1. Introduction
For
$\mu = (m_1, \ldots, m_n)$
with
$\sum_{i=1}^n m_i = k( 2g-2)$
, let
$\mathcal P_g^k(\mu)$
be the projectivised stratum of k-differentials
$\omega$
on smooth and connected complex curves of genus g that have labeled zeros and poles whose orders are specified by
$\mu$
. Equivalently,
$\mathcal P_g^k(\mu)$
parameterises k-canonical divisors of prescribed type
$\mu$
.
The study of differentials is important in surface dynamics and moduli theory. We refer to [
Reference ChenC2, Reference WrightW, Reference ZorichZo
] for an introduction to this subject. Despite various known results, the global geometry of
$\mathcal P_g^k(\mu)$
remains quite mysterious, e.g., the full structure of the tautological ring of
$\mathcal P_g^k(\mu)$
is largely unknown. Moreover, it is generally unclear whether the underlying coarse moduli space of
$\mathcal P_g^k(\mu)$
is an affine variety. The birational geometry of
$\mathcal P_g^k(\mu)$
is also less studied. Such questions are meaningful for understanding a moduli space, e.g., many exciting ideas and results have been discovered in the study of these questions for the moduli space
$\mathcal M_g$
of curves of genus g.
There has been some expectation among experts that the (rational) Picard group of
$\mathcal P_g^k(\mu)$
should behave similarly comparing to that of
$\mathcal M_g$
, which is of rank one (for
$g\geq 3$
) generated by the tautological divisor class
$\lambda$
. However, our result below shows that such an expectation should at least exclude the strata whose Teichmüller curves have nonvarying sums of Lyapunov exponents (as listed in [
Reference Chen and MöllerCM1, Reference Chen and MöllerCM2
] for about 30 of them). For simplicity we call them nonvarying strata. Note that a stratum can be disconnected due to spin and hyperelliptic structures ([
Reference Kontsevich and ZorichKZ, Reference LanneauL
]). Since different connected components can have distinct properties, when speaking of a stratum we mean a specific connected component according to the hyperelliptic and spin parity labelings.
The tautological ring of a stratum is defined in [
Reference ChenC4
] to be generated by the tautological classes pulled back from
$\mathcal M_{g,n}$
together with the tautological line bundle class of the projectivised Hodge bundle. We say that the tautological ring of a stratum is trivial if it is isomorphic to
$\mathbb Q$
generated by the fundamental class, i.e., if every tautological class of positive codimension is zero over
$\mathbb Q$
. Here a cycle class on a stratum is treated in the stacky sense when we perform intersection calculations, just like doing calculations on
$\mathcal M_{g,n}$
as a moduli stack. In contrast, when speaking of affine, birational, or topological properties, we refer to the underlying coarse moduli space of a stratum.
Theorem 1·1. The nonvarying strata of abelian and quadratic differentials listed in [ Reference Chen and MöllerCM1, Reference Chen and MöllerCM2 ] have trivial tautological rings and their coarse moduli spaces are affine varieties.
In contrast if a stratum is varying, then the tautological classes should not be all trivial (see Remarks 3·1 and 3·2).
Next we consider k-differentials of infinite area, i.e., when
$\mu$
contains at least one entry
$\leq -k$
(see [
Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM2
] for an introduction to k-differentials).
Theorem 1·2. The coarse moduli spaces of strata of k-differentials of infinite area are affine varieties.
Affine varieties are Stein spaces in analytic geometry. By [ Reference Andreotti and FrankelAF, Reference NarasimhanN ] we thus obtain the following result about the homology of an affine stratum of differentials.
Corollary 1·3.
Let
$\mathcal P_g^k(\mu)$
be one of the above affine strata of differentials. Then
$H_d(\mathcal P_g^k(\mu), \mathbb Z) = 0$
if
$d > \dim_{\mathbb C} \mathcal P_g^k(\mu)$
and
$H_d(\mathcal P_g^k(\mu), \mathbb Z)$
is torsion free if
$d = \dim_{\mathbb C} \mathcal P_g^k(\mu)$
.
Remark 1·4. For strata of holomorphic differentials with
$\mu = (m_1, \ldots, m_n)$
and
$m_i \geq 0$
for all i, a (strongly)
$(g+1)$
-convex exhaustion function was constructed on
$\mathcal P_g^1(\mu)$
by using the area and length functions on flat surfaces (see [M, proposition 3·17]), i.e.,
$\mathcal P_g^1(\mu)$
is a
$(g+1)$
-complete complex space. It follows that
$H_d(\mathcal P_g^1(\mu), \mathbb Z)$
is torsion and
$H_d(\mathcal P_g^1(\mu), \mathbb C) = 0$
for
$d> \dim_{\mathbb C} \mathcal P_g^1(\mu) + g = 3g-2+n$
(see [
Reference Ballico and BolondiBB
, theorem 0·3]).Footnote
1
In general little is known about the homology of strata of differentials (nevertheless see [
Reference Costantini, Möller and ZachhuberCMZ
] for computing the Euler characteristics and [
Reference ZykoskiZy
] for the
$L^{\infty}$
-isodelaunay decomposition). Even if one is interested in strata of holomorphic differentials in high genus, differentials in low genus and meromorphic differentials naturally appear in the boundary of compactified strata ([
Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM1, Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCGGM3
]). Therefore, we expect that the above results can be useful for inductively computing the homology of strata of differentials.
