flow and the pullback of . We adopt the convention to repeat the exponents according to the rank of such that we will always have of them, possibly some of them equal. Since is symplectic, the spectrum is symmetric, i.e. . The reader may consult [For06] or [Zor06] for a more detailed introduction to this subject.
Most of our results will be about the sum of Lyapunov exponents defined as
This sum depends, of course, on the measure chosen and we occasionally write to emphasize this dependence. In particular, one defines Lyapunov exponents for an -invariant suborbifold of carrying such a measure . We will focus on the case of a Teichmüller curve . Consequently, we use to denote the sum of its Lyapunov exponents.
The bridge between the ’dynamical’ definition of Lyapunov exponents and the ’algebraic’ method applied in the sequel is given by the following result. Note that if the VHS splits into direct summands one can apply Oseledec’s theorem to the summands individually. The full set of Lyapunov exponents is the union (with multiplicity) of the Lyapunov exponents of the summands.
4.2. Lyapunov exponents for loci of hyperelliptic flat surfaces
We recall a result of [EKZ, Section 2.3] that deals with the sum of Lyapunov exponents for Teichmüller curves generated by hyperelliptic curves, more generally for any invariant measure on loci of hyperelliptic flat surfaces. It implies immediately that hyperelliptic strata are non-varying.
Theorem 4.2 ([Ekz]).
Suppose that is a regular -invariant suborbifold in a locus of hyperelliptic flat surfaces of some stratum . Denote by the orders of singularities of the underlying quadratic differentials on the quotient projective line.
Then the sum of Lyapunov exponents for is
where, as usual, we associate the order to simple poles.
Hyperelliptic strata are non-varying. For a Teichmüller curve generated by we have
4.3. Siegel-Veech constants, slopes and the sum of Lyapunov exponents
Write for a partition of . Let