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admcycles - a Sage package for computations in the tautological ring of the moduli space of stable curves

Authors

Based on earlier implementation by Aaron Pixton

  • Johannes Schmitt

  • Jason van Zelm

  • Vincent Delecroix

How to get access

Website : https://gitlab.com/jo314schmitt/admcycles
Online tool : https://cocalc.com - currently slightly older version
User manual : https://arxiv.org/abs/2002.01709

The moduli space of stable curves

Definition/Theorem (Deligne, Mumford - 1969)
Let g,n0g,n \geq 0 be integers (with 2g2+n>02g-2+n>0). $$\overline{\mathcal{M}}_{g,n} = \left\{(C,p_1, \ldots, p_n) \colon \begin{array}{l} {C \text{ compact complex algebraic}\\ \text{curve of }\color{red}{\text{arithmetic}}\text{ genus }g \\ \text{with at worst }\color{red}{\text{nodal singularities}}}\\ {p_1, \ldots, p_n \in C \text{ distinct $\color{red}{\text{smooth}}ParseError: KaTeX parse error: Expected 'EOF', got '}' at position 8: points}̲}\\ {\colo…$ Then we have

  • Mg,n\overline{\mathcal{M}}_{g,n} is a smooth, irreducible, compact complex orbifold of dimension dimCMg,n=3g3+n\dim_\mathbb{C} \overline{\mathcal{M}}_{g,n} = 3g-3+n.

  • The subset Mg,n\mathcal{M}_{g,n} where the curve CC is smooth is dense and open and the complement Mg,n\partial \overline{\mathcal{M}}_{g,n} is a normal-crossing divisor.

Below we illustrate the space M3,2\overline{\mathcal{M}}_{3,2}.

Recursive boundary structure

To (C,p1,,pn)Mg,n(C,p_1, \ldots, p_n) \in \overline{\mathcal{M}}_{g,n} we can associate a stable graph Γ(C,p1,,pn)\Gamma_{(C,p_1, \ldots,p_n)}

Conversely, given a stable graph Γ\Gamma we have a gluing map ξΓ:vV(Γ)Mg(v),n(v)=M1,3×M2,1M3,2\xi_\Gamma : \prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g(v),n(v)} = \overline{\mathcal{M}}_{1,3} \times \overline{\mathcal{M}}_{2,1}\to \overline{\mathcal{M}}_{3,2}

Proposition
The map ξΓ\xi_\Gamma is finite with image equal to the closure MΓ\overline{\mathcal{M}}^\Gamma of MΓ={(C,p1,,pn):Γ(C,p1,,pn)=Γ}\mathcal{M}^\Gamma = \{(C,p_1, \ldots, p_n) : \Gamma_{(C,p_1, \ldots, p_n)} = \Gamma\}

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from admcycles import * # list_strata(g,n,e) returns the list of stable graphs in Mbar_g,n with e edges L = list_strata(3,2,1) L

Example

The output [1, 2] [[1, 2, 3], [4]] [(3, 4)] corresponds to the stable graph

Exercise

Execute the cell above (which says from admcycles import * ...) by clicking the grey box and then pressing Shift + Enter to start the computation.

Then, use the cell below to compute the number of stable graphs in M3,2\overline{\mathcal{M}}_{3,2} with precisely 2 edges.

(Hint: given a list L you can compute its length using len(L))

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The tautological ring of Mg,n\overline{\mathcal{M}}_{g,n}

Mg,n\overline{\mathcal{M}}_{g,n} compact space     \implies the singular cohomology H(Mg,n)H^*(\overline{\mathcal{M}}_{g,n}) is a finite-dimensional Q\mathbb{Q}-algebra

Generators of the tautological ring

Using the fact that Mg,n\overline{\mathcal{M}}_{g,n} is a moduli space, we can write down many natural cohomology classes.

Definition (ψ\psi-classes)
LiMg,n\mathbb{L}_i \to \overline{\mathcal{M}}_{g,n} complex line bundle, Li(C,p1,,pn)=TpiC\mathbb{L}_i|_{(C,p_1, \ldots, p_n)} = T_{p_i}^* C ψi=c1(Li)H2(Mg,n).\psi_i = c_1(\mathbb{L}_i) \in H^2(\overline{\mathcal{M}}_{g,n}).

Definition (κ\kappa-classes)
Forgetful morphism F:Mg,n+1Mg,n,(C,p1,,pn,pn+1)(C,p1,,pn)F: \overline{\mathcal{M}}_{g,n+1} \to \overline{\mathcal{M}}_{g,n}, (C,p_1, \ldots, p_n, p_{n+1}) \mapsto (C,p_1, \ldots, p_n) [CC smooth] κa=F((ψn+1)a+1)H2a(Mg,n).\kappa_a = F_* \left( (\psi_{n+1})^{a+1}\right) \in H^{2a}(\overline{\mathcal{M}}_{g,n}).

