The image in the moduli space of curves of strata of meromorphic and quadratic differentials

We compute the dimension of the image of the forgetful map from $Z$ to the moduli space of curves, where $Z$ is a connected component of the stratum of $k$-differentials with an assigned partition $\mu$, for the cases when $k=1$ with meromorphic partition and $k=2$ when the quadratic differentials have at worst simple poles.


Introduction
For positive integers g and k, let µ = (m 1 , . . . , m n ) a partition of (2g − 2)k and define H k g (µ) ⊆ M g,n to be the moduli space of k-canonical divisors of type µ, parametrizing pointed curves [C, p 1 , . . . , p n ] satisfying O C ( n i=1 m i p i ) = ω k C . It is natural to consider a connected component Z of H k g (µ) and ask what is the dimension of the image of the forgetful map π Z : Z → M g . We answer this question in the cases k = 1 with meromorphic partition and k = 2 when the quadratic differentials have at worst simple poles. Consequently, our results will provide new divisors along with higher codimension cycles on the moduli space M g . To underline the importance of such cycles we point out that for k = 1 and µ = (2, 2, . . . , 2) the image of the even component is the divisor of curves with a vanishing theta null, see [Tei88].
We start with the case k = 1 and drop the superscript k from the notation of the moduli of canonical divisors. It is obvious that if µ has a unique negative entry, equal to −1, the stratum H g (µ) is empty. We will assume in what follows that the partition µ is not of this form and prove the following result: Theorem 1. For g ≥ 2, let µ be a strictly meromorphic partition of 2g − 2 of length n and Z a connected, non-hyperelliptic component of H g (µ). Then the dimension of the image of the forgetful map π Z : Z → M g is the expected one, that is, min {2g + n − 3, 3g − 3}.
Together with the case of holomorphic differentials, treated by Gendron in [Gen18], and the obvious case of hyperelliptic components, Theorem 1 completely answers the question for strata with k = 1. As a consequence, every stratum H g (µ) has a connected component Z whose dimension of the image in M g is equal to min {dim(Z), 3g − 3}.
We prove a similar theorem for the case k = 2 when the poles are at worst of order 1. When we have µ = (2m 1 , . . . , 2m n ) a positive partition of 4g − 4, the stratum H 2 g (µ) contains the components of H g ( µ 2 ). We will denote by Q g (µ) the union of connected components of H 2 g (µ) that are not components of H g ( µ 2 ). When µ has at least one odd entry we make the convention Q g (µ) = H 2 g (µ). If µ is a partition of 4g − 4 as above, there are four cases when Q g (µ) is empty; namely g = 1 and µ = (1, −1), or µ = (0) and the cases when g = 2 and µ = (3, 1) or µ = (4). We assume in the next theorem that µ is not one of these partitions.
In this case, almost any non-hyperelliptic component of a stratum has dimension of the image in M g equal to min {2g − 3 + n, 3g − 3} with a unique exception in genus 4. It was proven by Chen and Möller in [CM12] that if µ = (3, 3, 3, 3), the stratum Q 4 (3, 3, 3, 3) has two non-hyperelliptic connected components distinguished by whether the value h 0 (C, p 1 + p 2 + p 3 + p 4 ) of a pointed curve [C, p 1 , p 2 , p 3 , p 4 ] in the stratum is 1 or 2. As this value plays a fundamental role in our proof, we get that the component Q Irr 4 (3, 3, 3, 3) with associated value 2, has dimension of the image one less than expected. We prove the following: Theorem 2. Let g ≥ 2 and µ be a partition of 4g − 4 with entries either positive or −1. Then if Z is a connected component of Q g (µ) that is not hyperelliptic, the dimension of the image of the forgetful map π Z : Z → M g is min {2g − 3 + n, 3g − 3}, with the unique exception Z = Q Irr 4 (3, 3, 3, 3) in genus 4, when the dimension of the image is 2g − 4 + n = 8.
Another case for which the answer is known is when the length of the partition µ is at least g ≥ 3 and H k g (µ) has a unique irreducible component, see [Bar18a] and [Bar18b]. This case will be used to simplify the proofs of the two theorems. In this paper we rely on the description of the forgetful map π µ : H k g (µ) → M g at the level of tangent spaces appearing in [BCGGM19], [Mon] and on a degeneration argument that ensures the locus where the tangent map is injective is non-empty.