Finally we turn to the stratification of the (ordinary) Hodge bundle of holomorphic differentials (up to scale) as the disjoint union of
$\mathcal P_g^1\{\mu\}$
, where
$\mu$
runs over all partitions of
$2g-2$
and
$\{ \mu \}$
denotes the version of strata with unlabeled zeros. Note that
$\mathcal P_g^1\{ \mu\}$
is a finite quotient of
$\mathcal P_g^1(\mu)$
under the group action permuting the zeros that have the same order. We also denote by
$\overline{\mathcal P}_g^1\{\mu\}$
the closure of
$\mathcal P_g^1\{\mu\}$
in the extended Hodge bundle over the Deligne–Mumford moduli space
$\overline{\mathcal M}_g$
of stable curves. We use similar notations for the stratification of the (quadratic) Hodge bundle of quadratic differentials with at worst simple poles and the extension over
$\overline{\mathcal M}_{g,s}$
, where s is a given number of simple poles to start with.
Theorem 1·5.
Let
$\mu = (m_1, \ldots, m_n)$
be a signature of abelian differentials with
$m_i \geq 1$
for all i, and let
$\mu' = (m_1 + m_2, m_3, \ldots, m_n)$
. Then the stratum
$\overline{\mathcal P}_g^1\{\mu'\}$
is an extremal effective divisor in
$\overline{\mathcal P}_g^1\{\mu\}$
.
Let
$\mu = (m_1, \ldots, m_n)$
be a signature of quadratic differentials with
$m_i \geq -1$
for all i, and let
$\mu' = (m_1 + m_2, m_3, \ldots, m_n)$
, where at least one of
$m_1, m_2$
is positive. Then the stratum
$\overline{\mathcal P}_g^2\{\mu'\}$
is an extremal effective divisor in
$\overline{\mathcal P}_g^2\{\mu\}$
.
Differentials of type
$\mu'$
as degeneration of differentials of type
$\mu$
can be realised geometrically by merging the two zeros of order
$m_1$
and
$m_2$
. Theorem 1·5 thus has an amusing interpretation that the stratification of the Hodge bundle (for abelian and quadratic differentials of finite area) is extremal, where merging two zeros leads to an extremal effective divisor in the boundary. We remark that if a stratum of type
$\mu$
is disconnected, then Theorem 1·5 holds for each connected component of type
$\mu$
and the component of type
$\mu'$
obtained by merging the zeros in the prescribed component of type
$\mu$
.
Strategies of the proofs
To prove Theorem 1·1 we show that the nonvarying property yields an extra relation between tautological divisor classes which forces them to be trivial. To prove Theorem 1·2 we observe the sign change of
$m_i+k$
for entries
$m_i\leq -k$
and
$m_i > -k$
in
$\mu$
, and use it to exhibit an ample divisor class which is trivial on strata of k-differentials of infinite area. To prove Theorem 1·5 we show that Teichmüller curves contained in
$\mathcal P_g^k\{\mu'\}$
have negative intersection numbers with the divisor class of
$\overline{\mathcal P}_g^k\{\mu'\}$
in
$\overline{\mathcal P}_g^k\{\mu\}$
for
$k = 1, 2$
.
Related works
Previously in [
Reference ChenC3
] the author showed that the nonvarying strata of abelian differentials for
$\mu = (4)^{\operatorname{odd}}$
, (3,1),
$(2,2)^{\operatorname{odd}}$
and
$(6)^{\operatorname{even}}$
are affine by using the geometry of canonical curves in low genus (see also [
Reference Looijenga and MondelloLM
] for a related discussion). This is now completed for all known nonvarying strata (including for quadratic differentials). Moreover in [
Reference ChenC3
] the author also showed that there is no complete curve in any stratum of k-differentials of infinite area, which now follows from Theorem 1·2. Note that in general the implication does not work the other way around, e.g.,
$\mathbb A^2$
removing a point contains no complete curves but it is not affine. For unprojectivised strata of holomorphic differentials, nonexistence of complete curves was shown in [
Reference GendronGe
] and [
Reference ChenC5
]. Finally the beginning case of Theorem 1·5 for the principal strata, i.e., for all zeros equal to one, was previously established in [
Reference GheorghitaGh
]. Now this extremal behaviour is shown to hold for all zero types. Therefore, our current results can further enhance the understanding of the tautological ring, affine geometry, topology, and birational geometry of the strata. We hope that the ideas employed in this paper can shed some light on related questions.
