Degenerations of Abelian Differentials
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smoothability criteria. As applications, we show that the spin parity of holomorphic and meromorphic differentials extends to distinguish twisted canonical divisors in the locus of stable pointed curves of pseudocompact type. We also justify whether zeros and poles on general curves in a stratum of differentials can be Weierstrass points. Moreover, we classify twisted canonical divisors on curves with at most two nodes in the minimal stratum in genus three. Our techniques combine algebraic geometry and flat geometry. Their interplay is a main flavor of the paper.
Key words and phrases:Abelian differential, translation surface, moduli space of curves, limit linear series, spin structure, admissible cover, Weierstrass point.
2010 Mathematics Subject Classification:14H10, 14H15, 14K20
An Abelian differential defines a flat metric on the underlying Riemann surface with conical singularities at its zeros. Varying the flat structure by induces an action on the moduli space of Abelian differentials, called Teichmüller dynamics. A number of questions about the geometry of a Riemann surface boil down to the study of its -orbit, which has provided abundant results in various fields. To name a few, Kontsevich and Zorich ([KZ]) classified connected components of strata of Abelian differentials with prescribed number and multiplicity of zeros. Surprisingly those strata can have up to three connected components, due to hyperelliptic and spin structures. Eskin and Okounkov ([EO]) used symmetric group representations and modular forms to enumerate special -orbits arising from covers of tori with only one branch point, which allows them to compute volume asymptotics of strata of Abelian differentials. Eskin and Masur ([EMa]) proved that the number of families of bounded closed geodesics on generic flat surfaces in a -orbit closure has quadratic asymptotics, whose leading term satisfies a formula of Siegel-Veech type. Eskin, Kontsevich, and Zorich ([EKZ]) further related a version of this Siegel-Veech constant to the sum of Lyapunov exponents under the Teichmüller geodesic flow. In joint work with Möller ([CM1, CM1]) the author applied intersection theory on moduli spaces of curves to prove a nonvarying phenomenon of sums of Lyapunov exponents for Teichmüller curves in low genus. A recent breakthrough by Eskin, Mirzakhani, and Mohammadi ([EMa, EMM]) showed that the closure of any -orbit is an affine invariant manifold, i.e. locally it is cut out by linear equations of relative period coordinates with real coefficients. More recently Filip ([F]) proved that all affine invariant manifolds are algebraic varieties defined over , generalizing Möller’s earlier work on Teichmüller curves ([Mö]).
Despite the analytic guise in the definition of Teichmüller dynamics, there is a fascinating and profound algebro-geometric foundation behind the story, already suggested by some of the results mentioned above. In order to borrow tools from algebraic geometry, the upshot is to understanding degenerations of Abelian differentials, or equivalently, describing a compactification of strata of Abelian differentials, analogous to the Deligne-Mumford compactification of the moduli space of curves by adding stable nodal curves. This is the focus of the current paper.
We use to denote the genus of a Riemann surface or a smooth, complex algebraic curve. Let be a partition of . Consider the space parameterizing pairs , where is a smooth, connected, compact complex curve of genus , and is a holomorphic Abelian differential on such that for distinct points . We say that is the stratum of (holomorphic) Abelian differentials with signature . For a family of differentials in , if the underlying smooth curves degenerate to a nodal curve, what is the corresponding limit object of the differentials? In other words, is there a geometrically meaningful compactification of and can we describe its boundary elements?
The space of all Abelian differentials on genus curves forms a vector bundle of rank , called the Hodge bundle, over the moduli space of smooth genus curves. Let be the Deligne-Mumford moduli space of stable nodal genus curves. The Hodge bundle extends to a rank vector bundle over . If is nodal, the fiber of over can be identified with , where is the dualizing line bundle of . Geometrically speaking, is the space of stable differentials such that has at worst simple pole at each node of with residues at the two branches of every node adding to zero (see e.g. [HMo, Chapter 3.A]).
Thus it is natural to degenerating Abelian differentials to stable differentials, i.e. compactifying in . Denote by the closure of in . For , let be the normalization of . First consider the case when has isolated zeros and simple poles, i.e. it does not vanish entirely on any irreducible component of . Identify with a stable differential on . Suppose that
where the are the zeros of in the smooth locus of , the are the preimages of the node which is not a pole of , and the are the simple poles of on the preimages of the node . Moreover, is the vanishing order of at , and are the vanishing orders of on , respectively. Our first result describes which strata closures in contain such .