It is known that the components of the Deligne-Mumford strata H k g (µ) are smooth of dimension 2g − 2 + n when k = 1 and µ is holomorphic and 2g − 3 + n if the component is not one corresponding to k-th powers of holomorphic abelian differentials. We refer the reader to [BCGGM19] and [Sch18] for an account of these results using deformation theory.
The Deligne-Mumford compactification of the strata H k g (µ) will play an important role in this article, as we will often degenerate to a singular curve in the boundary of M g,n inside M g,n . Results in this direction appear in [FP18], where it is shown that the compactification is contained as a component in space of twisted canonical divisors. An exhaustive description of the curves in the boundary of the strata was achieved in [BCGGM18] for k = 1 and [BCGGM19] for k at least 2.
The last important ingredient we need is the description of the connected components of H k g (µ). Unfortunately, the answer is not known for k ≥ 3. The case of holomorphic abelian differentials was treated by Kontsevich and Zorich in [KZ03], the meromorphic case was treated by Boissy in [Boi12] and the case of k = 2 when the partition µ has entries ≥ −1 was studied by Lanneau in [Lan05]. Lanneau's list of sporadic strata was found to be inexhaustive and was completed by Chen and Möller in [CM12]. We know that in the case k = 1, additional components appear because of the hyperelliptic and spin structure while for k = 2 with entries at least −1, such components appear due to hyperellipticity. Acknowledgement I would like to thank my advisor Gavril Farkas for all the support he offered me along the way and for introducing me to this beautiful topic. I am also grateful to Ignacio Barros and Dawei Chen for contributing with insightful ideas.

General approach
Let µ = (m 1 , . . . , m n ) a partition of (2g − 2)k of length n. We consider the Deligne-Mumford substack H k g (µ) ⊂ M g given by We see from Theorem 1 and Theorem 2 that the majority of connected components have image of dimension min {2g − 3 + n, 3g − 3} in M g . As we will encounter such components a lot, we make the convention: a connected component of the stratum and π Z : Z → M g the forgetful map. We say that Z is of the expected image dimension if the image of the forgetful map π Z has dimension equal to min {dim(Z), 3g − 3} The next proposition is fundamental in the proofs of Theorems 1 and 2, as it offers information about the behaviour of the map from H k g (µ) forgetting any of the sections.
where π is the map forgetting the marked points p i for i ∈ B.
For a point [C, p 1 , . . . , p n ] ∈ H k g (µ), the map at the level of tangent spaces We will now use the characterization in [BCGGM19] and [Mon] of the tangent space of H k g (µ) at a point. Take the map: Then the tangent map is the dashed map in the following diagram induced by the map β: By Serre duality, the dashed map above has d-dimensional kernel if and only if the dashed map in the next diagram has d-dimensional cokernel: The map β ∨ : We consider the case when ϕ is not the k-th power of a holomorphic abelian differential. The same method with slight and obvious modifications can be applied to the holomorphic abelian case.
We denote by a and b the maps labelled as such in the second diagram. As the map a is injective, we use the exactness of the diagram to deduce that coker The problem translates into understanding when elements of the space H 0 (C, ω 1−k C ( n i=1 m i p i )) are mapped to quadratic differentials that are holomorophic outside the points p i for i ∈ A.
Let s ∈ H 0 (C, ω 1−k C ( n i=1 m i p i )) and denote by −n i its order of vanishing at p i , where the order is negative for poles. We also have from the definition that the k-differential ϕ has order −m i at p i . It is a well known fact that if a function f has order n = 0 at a point p, then the differential df has order n − 1 at p. It follows from this discussion and from the definition of β ∨ that β ∨ (s) has order at p i equal to it follows that m i ≥ n i for all i = 1, . . . , n. The quadratic form β ∨ (s) is holomorphic at the point p j for j ∈ B if and only if we have n j ≤ m j − 1. It follows that s is sent by H 0 (β ∨ ) to Im(a) if and only if The conclusion follows, as the Riemann-Roch Theorem implies that the right-hand-side is Using the well-known relation between the dimension of the generic fiber and the rank of the generic tangent map, we deduce as a straight forward application of 4 the following: Then the dimension of the generic fiber of the map π Z,A : As it is not clear how to find smooth n-pointed curves [C, p 1 , . . . , p n ] satisfying h 0 (C, p 1 + . . . + p n ) = d, we degenerate to a nodal curve with one node and two irreducible components. We explain below, using the Riemann-Roch theorem, how this condition degenerates in this case.
By taking the long exact sequence associated to the short exact sequence This observation is essential as this equivalent form is a rank condition for a vector bundle morphism and can be extended to M g,n . This paves the way for using a degeneracy argument to deduce the dimension of a generic fiber of π.