Definition (The tautological ring)
The tautological ring RH(Mg,n)H(Mg,n)RH^*(\overline{\mathcal{M}}_{g,n}) \subset H^*(\overline{\mathcal{M}}_{g,n}) is spanned as a Q\mathbb{Q}-vector subspace by elements [Γ,α]=(ξΓ)(product of κ,ψ-classesα on vV(Γ)Mg(v),n(v)),[\Gamma,\alpha]=(\xi_\Gamma)_* \left(\underbrace{\text{product of }\kappa,\psi\text{-classes}}_{\alpha}\text{ on }\prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g(v),n(v)} \right), called decorated strata classes.

Let's start playing around with tautological classes. We can display a list of all generators [Γ,α]RH2d(Mg,n)[\Gamma, \alpha] \in RH^{2d}(\overline{\mathcal{M}}_{g,n}) using the function list_tautgens(g,n,d) and get access to this list using tautgens(g,n,d).

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list_tautgens(3,2,1) T = tautgens(3,2,1)

Intersection products

One of the basic operations in singular cohomology is the cup product of cohomology classes. The following results allows us to compute such products for decorated strata classes.

Theorem (Graber, Pandharipande - 2003)
The set of tautological classes RH(Mg,n)RH^*(\overline{\mathcal{M}}_{g,n}) is closed under the intersection/cup product, and there exists an explicit formula [Γ,α][Γ,α]=μi[Γi,αi][\Gamma,\alpha] \cdot [\Gamma',\alpha'] = \sum \mu_i [\Gamma_i, \alpha_i] with Γi\Gamma_i running over graphs which are simultaneous specializations of Γ\Gamma and Γ\Gamma'.

We can look at some examples involving the generators above.

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T[3]*T[7] # classes of two boundary divisors, intersecting transversally

Comparing with the picture above, we can see that T[3] and T[7] are the classes of two boundary divisors, which intersect transversally in the codimension 22 stratum associated to the stable graph above.

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T[7]^2 # self-intersection formula of boundary divisor

Intersection numbers

We can also compute actual intersection numbers. For this, given a tautological class t contained in RH2(3g3+n)(Mg,n)RH^{2(3g-3+n)}(\overline{\mathcal{M}}_{g,n}), we can compute its degree Mg,nt=deg(t[Mg,n])\int_{\overline{\mathcal{M}}_{g,n}} \mathrm{t} = \mathrm{deg}(\mathrm{t} \frown [\overline{\mathcal{M}}_{g,n}]) using t.evaluate(). This is based on a conjecture by [Witten - 1991], proved by [Kontsevich - 1992].

Exercise

Use the cell below to compute the integral [M3,2]ψ16ψ22.\int_{[\overline{\mathcal{M}}_{3,2}]} \psi_1^6 \psi_2^2. Hint: You can get access to the classes ψi\psi_i either from the list T above, or using the function psiclass(i,g,n).

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Tautological relations

An important aspect of the tautological ring is that the generators [Γ,α][\Gamma, \alpha] are not necessarily linearly independent. A linear relation between them is called a tautological relation. A lot of work has gone into studying such relations ([Faber, Zagier - 2000]; [Pandharipande, Pixton - 2010]; [Pixton - 2012]; [Pandharipande, Pixton, Zvonkine - 2013]), finally leading to a conjectural description of all tautological relations, originally proposed by Pixton.

These so-called generalized Faber-Zagier relations RFZR^{\text{FZ}} have been implemented by Pixton in the predecessor of admcycles and are accessible in admcycles in two ways: given a tautological class t in RH2d(Mg,n)RH^{2d}(\overline{\mathcal{M}}_{g,n}) you can

  • check if it is contained in the system RFZR^{\text{FZ}} (and thus zero) by calling t.is_zero(),

  • express it in a basis of the vector space span([Γ,α]:Γ,α)/RFZ,\mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}}, which is conjecturally isomorphic to RH2d(Mg,n)RH^{2d}(\overline{\mathcal{M}}_{g,n}), by calling t.toTautbasis().

Exercise

There exists a linear relation between the classes

Find the relation using the function toTautbasis() and confirm it using is_zero.

Note: This task may require more than one command. You can create new cells for computations via the menu Insert->Insert Cell Above/Below at the top of the screen.

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kappa = kappaclass(1,1,2) # kappaclass(a,g,n) computes kappa_a on Mbar_{g,n} psi = psiclass(1,1,2) + psiclass(2,1,2) A = StableGraph([0,1],[[1,2,3],[4]],[(3,4)]).boundary_pushforward()
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The functions toTautbasis and is_zero are perfectly fine for confirming that classes are zero, since R.is_zero() == True means that R is a combination of known tautological relations, and thus indeed vanishes.