2. Triviality criteria
Marking the zeros and poles of differentials,
$\mathcal P_g^k(\mu)$
can be viewed as a subvariety in the moduli space
$\mathcal M_{g,n}$
of curves of genus g with n marked points. We introduce the following tautological divisor classes that will be used throughout the paper. Let
$\eta$
be the first Chern class of the tautological line bundle
$\mathcal O(-1)$
on
$\mathcal P_g^k(\mu)$
whose fibers are spanned by the underlying k-differentials
$\omega$
. Let
$\lambda$
be the first Chern class of the Hodge bundle,
$\kappa$
be the Miller–Morita–Mumford class, and
$\psi_i$
be the cotangent line bundle class associated to the ith marked point. When
$m_i \neq -k$
for all i, we will often use the following quantity
\begin{align*}\kappa_\mu = \frac{1}{k}(2g-2+n) -\sum_{i=1}^n \frac{1}{m_i+k}.\end{align*}
Recall that the tautological ring of
$\mathcal P_g^k(\mu)$
is defined to be generated by the tautological classes pulled back from
$\mathcal M_{g,n}$
together with
$\eta$
. If
$\mu$
does not contain
$-k$
as an entry, then it was shown in [
Reference ChenC4
, theorem 1·1] that in this case
$\eta$
can generate the entire tautological ring of
$\mathcal P_g^k(\mu)$
.
Proposition 2·1.
Suppose
$m_i \neq -k $
for all the entries of
$\mu$
. Let D be an effective divisor in
$\mathcal M_{g,n}$
with divisor class
$a\lambda + \sum_{i=1}^n b_i \psi_i$
. Then the pullback of D to
$\mathcal P_g^k(\mu)$
has divisor class
\begin{align*} \frac{1}{12}\Big(\frac{(2g-2+n)a}{k} + \sum_{i=1}^n \frac{12b_i -a }{m_i +k}\Big) \eta. \end{align*}
Moreover if the above coefficient of
$\eta$
is nonzero and if D and
$\mathcal P_g^k(\mu)$
are disjoint, then
$\eta$
is trivial on
$\mathcal P_g^k(\mu)$
and the tautological ring of
$\mathcal P_g^k(\mu)$
is trivial.
For the nonvarying strata of abelian and quadratic differentials, the desired divisors D in Proposition 2·1 arise from those used in [
Reference Chen and MöllerCM1, Reference Chen and MöllerCM2
] which are disjoint from the respective strata. Occasionally we will use some variants of
$\mathcal M_{g,n}$
by marking fewer points or lifting to the spin moduli spaces, for which the above reasoning still works as we only consider the interior of these moduli spaces.
Proof. The following relations hold on
$\mathcal P_g^k(\mu)$
(see [
Reference ChenC4
, proposition 2·1]):
\begin{align*}\eta = (m_i + k) \psi_i, \quad 12\lambda = \kappa = \kappa_\mu \eta. \end{align*}
Then the claim on the pullback divisor class follows from these relations.
For the other claim, we only need to show that in this case
$\eta$
being trivial implies that the tautological ring of
$\mathcal P_g^k(\mu)$
is trivial, which indeed follows from the fact that
$\eta$
generates the tautological ring when
$\mu$
contains no entries equal to
$-k$
(see [
Reference ChenC4
, theorem 1·1]).
Proposition 2·2.
If all tautological divisor classes are trivial on
$\mathcal P_g^k(\mu)$
, then the coarse moduli space of
$\mathcal P_g^k(\mu)$
is an affine variety.
Proof. The tautological divisor class
$\kappa + \sum_{i=1}^n \psi_i$
is ample on
$\overline{\mathcal M}_{g,n}$
(see [
Reference CornalbaCo
, theorem (2·2)]) and is trivial on
$\mathcal P_g^k(\mu)$
by the assumption. Let
$\delta$
be the total boundary of
$\overline{\mathcal M}_{g,n}$
and
$0 < \epsilon \ll 1$
. The divisor class
$\kappa + \sum_{i=1}^n \psi_i + \epsilon \delta$
remains ample and its restriction to the closure of
$\mathcal P_g^k(\mu)$
can be represented by a divisor fully supported on the boundary. This proves the claim.
3. Nonvarying strata
In this section we prove Theorem 1·1. We use the effective divisors described in [ Reference Chen and MöllerCM1, Reference Chen and MöllerCM2 ] and then apply Propositions 2·1 and 2.2.