Consider a curve C = C 1 ∪ C 2 consisting of two irreducible components glued together at exactly one node. We know that In particular, by denoting We will only be interested in the case d = 1, where we have the following: Lemma 6. With the above notations the map H 0 (C, ω C ) → H 0 (C, ω C /ω C (− i∈A p i )) is surjective if and only if the following two maps are surjective:

Meromorphic strata
As explained previously, the condition h 0 (C, i∈A p i ) = 1 can be interpreted as a maximal rank condition for a morphism of vector bundles on M g,n , which is an open condition. To deduce that this condition is generically satisfied on a component Z of H k g (µ) it is enough to find a nodal curve C on Z respecting the surjectivity in Lemma 6. We make this argument explicit in this next proposition, which will be fundamental in proving Theorem 1.
Proposition 7. Let µ a strictly meromorphic partition of 2g − 2 of length n such that H g (µ) is nonempty. Then for every subset A ⊆ {1, 2, . . . , n} of cardinality |A| ≤ g there exists an n-pointed marked curve Proof. We prove this result by induction on the genus. The proposition is clear for genus g = 1.
The case of g = 2 is treated in the Appendix and we assume that g ≥ 3. Using Proposition 4.20 from [Bar18a] and Corollary 5 we observe that the proposition is true for the case n ≥ g. In particular, we can assume without loss of generality that n ≤ g − 1 and A = {1, 2, . . . , n}. As previously explained, what we need to prove is equivalent to the surjectivity of the map By Lemma 6, it is enough to find a pointed curve with an unique node [C = C 1 ∪ C 2 , p 1 , . . . , p n ] ∈ H g (µ) that satisfies the conditions: In order to see that such a curve exists in the compactification, observe that there exists an m i ≥ 3 and use the clutching: Here we glue along the last markings of the two strata. By the description of the compactification in [BCGGM18] we conclude that the image of the clutching is contained in the closure of H g (µ). The conclusion follows by the induction hypothesis.
We say a partition µ is of even type if all its entries are even or if it is of the form (2m 1 , . . . , 2m n , −1, −1) with m i positive for all i. We study the case when µ is of even type and deduce a similar statement as the one before. This will require a little more care, as we would need to use a degeneration argument that keeps track of the parity of the spin structure.
Proposition 8. Let µ a partition of length n and even type. Take Z a connected component of H g (µ) and A ⊆ {1, 2, . . . , n} a subset of cardinality |A| ≤ g. Assume we are not in one of the following exceptional cases: i) Z is a hyperelliptic component ii) Z = H even g (2, 2, . . . , 2, −1, −1) and A = {1, 2, . . . , g} iii) Z = H odd g (2, 2, . . . , 2, −2) and A = {1, 2, . . . , g} Then there exists a pointed curve [C, p 1 , . . . , p n ] ∈ Z such that: Proof. We proceed as in the previous proof. We first observe that the statement is obviously true for the case g = 1 and assume it is true by induction for the case of genus up to g − 1. Without any loss of generality we can assume |A| = min {g, n}.
In the cases ii) and iii), the proposition is true for the other connected component by Proposition 7, so we can assume we are not in this setting for µ and A.
As we excluded cases i-iii) for Z, we are in one of the following three cases: a) There exists i ∈ A such that m i ≥ 4 b) g = 2 with µ = (4, −1, −1) and A = {2, 3} or µ = (2, 2, −1, −1) and A = {3, 4} c) There exist i ∈ A and B ⊂ {1, 2, . . . , n} in the complement of A such that m i is even, m i + j∈B m j ≥ 4 and m j ≥ 2 for all j ∈ B.
Notice first that the proposition is true for the partitions in case b) by the previous proposition, as the strata have two connected components and the statement is false for the hyperelliptic one. For the other cases we take the following clutchings where the glueings are along the last markings. They correspond to the cases a) and c) respectively: The strata in genus 1 on the left of the clutchings have both even and odd components and the strata in genus g − 1 on the right satisfy the induction hypothesis for at least one component by Proposition 7. The conclusion follows as long as we can obtain singular curves as in Lemma 6 that lay in the even and odd components respectively. Using the parity description for spin structures on curves with a unique node in [Cor89], this is clearly the case when H g (µ) has exactly two connected components that are distinguished by parity. This is the case when µ is not of hyperelliptic type or when g = 2 and µ is hyperelliptic.