In the above setting, we have
Comparing to the signature of , the notation means keeping all unchanged, adding one to all , and getting rid of all . We remark that when vanishes on a component of , we prove a similar result (see Corollary 3.4).
Despite that has a nice vector bundle structure, a disadvantage of compactifying in is that sometimes it loses information of the limit positions of the zeros of , especially if vanishes on a component of the underlying curve. Alternatively, we can consider degenerations in the Deligne-Mumford moduli space of stable genus curves with ordered marked points by marking the zeros of differentials in .
For an ordered partition of , let parameterize pointed stable curves , where is a canonical divisor on a smooth curve . We say that is the stratum of (holomorphic) canonical divisors with signature . If we do not order the zeros, then is just the projectivization of , parameterizing differentials modulo scaling. Denote by the closure of in .
Inspired by the theory of limit linear series [EH1], we focus on nodal curves of the following type in . A nodal curve is of compact type if every node is separating, i.e. removing it makes the whole curve disconnected. A nodal curve is of pseudocompact type if every node is either separating or a self-intersection point of an irreducible component. We call a node of the latter type a self-node or an internal node, since both have been used in the literature. Note that curves of compact type are special cases of pseudocompact type, where all irreducible components are smooth.
For the reader to get a feel, let us first consider curves of compact type with only one node. Suppose such that , where is smooth and has genus , and is a node connecting and . In particular, the marked points are different from . Define
as the sum of zero orders in each component of . Our next result determines when the stratum closure in contains such .
In the above setting, if and only if
for , where stands for linear equivalence.
For curves of (pseudo)compact type with more nodes, we prove a more general result (see Theorem 4.13 and Remark 4.14). In general, we remark that for a pointed nodal curve of pseudocompact type to be contained in , the linear equivalence condition as above is necessary, but it may fail to be sufficient (see Example 4.5 and Proposition 4.6).
For curves of non-pseudocompact type, extra complication comes into play when blowing up a non-separating node and inserting chains of rational curves in order to obtain a regular smoothing family of the curve. We explain this issue and discuss a possible solution in Section 4.5. We also treat certain curves of non-pseudocompact type in low genus by an ad hoc method (see Section 7).
A useful idea is to thinking of the pair
appearing in Theorem 1.2 as a twisted canonical divisor (see Section 4.1), in the sense that each entry is an ordinary canonical divisor on . Note that if , then it is not effective, i.e. the corresponding differential on is meromorphic with a pole. In general, we call such a polar component. Conversely if on a twisted canonical divisor is effective, we call it a holomorphic component.
Therefore, it is nature to enlarge our study by considering meromorphic differentials and their degenerations, also for the sake of completeness. Take a sequence of integers such that , and
We still use to denote the stratum of meromorphic differentials with signature , parameterizing meromorphic differentials on connected, closed genus Riemann surfaces such that
for distinct . We sometimes allow the case by treating as a marked point irrelevant to the differential. Let be the corresponding stratum of meromorphic canonical divisors with signature . As in the case of holomorphic differentials, ordering and marking the zeros and poles, we denote by the closure of in with . As an analogue of Theorem 1.2, we have the following result.
Suppose is a curve of compact type with one node such that has genus and both are polar components. Then if and only if
for , where .
Again, here we treat the pair as a twisted meromorphic canonical divisor on . For curves of (pseudo)compact type with more nodes, we prove a more general result for twisted meromorphic canonical divisors (see Theorem 4.19). We remark that in both holomorphic and meromorphic cases, the upshot of our proof is to establishing certain dimension bounds for irreducible components of moduli spaces of twisted canonical divisors (see Section 4.2).
Note that for special signatures , can be disconnected. Kontsevich and Zorich ([KZ]) classified connected components for strata of holomorphic differentials. In general, may have up to three connected components, distinguished by hyperelliptic, odd or even spin structures. When these components exist, we adapt the same notation as [KZ], using “”, “” and “” to distinguish them. Recently Boissy ([Bo]) classified connected components for strata of meromorphic differentials, which are similarly distinguished by hyperelliptic and spin structures. Therefore, when has more than one connected component, one can naturally ask how to distinguish the boundary points in the closures of its connected components.