3·1. The hyperelliptic strata
Note that
$\lambda$
is trivial on the hyperelliptic locus (see [
Reference Cornalba and HarrisCH
]), which implies that the hyperelliptic strata have trivial tautological rings and are affine varieties.
3·2.
$\mathcal P_3^1(4)^{\operatorname{odd}}$
and
$\mathcal P_3^1(3,1)$
The divisor H parameterising hyperelliptic curves in
$\mathcal M_3$
is disjoint from these strata and has divisor class
$9\lambda$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (1)]).
3·3.
$\mathcal P_3^1(2,2)^{\operatorname{odd}}$
For
$(X, p_1, p_2)\in \mathcal P_3^1(2,2)^{\operatorname{odd}}$
, by definition
$p_1 + p_2$
is an odd theta characteristic, which maps
$\mathcal P_3^1(2,2)^{\operatorname{odd}}$
to the odd spin moduli space
$\mathcal S_3^-$
. The divisor
$Z_3$
in
$\mathcal S_3^-$
parameterises odd theta characteristics with a double zero. It is disjoint from
$\mathcal P_3^1(2,2)^{\operatorname{odd}}$
and has divisor class
$11\lambda$
(see [
Reference Chen and MöllerCM1
, section 5·3]).
3·4.
$\mathcal P_3^1(2,1,1)$
Marking the first two zeros maps
$\mathcal P_3^1(2,1,1)$
to
$\mathcal M_{3,2}$
. The Brill–Noether divisor
$BN_{3,(1,2)}^1$
in
$\mathcal M_{3,2}$
parameterises curves that admit a
$g^1_3$
given by
$p_1 + 2p_2$
. It is disjoint from
$\mathcal P_3^1(2,1,1)$
and has divisor class
$-\lambda + \psi_1 + 3\psi_2$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (5)]).
3·5.
$\mathcal P_4^1(6)^{\operatorname{even}}$
For
$(X, p)\in \mathcal P_4^1(6)^{\operatorname{even}}$
, by definition 3p is an even theta characteristic. The theta-null divisor
$\Theta_4$
in
$\mathcal M_{4,1}$
parameterises curves that admit an odd theta characteristic whose support contains the marked point. It is disjoint from
$\mathcal P_4^1(6)^{\operatorname{even}}$
and has divisor class
$30\lambda + 60 \psi_1$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (3)]).
3·6.
$\mathcal P_4^1(6)^{\operatorname{odd}}$
and
$\mathcal P_4^1(5,1)$
Marking the zero of the largest order, these strata map to
$\mathcal M_{4,1}$
. The Brill–Noether divisor
$BN_{3,(2)}^1$
in
$\mathcal M_{4,1}$
parameterises curves that admit a
$g^1_3$
ramified at the marked point. It is disjoint from both strata and has divisor class
$8\lambda + 4\psi_1$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (7)]).
3·7.
$\mathcal P_4^1(3,3)^{\operatorname{nonhyp}}$
The divisor
$\operatorname{Lin}_3^1$
in
$\mathcal M_{3,2}$
parameterises curves that admit a
$g^1_3$
with a fiber containing both marked points. It is disjoint from
$\mathcal P_4^1(3,3)^{\operatorname{nonhyp}}$
and has divisor class
$8\lambda - \psi_1 - \psi_2$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (8)]).
3·8.
$\mathcal P_4^1(2,2,2)^{\operatorname{odd}}$
For
$(X, p_1, p_2, p_3)\in \mathcal P_4^1(2,2,2)^{\operatorname{odd}}$
, by definition
$p_1 + p_2 + p_3$
is an odd theta characteristic. The divisor
$Z_4$
in the odd spin moduli space
$\mathcal S_4^-$
parameterises odd theta characteristics with a double zero. It is disjoint from
$\mathcal P_4^1(2,2,2)^{\operatorname{odd}}$
and has divisor class
$12\lambda$
(see [
Reference Chen and MöllerCM1
, section 5·3]).
3·9.
$\mathcal P_4^1(3,2,1)$
The Brill–Noether divisor
$BN_{4,(1,1,2)}^1$
in
$\mathcal M_{4,3}$
parameterises curves that admit a
$g^1_4$
given by
$p_1 + p_2 + 2p_3$
. It is disjoint from
$\mathcal P_4^1(3,2,1)$
and has divisor class
$-\lambda + \psi_1 + \psi_2 + 3\psi_3$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (6)]).
3·10.
$\mathcal P_5^1(8)^{\operatorname{even}}$
The Brill–Noether divisor
$BN_{3}^1$
in
$\mathcal M_{5,1}$
parameterises curves with a
$g^1_3$
. It is disjoint from
$\mathcal P_5^1(8)^{\operatorname{even}}$
and has divisor class
$8\lambda$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (4)]).
3·11.