As the genus 2 case is completely covered, we assume g ≥ 3. The proposition is also true in the cases n = 3, 4 with µ of hyperelliptic type, as otherwise the hyperelliptic component would have expected image dimension. We are left with the cases µ of hyperelliptic type and n = 1 or 2. The case of n = 1 is obvious and we now treat the second case: Assume that a component Z of H g (2n, −2m) that is not hyperelliptic satisfy h 0 (C, p 1 + p 2 ) = 2 for all points [C, p 1 , p 2 ] ∈ Z. It follows that C is hyperelliptic and p 1 + p 2 is a g 1 2 . By hypothesis we have 2np 1 − 2mp 2 = (g − 1)g 1 2 Adding 2mp 1 + 2mp 2 = 2mg 1 2 to both sides we get 2(n + m)p 1 = (g − 1 + 2m)g 1 2 Hence we have a finite number of choices for p 1 and p 2 . From this it follows that the map π Z is finite, contradicting the assumption because of Corollary 5 It is useful to observe that the proposition is true for µ = (2, 2, . . . , 2, −1, −1) or (2, 2, . . . , 2, −2) when A = {1, 2, . . . , g − 1, g + 1}.
Let us now explain how the condition in propositions 7 and 8 is sufficient to deduce that the component Z is of the expected image dimension. This will follow as a trivial application of the Riemann-Roch Theorem.
Proof of Theorem 1: Take Z a nonhyperelliptic component of H g (µ). Using Propositions 7 and 8 it follows that we can find A ⊆ {1, 2, . . . , n} of cardinality min {g, n} and a smooth pointed curve [C, p 1 , . . . , p n ] in Z such that h 0 (C, It then follows that: Theorem 1 follows from Corollary 5.

Quadratic strata
There is nothing fundamentally different in this case from the meromorphic one. However, this case has its particularities as there is no parity to be taken into account, there are some sporadic strata that have to be considered separately and there are some empty strata in low genus that need to be accounted when applying the induction. All this information can be found in [CM12] and [Lan05]. We proceed to prove the analogue in the quadratic case of Proposition 7: Proposition 9. Let µ = (m 1 , . . . , m n ) be a partition of 4g − 4 with all entries either -1 or positive, such that Q g (µ) is non-empty. Then for every subset A ⊆ {1, 2, . . . , n} of cardinality |A| ≤ g there exists an element [C, p 1 , . . . , p n ] ∈ Q g (µ) such that: with the exception of the case µ = (2, 1, 1) and A = {2, 3} when the only component is the hyperelliptic one.
Proof. We first observe that for any partition µ of 0 different from (0) and (1, −1), the stratum Q 1 (µ) is non-empty. Again, we consider this case to be the initial step and proceed by induction on the genus g. The genus 2 case will be treated in the Appendix.
Suppose that the proposition is true for genus up to g − 1 and let us prove it for g ≥ 3. Again, by proposition 4.20 in [Bar18a] we see that the proposition is true when n ≥ g and hence it is enough to treat the case n ≤ g − 1 and A = {1, 2, . . . , n}.
In this case it follows that there exists m i ≥ 4 and without loss of generality we assume that i = 1. Then we take the clutching along the last markings , the image is in the closure of Q g (µ). This map is well-defined as long as both the strata in the clutching are non-empty. If the stratum in genus g − 1 on the right is non-empty and not Q 2 (1, 1, 2), we can apply the induction hypothesis as in Proposition 7 and the conclusion follows.
As we will see in the Appendix that the proposition is true in these two cases and in the case g = 2, the conclusion follows.
Proof of Theorem 2: Analogously to the proof of Theorem 1, it follows from Proposition 9 that every stratum has a connected component of the expected image dimension. In particular, the only cases left to consider are those when H g (µ) has more than one connected component.
The cases when n ≥ 3 and µ is hyperelliptic also follow as the hyperelliptic component is not of the expected image dimension.
We claim that Theorem 2 holds for these sporadic strata and hence the conclusion follows.

Sporadic quadratic strata
It is fruitful to consider for these cases a proof by contradiction. This is due to Corollary 5, as an assumption on the dimension of the fibers provides an extra divisorial information for the marked points.
Using the Riemann-Roch Theorem we deduce that h 0 (C, K C − p − q − r) = 1. In particular, there exists a point s ∈ C such that: p + q + r + s = K C Using this relation, together with 6p + 3q − r = 2K C we can deduce: There are three possible cases: i) Generically s = p, case in which we have 4r = K C and hence we have finite number of choices for r and by our assumption, the composition map Q 3 (6, 3, −1) → H 3 (4) → M 3 has one-dimensional fibers, and hence by Corollary 5 we have p + q = g 1 2 . It follows that 3p − r = g 1 2 and hence p is a Weierstrass point. Then p = q, contradicting the fact that they are distinct. ii) Generically s = r and then the map from Q 3 (6, 3, −1) → M 3,2 forgetting the marking q has image H 3 (7, −3) which is of the expected image dimension by Theorem 1.