For hyperelliptic components, it is well-known that a degenerate hyperelliptic curve in can be described explicitly using the theory of admissible covers ([HMu]), by comparing to the moduli space of stable genus zero curves with marked points, where the marked points correspond to the branch points of a hyperelliptic double cover. In this way we have a good understanding of compactifications of hyperelliptic components. For spin components, the following result distinguishes their boundary points in the locus of curves of pseudocompact type.
Let be a stratum of holomorphic or meromorphic differentials with signature that possesses two spin components and . Then and are disjoint in the locus of curves of pseudocompact type.
However, we remark that in the locus of curves of non-pseudocompact type in , these components can intersect (see Theorem 5.3).
For a point on a genus Riemann surface , if , we say that is a Weierstrass point. The study of Weierstrass points has been a rich source for understanding the geometry of Riemann surfaces (see e.g. [ACGH, Chapter I, Exercises E]). In the context of strata of holomorphic differentials, for example, if is a canonical divisor of such that , then it is easy to see that is a Weierstrass point. Furthermore, the Weierstrass gap sequences of the unique zero of general differentials in the minimal strata were calculated by Bullock ([Bu]). Using techniques developed in this paper, we can prove the following result.
Let be a general curve parameterized in (the non-hyperelliptic components of) . Then is not a Weierstrass point.
This paper is organized as follows. In Section 2, we introduce basic tools that are necessary to prove our results. In Section 3, we consider degenerations of Abelian differentials in the Hodge bundle and prove Theorem 1.1. In Section 4, we consider degenerations of canonical divisors in and prove Theorems 1.2 and 1.3. In Section 5, we consider boundary points of connected components of and prove Theorem 1.4. In Section 6, we study Weierstrass point behavior for general differentials in and prove Theorem 1.5. Finally in Section 7, we carry out a case study by analyzing the boundary of in in detail.
Our techniques combine both algebraic geometry and flat geometry. The interplay between the two fields is a main flavor throughout the paper. For that reason, we will often identify smooth, complex algebraic curves with Riemann surfaces and switch our language back and forth.
Acknowledgements. The author is grateful to Madhav Nori and Anand Patel for many stimulating discussions. The author also wants to thank Matt Bainbridge, Gabriel Bujokas, Izzet Coskun, Alex Eskin, Simion Filip, Sam Grushevsky, Joe Harris, Yongnam Lee, Martin Möller, Nicola Tarasca, and Anton Zorich for relevant conversations and their interests in this work. Quentin Gendron informed the author that he has obtained some of the results in Sections 5 and 7 independently ([G]), the methods being in some cases related, in some cases disjoint, and the author thanks him for communications and comments on an earlier draft of this work. Results in this paper were announced at the conference “Hyperbolicity in Algebraic Geometry”, Ilhabela, January 2015. The author thanks the organizers Sasha Anan’in, Ivan Cheltsov, and Carlos Grossi for their invitation and hospitality.
In this section, we review basic background material and introduce necessary techniques that will be used later in the paper.
2.1. Abelian differentials and translation surfaces
A translation surface (also called a flat surface) is a closed, topological surface together with a finite set such that:
There is an atlas of charts from with transition functions given by translation.
For each , under the Euclidean metric of the total angle at is for some .
We say that is a saddle point of cone angle .
Equivalently, a translation surface is a closed Riemann surface with a holomorphic Abelian differential , not identically zero:
The set of zeros of corresponds to in the first definition.
If is a zero of of order , then the cone angle at is .
Let us briefly explain the equivalence between translation surfaces and Abelian differentials. Given a translation surface, away from its saddle points differentiating the local coordinates yields a globally defined holomorphic differential. Conversely, integrating an Abelian differential away from its zeros provides an atlas of charts with transition functions given by translation. Moreover, a saddle point has cone angle if and only if locally for a suitable coordinate , hence if and only if is a zero of of order . We refer to [Z] for a comprehensive introduction to translation surfaces.