$\mathcal P_5^1(8)^{\operatorname{odd}}$
The divisor
$\operatorname{Nfold}_{5,4}^1(1)$
in
$\mathcal M_{5,1}$
parameterises curves that admit a
$g^1_4$
with a fiber containing 3p. It is disjoint from
$\mathcal P_5^1(8)^{\operatorname{odd}}$
and has divisor class
$ 7\lambda + 15\psi_1$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (9)]).
3·12.
$\mathcal P_5^1(5,3)$
The Brill–Noether divisor
$BN_{4,(1,2)}^1$
in
$\mathcal M_{5,2}$
parameterises curves that admit a
$g^1_4$
with a fiber containing
$p_1 + 2p_2$
. It is disjoint from
$\mathcal P_5^1(5,3)$
and has divisor class
$7\lambda + 7\psi_1 + 2\psi_2$
(see [
Reference Chen and MöllerCM1
, section 2·5, equation (10)]).
3·13.
$\mathcal P_4^1(4,2)^{\operatorname{even}}$
,
$\mathcal P_4^1(4,2)^{\operatorname{odd}}$
, and
$\mathcal P_5^1(6,2)^{\operatorname{odd}}$
The nonvarying property of these strata was proved in [
Reference Yu and ZuoYZ
] by using the Harder–Narasimhan filtration of the Hodge bundle (see also [
Reference Chen and MöllerCM2
, appendix A]). Note that the filtration holds not only on a Teichmüller curve but also on (the interior of) each stratum. Let
$L_\mu$
be the sum of (nonnegative) Lyapunov exponents which satisfies that
$\lambda = L_\mu \eta$
from the Harder–Narasimhan filtration on these strata. In these cases we know that
$L_\mu \neq \kappa_\mu / 12$
. Then the triviality of
$\eta$
on these strata follows from the two distinct relations
$\lambda = (\kappa_\mu/12) \eta$
and
$\lambda = L_\mu \eta$
.
3·14. Nonvarying strata of quadratic differentials in genus one and two
Strata in genus
$\leq 2$
can be generally dealt with as in Sections 4·2 and 4·3 below.
3·15.
$\mathcal P^2_3(9,-1)^{\operatorname{irr}}$
This stratum is disjoint from the hyperelliptic divisor H in
$\mathcal M_3$
whose divisor class is
$9\lambda$
(see [
Reference Chen and MöllerCM2
, section 3·1, equation (6)]).
3·16.
$\mathcal P^2_3(8)$
,
$\mathcal P^2_3(7,1)$
,
$\mathcal P^2_3(8,1,-1)$
,
$\mathcal P^2_3(10, -1, -1)^{\operatorname{nonhyp}}$
and
$\mathcal P^2_3(9,-1)^{\operatorname{reg}}$
Marking the zero of the largest order, these strata map to
$\mathcal M_{3,1}$
. The divisor W of Weierstrass points in
$\mathcal M_{3,1}$
is disjoint from these strata and has divisor class
$ -\lambda + 6\psi_1$
(see [
Reference Chen and MöllerCM2
, section 3, equation (5)]).
3·17.
$\mathcal P^2_3(6,2)^{\operatorname{nonhyp}}$
,
$\mathcal P^2_3(6,1,1)^{\operatorname{nonhyp}}$
,
$\mathcal P^2_3(5,3)$
,
$\mathcal P^2_3(5,2,1)$
,
$\mathcal P^2_3(4,4)$
,
$\mathcal P^2_3(4,3,1)$
,
$\mathcal P^2_3(5,4,-1)$
,
$\mathcal P^2_3(5,3,1,-1)$
,
$\mathcal P^2_3(7,2,-1)$
,
$\mathcal P^2_3(7,3, -1, -1)$
and
$\mathcal P^2_3(6,3,-1)^{\operatorname{reg}}$
Marking the first two zeros, these strata map to
$\mathcal M_{3,2}$
. The Brill–Noether divisor
$BN^1_{3,(2,1)}$
in
$\mathcal M_{3,2}$
parameterises curves that admit a
$g^1_3$
given by
$2p_1 + p_2$
. It is disjoint from these strata and has divisor class
$-\lambda + 3\psi_1 + \psi_2$
(see [
Reference Chen and MöllerCM2
, section 3·1, equation (8)]).
3·18.
$\mathcal P^2_3(4,2,2)$
,
$\mathcal P^2_3(3,3,2)^{\operatorname{nonhyp}}$
,
$\mathcal P^2_3(4,3,2,-1)$
,
$\mathcal P^2_3(3,2,2,1)$
,
$\mathcal P^2_3(3,3,1,1)^{\operatorname{nonhyp}}$
and
$\mathcal P^2_3(3,3,3,-1)^{\operatorname{reg}}$
Marking the first three zeros, these strata map to
$\mathcal M_{3,3}$
. The Brill–Noether divisor
$BN^1_{3,(1,1,1)}$
in
$\mathcal M_{3,3}$
parameterises curves that admit a
$g^1_3$
given by
$p_1 + p_2 + p_3$
. It is disjoint from these strata and has divisor class
$-\lambda + \psi_1 + \psi_2 + \psi_3$
(see [
Reference Chen and MöllerCM2
, section 3·1, equation (7)]).