Since all possible cases yield the same result, we conclude that the connected components of Q 3 (6, 3, −1) have image of dimension 2g − 3 + n = 6 in M 3 .
Case 2: µ = (3, 3, 3, −1) We take Z a connected component of Q 3 (3, 3, 3, −1) and assume that the map π Z : Z → M 3 does not have one dimensional fibers as expected. It follows that it must have two dimensional fibers and then, by Corollary 5 we have that for all points [C, p, q, r, s] ∈ Z we have: h 0 (C, p + q + r + s) = 3 By Clifford's Theorem, it follows that C is hyperelliptic and p + q + r + s = 2g 1 2 . Without loss of generality assume that p, q and r, s are pairs of conjugate points for the hyperelliptic involution. It then follows that 3r − s = g 1 2 and hence r is a Weierstrass point, hence r = s, contradicting the fact that they are distinct.
Case 3: µ = (3, 3, 3, 3) In this case, according to [CM12] the two non-hyperelliptic connected components are differentiated by whether the value of h 0 (C, p 1 + p 2 + p 3 + p 4 ) is 1 or 2. This value is exactly what we need to deduce the dimension of the generic fiber. Hence one of the two components is dominant over M 4 while the other, denoted Q Irr 4 (3, 3, 3, 3) has generically one dimensional fibers.
Remark 10. The image of the component Q Irr 4 (3, 3, 3, 3) provides an interesting divisor on the moduli space M 4 , namely the locus of curves [C] that have a g 1 4 denoted by A with the property that 3A = 2K C . By observing that h 0 (K C − A) = 1 we deduce that the locus is also the image of the map H nonhyp 4 (3, 3) → M 4 . The class in Pic(M 4 ) ⊗ Q of its closure was computed in [Mul16].
In the case when h 0 (C, 2p 1 + p 2 + p 3 ) = 2 we observe that for every curve in the divisor in Remark 10 there exist p 1 , p 2 , p 3 ∈ C such that 2p 1 + p 2 + p 3 = A. As both the component and the divisor are irreducible of the same dimension and the divisor is in the image of the projection to M 4 it follows that it is the image of the forgetting map and hence the map is generically finite.

Appendix: Particular cases in low genus
Proposition. Proposition 7 is true in the case g = 2.
When |A| = 1 the result is obvious. We will consider the case |A| = 2 . Take first the case when there exists i ∈ A such that m i ≥ 2. Then, we can take the clutching along the last markings: The image of this morphism is in H 2 (µ) by [BCGGM18], hence the conclusion follows by Lemma 6 and induction. When both markings in A correspond to elements m i ≤ 1, take B ⊂ {1, 2, . . . , n} consisting of the elements j in the complement of A such that m j > 0. Without any loss of generality assume that A = {1, 2} and m 1 ≥ m 2 . Then, take the following partitions of 0: Then we take the clutching along the last markings: The conclusion follows again by induction. The only case when we cannot apply induction is when one of the strata is empty. This happens only if m 1 = m 2 = −1 and all other entries are positive. Assume in this case that p 1 + p 2 = g 1 2 for all curves [C, p 1 , . . . , p n ] ∈ H 2 (µ). We observe that the image of the map forgetting p 1 and p 2 is in H 2 2 (m 3 , . . . , m n ) and the generic fiber is one-dimensional. By dimension reasons, it follows that the image must be contained in a component corresponding to abelian holomorphic differentials. This is possible if and only if m 3 , . . . , m n are all even. Hence we are left with the cases µ = (−1, −1, 2, 2) and µ = (−1, −1, 4), and we see that in the nonhyperelliptic component, it is not true that p 1 + p 2 is always a g 1 2 .
assumption p 1 + p 2 = g 1 2 for all points [C, p 1 , . . . , p n ] in the stratum. We see that by forgetting the point p 1 we get a finite map: Q 2 (µ) → H 2−m1 2 (m 2 − m 1 , m 3 , . . . , m n ) There is no component corresponding to holomorphic abelian differentials as there are both positive and negative entries. Hence we obtain a contradiction by dimension reasons.
Our proof by contradiction is now complete, and hence the proposition is true when g = 2.