2.2. Strata of Abelian differentials and canonical divisors
Take a sequence of positive integers such that . We say that is a partition of . Define
We say that is the stratum of (holomorphic) Abelian differentials with signature . Using the description in Section 2.1, equivalently parameterizes translation surfaces with saddle points, each having cone angle . By using relative period coordinates (see e.g. [Z, Section 3.3]), can be regarded as a complex orbifold of dimension
where is the number of entries in .
For special partitions , can be disconnected. Kontsevich and Zorich ([KZ, Theorems 1 and 2]) classified connected components of for all . If a translation surface has being hyperelliptic, or , where is a Weierstrass point of in the former or and are conjugate under the hyperelliptic involution of in the latter, we say that is a hyperelliptic translation surface. Note that being a hyperelliptic translation surface not only requires to be hyperelliptic, but also imposes extra conditions to (see [KZ, Definition 2 and Remark 3]).
In addition, for a nonhyperelliptic translation surface , if , then the line bundle
is a square root of , which is called a theta characteristic. Such a theta characteristic along with its parity, i.e.
is called a spin structure. In general, may have up to three connected components, distinguished by possible hyperelliptic and spin structures.
Note that two Abelian differentials are multiples of each other if and only if their associated zero divisors are the same. Therefore, it makes sense to define the stratum of canonical divisors with signature in , denoted by , parameterizing such that is a canonical divisor in . Here we choose to order the zeros only for the convenience of stating related results. Alternatively if one considers the corresponding stratum of canonical divisors without ordering the zeros, it is just the projectivization of . In particular,
2.3. Meromorphic differentials and translation surfaces with poles
One can also consider the flat geometry associated to meromorphic differentials on Riemann surfaces. In this case we obtain flat surfaces with infinite area, called translation surfaces with poles.
For such that , denote by
the stratum of meromorphic differentials parameterizing , where is a meromorphic differential on a closed, connected genus Riemann surface such that has zeros of order and poles of order , respectively. The dimension and connected components of have been determined by Bossy ([Bo, Theorems 1.1, 1.2 and Lemma 3.5]), using an infinite zippered rectangle construction. In particular, if , i.e. if there is at least one pole, then
If we consider meromorphic differentials modulo scaling, i.e. meromorphic canonical divisors, then the corresponding stratum has dimension
As in the case of holomorphic Abelian differentials, can be disconnected due to possible hyperelliptic and spin structures ([Bo, Section 5]), but all connected components of a stratum have the same dimension.
A special case is when has a simple pole at . Under flat geometry, the local neighborhood of can be visualized as a half-infinite cylinder (see [Bo, Figure 3]). The width of the cylinder corresponds to the residue of at .
For a pole of order , one can glue basic domains appropriately to form a flat-geometric presentation (see [Bo, Section 3.3]). Each basic domain is a “broken half-plane” whose boundary consists of a half-line to the left and a paralell half-line to the right, connected by finitely many broken line segments. In particular, the residue of a pole can be read off from the complex lengths of the broken line segments and the gluing pattern.
For example, for the differential gives a zero of order , so locally one can glue half-disks consecutively to form a cone of angle , see Figure 1.
Now let , and the differential with respect to has a pole of degree with zero residue. In terms of the flat-geometric language, the half-disks transform to half-planes (with the disks removed), where the newborn left and right half-line boundaries are identified in pairs by the same gluing pattern, see Figure 2.
Furthermore, varying the positions of the half-line boundaries with suitable rotating and scaling can produce poles of order with arbitrary nonzero residues (see [Bo, Section 2.2]).
2.4. Deligne-Mumford stable curves and stable one-forms
Let be the Deligne-Mumford moduli space of stable nodal genus curves with ordered marked points . The stability condition means that is finite, or equivalently, the normalization of every rational component of contains at least three special points (preimages of a node or marked points). For , denote by the boundary component of whose general point parameterizes two smooth curves of genus and , respectively, glued at a node such that the genus component only contains the marked points labeled by in the smooth locus. For (resp. ), we require that (resp. ) to fulfill the stability condition. The codimension of in is one, so we call it a boundary divisor.