3·19.
$\mathcal P^2_4(13, -1)$
,
$\mathcal P^2_4(11, 1)$
and
$\mathcal P^2_4(12)^{\operatorname{reg}}$
Marking the zero of the largest order, these strata map to
$\mathcal M_{4,1}$
. The divisor W of Weierstrass points in
$\mathcal M_{4,1}$
is disjoint from these strata and has divisor class
$-\lambda + 10\psi_1$
(see [
Reference Chen and MöllerCM2
, lemma 3·2]).
3·20.
$\mathcal P^2_4(10, 2)^{\operatorname{nonhyp}}$
,
$\mathcal P^2_4(8,4)$
,
$\mathcal P^2_4(8,3,1)$
and
$\mathcal P^2_4(9,3)^{\operatorname{reg}}$
Marking the first two zeros, these strata map to
$\mathcal M_{4,2}$
. The Brill–Noether divisor
$BN^1_{4,(3,1)}$
in
$\mathcal M_{4,2}$
parameterises curves that admit a
$g^1_4$
given by
$3p_1 + p_2$
. It is disjoint from these strata and has divisor class
$-\lambda + 6\psi_1 + \psi_2 $
(see [
Reference Chen and MöllerCM2
, lemma 3·2]).
3·21.
$\mathcal P^2_4(7,5)$
and
$\mathcal P^2_4(6,6)^{\operatorname{reg}}$
The Brill–Noether divisor
$BN^1_{4,(2,2)}$
in
$\mathcal M_{4,2}$
parameterises curves that admit a
$g^1_4$
given by
$2p_1 + 2p_2$
. It is disjoint from these strata and has divisor class
$-\lambda + 3\psi_1 + 3\psi_2$
(see [
Reference Chen and MöllerCM2
, lemma 3·2]).
3·22.
$\mathcal P^2_4(7,3,2)$
,
$\mathcal P^2_4(5,4,3)$
and
$\mathcal P^2_4(6,3,3)^{\operatorname{reg}}$
The Brill–Noether divisor
$BN^1_{4,(2,1,1)}$
in
$\mathcal M_{4,3}$
parameterises curves that admit a
$g^1_4$
given by
$2p_1 + p_2 + p_3$
. It is disjoint from these strata and has divisor class
$-\lambda + 3\psi_1 + \psi_2 + \psi_3 $
(see [
Reference Chen and MöllerCM2
, lemma 3·2]).
3·23.
$\mathcal P^2_4(3,3,3,3)^{\operatorname{reg}}$
It is disjoint from the Brill–Noether divisor
$BN^1_{4,(1,1,1,1)}$
in
$\mathcal M_{4,4}$
whose divisor class is
$-\lambda + \psi_1 + \psi_2 + \psi_3 + \psi_4$
(see [
Reference Chen and MöllerCM2
, lemma 3·2]).
3·24.
$\mathcal P^2_3(6,3,-1)^{\operatorname{irr}}$
,
$\mathcal P^2_4(12)^{\operatorname{irr}}$
and
$\mathcal P^2_4(9,3)^{\operatorname{irr}}$
We can use the Harder–Narasimhan filtration of the quadratic Hodge bundle for these strata (see [ Reference Chen and MöllerCM2 , appendix A] and [ Reference Chen and YuCY , section 4·2]) and argue as in Section 3·13.
Remark 3·1. We say that a stratum of abelian differentials
$\mathcal P^1_g(\mu)$
is varying, if it contains two Teichmüller curves that have distinct sums of Lyapunov exponents. In this case we claim that
$\eta$
is nontrivial on
$\mathcal P^1_g(\mu)$
.
To see this, let
$\Delta_0$
be the locus of stable differentials of type
$\mu$
on curves with separating nodes only, where the differentials have simple poles with opposite residues at the nodes. If
$\mathcal P^1_g(\mu)$
is irreducible, then
$\Delta_0$
is also irreducible and modeled on
$\mathcal P^1_{g-1}(\mu, \{-1,-1\})$
. Moreover, every Teichmüller curve in
$\mathcal P^1_g(\mu)$
only intersects
$\Delta_0$
in the boundary (see [
Reference Chen and MöllerCM1
, corollary 3·2]).