The Hodge bundle is a rank vector bundle on (in the orbifold sense). Formally it is defined as
where is the universal curve and is the relative dualizing line bundle of . Geometrically speaking, the fiber of over is , where is the dualizing line bundle of . If is nodal, then can be identified with the space of stable differentials on the normalization of . A stable differential on is a meromorphic differential that is holomorphic away from preimages of nodes of and has at worst simple pole at the preimages of a node, with residues on the two branches of a polar node adding to zero (see e.g. [HMo, Chapter 3.A]).
2.5. Admissible covers
Harris and Mumford ([HMu]) developed the theory of admissible covers to deal with degenerations of branched covers of smooth curves to covers of nodal curves. Let be a finite morphism of nodal curves satisfying the following conditions:
maps the smooth locus of to the smooth locus of and maps the nodes of to the nodes of .
Suppose for a node and a node . Then there exist suitable local coordinates for the two branches at , and local coordinates for the two branches at , such that
for some , see Figure 3.
We say that such a map is an admissible cover. The reader can refer to [HMo, Chapter 3.G] for a comprehensive introduction to admissible covers. In this paper we will only use admissible double covers of rational curves as degenerations of hyperelliptic coverings of . In particular, the closure of the locus of hyperelliptic curves in is isomorphic to the moduli space of stable genus zero curves with unordered marked points.
2.6. Limit linear series
A linear series on a smooth curve consists of a degree line bundle with a subspace such that . For a point , take a basis of such that the vanishing orders are strictly increasing. We say that is the vanishing sequence of at , which is apparently independent of the choices of a basis. Set . The sequence is called the ramification sequence of .
Now consider a nodal curve . Recall that if removing any node makes the whole curve disconnected, is called of compact type. Equivalently, a nodal curve is of compact type if and only if its Jacobian is compact, which is then isomorphic to the product of Jacobians of its connected components. One more equivalent definition uses the dual graph of a nodal curve, whose vertices correspond to components of the curve and two vertices are linked by an edge if and only if the corresponding two components intersect at a node. It is easy to see that a curve is of compact type if and only if its dual graph is a tree.
Eisenbud and Harris ([EH1]) established a theory of limit linear series as a powerful tool to study degenerations of linear series from smooth curves to curves of compact type. If is a curve of compact type with irreducible components , a (refined) limit linear is a collection of ordinary ’s on each such that if and intersect at a node and if and are the vanishing sequences of and at , respectively, then for all .
Eisenbud and Harris showed that if a family of ’s on smooth curves degenerate to a curve of compact type, then the limit object is a limit linear . Furthermore, they constructed a limit linear series moduli scheme that is compatible with imposing ramification conditions to points in the smooth locus of a curve, came up with a lower bound for any irreducible component of , and used it to study smoothability of limit linear series. They also remarked that the method works for a larger class of curves, called tree-like curves, which we call of pseudocompact type in our context. Recall that a curve is of pseudocompact type, if every node is either separating or a self-node, i.e. arising from the self-intersection of an irreducible component of the curve. Equivalently, a curve is of pseudocompact type if any closed path in its dual graph is a loop connecting a vertex to itself.
We want to apply limit linear series to the situation when canonical divisors with distinct zeros with prescribed vanishing orders degenerate in the Deligne-Mumford moduli space . In this context we need to treat the case of limit canonical series , because on a smooth genus curve a is uniquely given by the canonical line bundle along with the space of holomorphic Abelian differentials. We illustrate its application in some cases (see Example 4.5 and Proposition 4.6). Nevertheless, in general our situation is slightly different, since an element in a stratum of differentials is a single section of the canonical line bundle, not the whole space of sections. In principle keeping track of degenerations of along with a special section could provide finer information, but in practice it seems complicated to work with. Instead, in Section 4.1 we introduce the notion of twisted canonical divisors that play the role of “limit canonical divisors” on curves of pseudocompact type. We also discuss a possible extension of twisted canonical divisors to curves of non-pseudocompact type in Section 4.5.
2.7. Moduli of spin structures
Recall that a theta characteristic is a line bundle on a smooth curve such that , i.e. is a square root of the canonical line bundle. A theta characteristic is also called a spin structure, whose parity is given by . In particular, a spin structure is either even or odd, and the parity is deformation invariant (see [A, Mu]). Cornalba ([Co]) constructed a compactified moduli space of spin curves over , which defines limit spin structures and further distinguishes odd and even parities.
Let us first consider spin structures on curves of compact type. Take a nodal curve with two smooth components and union at a node . Blow up to insert a between and with new nodes for . Such is called an exceptional component. Then a spin structure on consists of the data
where is an ordinary theta characteristic on and is a line bundle of degree one on the exceptional component. Note that the total degree of is
which remains to be one half of the degree of . Since , the parity of is determined by
In other words, is even (resp. odd) if and only if and have the same (resp. opposite) parity. If there is no confusion, we will simply drop the exceptional component and treat as a limit theta characteristic. The same description works for spin structures on a curve of compact type with more nodes, by inserting an exceptional between any two adjacent components, and the parity is determined by the sum of the parities on each non-exceptional component.
If is a nodal curve of non-compact type, say, by identifying to form a non-separating node , there are two kinds of spin structures on . The first kinds are just square roots of , which can be obtained as follows. Take a line bundle on such that . For each parity, there is precisely one way to identify the fibers of over and , such that it descends to a square root of with the desired parity. The second kinds are obtained by blowing up to insert a attached to at and , and suitably gluing a theta characteristic on to on the exceptional component. In this case the parity is the same as that of .
3. Degenerations in the Hodge bundle
In this section we consider degenerations of holomorphic Abelian differentials in the Hodge bundle over . Let us first prove Theorem 1.1. Recall that is the normalization of . Identify with a stable differential on satisfying that
where the are the zeros of in the smooth locus of , the are the preimages of the node which is not a pole of , and the are the simple poles of on the preimages of the node , see Figure 4.
Moreover, is the vanishing order of at , and are the vanishing orders of on , respectively. Then Theorem 1.1 states that is contained in the closure of in the Hodge bundle over .
Proof of Theorem 1.1.
We will carry out two local operations. First, we need to smooth out a holomorphic node with zero order and on the two branches of to two smooth points of zero order and , respectively. Secondly, we need to smooth out simple poles.
Let us describe the first operation. Recall the notation that the preimages of in the normalization are and . In , take two sufficiently small parallel intervals of equal length to connect to a nearby point and connect to a nearby point in reverse directions, cut along the intervals, and finally identify the edges by translation as in Figure 5.
Locally we obtain two new zeros and of order and , respectively. The zero orders increase by one for each, because the cone angles at and at are both , so after this operation the new cone angle at each zero gain an extra . In particular, as long as the interval is small enough but nonzero, it gives rise to a differential on a genus Riemann surface, which preserves the other zero and pole orders of . Now shrinking the interval to a point, this operation amounts to identifying and , thus recovering the stable differential .
The condition implies that both cylinders have the same width (in opposite direction). Truncate the half-infinite cylinders by two parallel vectors (given by the residues) and identify the top and bottom by translation as in Figure 7.
The cylinders become of finite length, i.e., locally the simple pole disappears. This operation is called plumbing a cylinder in the literature, see Figure 8.
In particular, the plumbing operation does not produce any new zeros nor poles. Conversely, extending the two finite cylinders to infinity on both ends, we recover the pair of simple poles. The reader can refer to [W, Section 6.3] for an explicit example of analytically plumbing an Abelian differential at a simple pole.
Now carrying out the two operations locally for all holomorphic nodes and simple poles one by one, we thus conclude that the stable differential can be realized as a degeneration of holomorphic differentials in the desired stratum. ∎
Let on a nodal curve . If such that every is in the smooth locus of , then
We first remark that by assumption cannot have separating nodes. Otherwise if was a connected component of separated by such a node , since the restriction of to is , it would have a base point at , contradicting that has no zero in the nodal locus of . Now let us proceed with the proof of the corollary. Identify with a stable differential on the normalization of . By assumption, has simple poles at preimages of each node of , and hence there is no holomorphic node. The desired result thus follows as a special case of Theorem 1.1. ∎
Nori ([N]) informed the author that the above corollary can also be proved by studying first-order deformations of such .