Suppose that
$\eta$
is trivial on
$\mathcal P^1_g(\mu)$
. Then
$\eta = e \delta_0$
on the partial compactification
$\mathcal P^1_g(\mu)\cup \Delta_0$
for some constant e. Since
$\lambda = (\kappa_\mu / 12)\eta$
on
$\mathcal P^1_g(\mu)$
, we have
$\lambda = \ell \delta_0$
on
$\mathcal P^1_g(\mu)\cup \Delta_0$
for some constant
$\ell$
. Then the sum of Lyapunov exponents of a Teichmüller curve C is given by
$(\lambda \cdot C) / (\eta\cdot C) = \ell / e$
which is independent of C, contradicting that
$\mathcal P^1_g(\mu)$
is a varying stratum.
Note that the above criterion is practically checkable for arithmetic Teichmüller curves generated by square-tiled surfaces by using the combinatorial description of area Siegel–Veech constants and sums of Lyapunov exponents (see [ Reference Eskin, Kontsevich and ZorichEKZ , section 2·5·1] and also [ Reference ChenC1 , theorem 1·8] for the relation with slopes of Teichmüller curves).
The same conclusion would hold for a varying stratum of quadratic differentials, assuming that all Teichmüller curves contained in the stratum intersect only one irreducible boundary divisor (see [ Reference Chen and MöllerCM2 , remark 4·7] for this assumption).
Remark 3·2. We speculate that the rational Picard group (and the rational second cohomology group) of any varying stratum of holomorphic differentials
$\mathcal P^1_g(\mu)$
should be of rank one generated by
$\eta$
.
4. Differentials of infinite area
In this section we prove Theorem 1·2 for strata of k-differentials of infinite area. For completeness we also include the discussion for some strata in low genus.
4·1. Strata in genus zero
Every stratum in genus zero is isomorphic to the corresponding moduli space of pointed smooth rational curves, which has a trivial Chow ring and is affine.
4·2. Strata in genus one
Since
$\psi_i = 0$
on
$\mathcal M_{1,n}$
for all i (see [
Reference Arbarello and CornalbaAC
, theorem 2·2 (c)]), every stratum
$\mathcal P^k_1(\mu)$
in genus one (including the case of some
$m_i = -k$
in
$\mu$
) has a trivial tautological ring and is affine by Proposition 2·2. The affinity also follows from the fact that the ambient space
$\mathcal M_{1,n}$
is affine (see [
Reference ChenC3
, theorem 3·1]).
4·3. Strata in genus two
Since
$\lambda$
is trivial on
$\mathcal M_2$
, every stratum
$\mathcal P^k_2(\mu)$
in genus two with
$m_i \neq -k$
for all i has a trivial tautological ring and is affine by Proposition 2·2.
4·4. Strata with poles of order
$\leq - k$
Suppose
$m_1 \leq -k$
. By the relations
$\eta = (m_i + k)\psi_i$
for all i and
$(2g-2)\eta = k \kappa - \sum_{i=1}^n m_i \psi_i$
, we obtain that
\begin{eqnarray*}k\Big(\kappa + \sum_{i=1}^n \psi_i\Big) & = & (2g-2+n) \eta \\[5pt] & = & (2g-2+n)(m_1+k) \psi_1.\end{eqnarray*}
It implies that
$\kappa + \sum_{i=1}^n \psi_i + a \psi_1$
is trivial on
$\mathcal P^k_g(\mu)$
where
$a = - (2g-2+n)(m_1+k) / k \geq 0$
. Since
$\kappa + \sum_{i=1}^n \psi_i + a \psi_1$
(as ample plus nef) remains ample on
$\overline{\mathcal M}_{g,n}$
, it follows that
$\mathcal P^k_g(\mu)$
is affine and Theorem 1·2 is justified.
5. Extremal stratification of the Hodge bundle
In this section we prove Theorem 1·5. Recall the notation
$\mu = (m_1, \ldots, m_n)$
,
$\mu' = (m_1+m_2, m_3, \ldots, m_n)$
, and we work with unordered zeros. The idea is to show that Teichmüller curves in
$\mathcal P_g^k\{\mu'\}$
has negative intersection numbers with the divisor class of
$\overline{\mathcal P}_g^k\{\mu'\}$
in
$\overline{\mathcal P}_g^k\{\mu\}$
for
$k = 1, 2$
. In order to verify that, we make use of the aforementioned fact that degenerate differentials in the boundary of a Teichmüller curve can have only simple polar nodes (after taking a local square root in the case of quadratic differentials).
Proof of Theorem
1·5. First consider the case of abelian differentials. The following relation of divisor classes holds on
$\overline{\mathcal P}^1_g\{\mu\}$
:
\begin{align*} 12 \lambda - D_h - \kappa_\mu \eta = (m_1+m_2+1) \Big(1 - \frac{1}{m_1+1} - \frac{1}{m_2+1} + \frac{1}{m_1+m_2+1}\Big) \overline{\mathcal P}^1_g\{\mu'\}\end{align*}
modulo the other boundary divisors, where
$D_h$
is the boundary divisor generically parameterising stable differentials with simple polar nodes. This relation follows from [
Reference Chen, Costantini and MöllerCCM
, proposition 6·3], where
$\overline{\mathcal P}^1_g\{\mu\}$
can be identified with the boundary divisor
$D_\Gamma$
having
$\Gamma_{\bot}$
as a rational vertex containing the two merged zeros and
$\ell_\Gamma = m_1 + m_2 + 1$
therein. Note that the coefficient on the right of the above equality is positive for
$m_1, m_2 \geq 1$
.
Let C be the closure of a Teichmüller curve in
$\mathcal P^1_g\{\mu'\}$
. Since C does not intersect the part of the boundary of
$\overline{\mathcal P}^1_g\{\mu\}$
away from
$D_h$
and
$\overline{\mathcal P}^1_g\{\mu'\}$
, we obtain
\begin{eqnarray*} \frac{(12 \lambda - D_h - \kappa_\mu \eta)\cdot C}{\eta \cdot C} & = & \kappa_{\mu'} - \kappa_{\mu} \\[5pt] & = & -1 + \frac{1}{m_1+1} + \frac{1}{m_2+1} - \frac{1}{m_1+m_2+1},\end{eqnarray*}
where
$12\lambda - D_h = \kappa_{\mu'} \eta$
holds on the stratum
$\overline{\mathcal P}^1_g\{\mu'\}$
(modulo the other irrelevant boundary divisors of
$\overline{\mathcal P}^1_g\{\mu'\}$
which do not meet C). Consequently, this implies
\begin{align*}\frac{C\cdot \overline{\mathcal P}^1_g\{\mu'\}}{C\cdot \eta} = - \frac{1}{m_1 + m_2 + 1} < 0. \end{align*}
Since
$\eta$
has positive degree on every Teichmüller curve (e.g., from the positive sign of the area Siegel–Veech constant), it follows that
$C\cdot \overline{\mathcal P}^1_g\{\mu'\} < 0$
in
$\overline{\mathcal P}^1_g\{\mu\}$
. Since Teichmüller curves are dense in
$\overline{\mathcal P}^1_g\{\mu'\}$
and the negative ratio above is independent of each individual Teichmüller curve, the desired claim follows by using the same argument as in [
Reference GheorghitaGh
, section 4].
The case of quadratic differentials is similar. The following relation of divisor classes holds on
$\overline{\mathcal P}^2_g\{\mu\}$
:
\begin{align*} 12 \lambda - D_h - \kappa_\mu \eta = (m_1+m_2+2)\Big(\frac{1}{2} - \frac{1}{m_1+2} - \frac{1}{m_2+2} + \frac{1}{m_1+m_2+2}\Big) \overline{\mathcal P}^2_g\{\mu'\}\end{align*}
modulo the other boundary divisors, where the coefficient on the right is nonzero by the assumption that
$m_1, m_2 \geq -1$
and at least one of them is positive. Let C be the closure of a Teichmüller curve in
$\mathcal P^2_g\{\mu'\}$
. Since C does not intersect the part of the boundary of
$\overline{\mathcal P}^2_g\{\mu\}$
away from
$D_h$
and
$\overline{\mathcal P}^2_g\{\mu'\}$
, we obtain
\begin{eqnarray*} \frac{(12 \lambda - D_h - \kappa_\mu \eta)\cdot C}{\eta \cdot C} = -\frac{1}{2} + \frac{1}{m_1+2} + \frac{1}{m_2+2} - \frac{1}{m_1+m_2+2}.\end{eqnarray*}
It follows that
\begin{align*}\frac{C\cdot \overline{\mathcal P}^2_g\{\mu'\}}{C\cdot \eta} = - \frac{1}{m_1 + m_2 + 2} < 0. \end{align*}
The rest of the argument is the same as before.
Remark 5·1. If a stratum in Theorem 1·5 is disconnected, since each of its connected components contains a dense collection of Teichmüller curves, the conclusion of Theorem 1·5 still holds for each connected component.
Remark 5·2. For k-differentials with
$k > 2$
, since there is no meaningful way to define Teichmüller curves in this case, our method cannot be directly adapted. We leave it as an interesting question to determine whether the stratification of the kth Hodge bundle is extremal for
$k > 2$
.
Acknowledgements
The author would like to thank Frederik Benirschke, Matteo Costantini, Iulia Gheorghita, Samuel Grushevsky, Richard Hain, Martin Möller, Gabriele Mondello and Scott Mullane for helpful discussions on relevant topics. The author also sincerely thanks the referee for very useful comments.
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