Let consist of two elliptic curves and meeting at two nodes and . Let and be the preimages of in the normalization of . Let be a section of such that has a double zero at a smooth point and two simple poles at and has a double zero at a smooth point and two simple poles at , see Figure 9.
In other words, in and in with the residue condition for . It follows from Theorem 1.1 that . Note that has two connected components and . In this example, is in the closure of locus of genus three hyperelliptic curves. However, by assumption and are ramification points of the corresponding admissible double cover, hence they are not conjugate under the hyperelliptic involution. We thus conclude that and .
In Theorem 1.1, the zero orders and on both branches of a holomorphic node matter, not only their sum. For example, translation surfaces in cannot degenerate to two flat tori and attached at one point such that both have nonzero area (in this case , hence Theorem 1.1 only implies smoothing into ). This is because the dualizing line bundle restricted to is , which has degree one. In particular, it cannot have a double zero, unless the stable differential vanishes entirely on . However, if we forget the flat structure and only keep track of the limit position of the double zero, using the notion of twisted canonical divisors (Section 4.1) we will see that points in that are -torsions to appear as all possible limits of the double zero.
We have discussed the case when has isolated zeros. If vanishes on a component of , we can obtain a similar result by tracking the zero orders on the branches of the nodes contained in the complement of the vanishing component. For ease of statement, let us deal with the case when vanishes on only one component , where is a connected subcurve of . The general case that vanishes on more components can be similarly tackled without further difficulty.
Let and . Since vanishes on , all are holomorphic nodes. In the normalization , let and be the preimages of contained in and in , respectively, for .
Consider restricted to such that
where are the isolated zeros of in the smooth locus of , are the preimages of the node that is not a pole of and not contained in , and are the simple poles of on the preimages of the node . As before, is the vanishing order of at , are the vanishing orders of on the preimages of the holomorphic node , respectively, and is the vanishing order of on the preimage of contained in .
Suppose the arithmetic genus of is and let be a partition of such that admits a differential with signature , where is the vanishing order of at and are the vanishing orders of at the zeros other than the in .
In the above setting, we have
Consider the nodal flat surface given by on and on union at . Apply the local operations as in the proof of Theorem 1.1 to smooth out and into the desired stratum. Meanwhile, scaling by as , the area of the flat surface tends to zero while on remains unchanged. The limit flat surface restricted to corresponds to the identically zero differential on , hence equal to . We thus obtain as a degeneration of differentials in the desired stratum. ∎
Let consist of two smooth curves and , both of genus two, attached at a node . Let and be the preimages of in the normalization of . Let be a section of , identified with a stable differential on the normalization of , such that and for a smooth point . In this case, . Take and on such that for a smooth point , see Figure 10.
Then we have and in the above notation. By Corollary 3.4, we conclude that .
4. Degenerations in the Deligne-Mumford space
As we have seen, a stable differential in the closure may vanish entirely on a component of the underlying curve. In this case when Abelian differentials in degenerate to it, we lose the information about the vanishing component as well as the limit positions of the zeros. Below we describe a refined compactification that resolves this issue. Note that modulo scaling, an Abelian differential is uniquely determined by its zeros, i.e. the corresponding canonical divisor. Recall that parameterizes canonical divisors with signature . Viewing it as a subset in by marking the zeros, we can take the closure . The question reduces to analyzing which stable pointed curves appear in the boundary of .
Inspired by the theory of limit linear series, we introduce the notion of twisted canonical divisors, first on curves of pseudocompact type. The upshot is that when canonical divisors in degenerate from underlying smooth curves to a curve of pseudocompact type, the limit object in must be a twisted canonical divisor. Conversely, in a number of cases twisted canonical divisors do appear as such limits, but not always. In the end we also discuss how to extend this notion to curves of non-pseudocompact type.
4.1. Twisted canonical divisors
For the reader to get a feel, let us begin with curves of compact type with only one node. Suppose that a curve has a node connecting two smooth components and of genera and , respectively, with . Moreover, suppose that is the limit of a family of smooth genus curves over a punctured disk , see Figure 11.
Let be the universal curve. The dualizing line bundle serves as a limit of canonical line bundles as . However, this limit is not unique. Becuase for any ,
is also a limit of .