But what if we want to show that things don't vanish? Since the generalized Faber-Zagier relations are only conjecturally complete, an output R.is_zero() == False just tells us that R is not zero assuming the conjecture.

To get a non-conditional result, i.e how to show that in the exact sequence 0Kspan([Γ,α]:Γ,α)/RFZRH2d(Mg,n)0()0 \to K \to \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} \to RH^{2d}(\overline{\mathcal{M}}_{g,n}) \to 0 \quad (*) the kernel KK vanishes (and thus that RFZR^{\text{FZ}} indeed give all relations)? We can use the following strategy: there exists an intersection pairing RH2d(Mg,n)RH2(3g3+nd)(Mg,n)Q,[Γ1,α1][Γ2,α2][Mg,n][Γ1,α1][Γ2,α2].\begin{align*} RH^{2d}(\overline{\mathcal{M}}_{g,n}) \otimes RH^{2(3g-3+n-d)}(\overline{\mathcal{M}}_{g,n}) &\to \mathbb{Q},\\ [\Gamma_1, \alpha_1] \otimes [\Gamma_2, \alpha_2] &\mapsto \int_{[\overline{\mathcal{M}}_{g,n}]} [\Gamma_1, \alpha_1] \cdot [\Gamma_2, \alpha_2]\,. \end{align*}

As we have seen in the last sections, we can explicitly compute all pairings appearing above using admcycles and obtain a matrix MM whose entries are M[Γ1,α1],[Γ2,α2]=[Mg,n][Γ1,α1][Γ2,α2], M_{[\Gamma_1, \alpha_1], [\Gamma_2, \alpha_2]} = \int_{[\overline{\mathcal{M}}_{g,n}]} [\Gamma_1, \alpha_1] \cdot [\Gamma_2, \alpha_2]\,, and this matrix satisfies rankMdimRH2d(Mg,n).\mathrm{rank}\, M \leq \dim RH^{2d}(\overline{\mathcal{M}}_{g,n}). Then if we check rankM=dimspan([Γ,α]:Γ,α)/RFZ\mathrm{rank}\, M = \dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} it follows that in ()(*) indeed the kernel KK must vanish.

Example : RH2(M1,2)RH^2(\overline{\mathcal{M}}_{1,2})

Let's see this in practice: for the classes in RH2(M1,2)RH^2(\overline{\mathcal{M}}_{1,2}) considered above, we saw that toTautbasis() gave out vectors of length 22, so dimspan([Γ,α]:Γ,α)/RFZ=2\dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} = 2 in this case. Since 3g3+n=23g-3+n = 2 in this case, the relevant pairing is

RH2(Mg,n)RH2(Mg,n)Q.RH^{2}(\overline{\mathcal{M}}_{g,n}) \otimes RH^{2}(\overline{\mathcal{M}}_{g,n}) \to \mathbb{Q}\,.

Let's compute the matrix MM and its rank:

In [0]:
D = tautgens(1,2,1) # tautgens(g,n,d) lists of all generators [Gamma_i, alpha_i] in RH^{2d}(Mbar_{g,n}) pairings = [[(a*b).evaluate() for a in D] for b in D] M = matrix(QQ, pairings) M
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M.rank()

So indeed we get the equality rankM=dimspan([Γ,α]:Γ,α)/RFZ=2\mathrm{rank}\, M = \dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} = 2 and thus we can be sure that the system RFZR^{\text{FZ}} is complete in this case.

Natural cycle classes on the moduli space of stable curves

Using the fact that Mg,n\overline{\mathcal{M}}_{g,n} is a moduli space, there are many ways to construct interesting cohomology classes on it. Many (though not all) of them actually result in tautological classes, and again many (though not all) of these have been implemented in admcycles.

A) Cycles associated to closed subsets

For SMg,nS \subset \overline{\mathcal{M}}_{g,n} a closed, algebraic subset of C\mathbb{C}-codimension dd, there exists a fundamental class [S]Hdim2d(Mg,n)Hdim2d(Mg,n)PDH2d(Mg,n),[S] \in H_{\dim - 2d}(\overline{\mathcal{M}}_{g,n}) \cong H^{\dim - 2d}(\overline{\mathcal{M}}_{g,n})^\vee \underset{\mathrm{PD}}{\cong} H^{2d}(\overline{\mathcal{M}}_{g,n}), where PD\mathrm{PD} is the isomorphism from Poincaré duality.

Example : Hyperelliptic cycles

An important example are the loci of hyperelliptic curves

Hypg,n,2m={(C,p1,,pn,q1,q1,,qm,qm):C hyperelliptic,pi Weierstrass points,qj,qj hyperelliptic conjugate}Mg,n+2m\begin{equation*} \mathrm{Hyp}_{g,n,2m} = \left\{(C,p_1,\ldots, p_n, q_1, q_1', \ldots, q_m, q_m') : \begin{array}{c} C \text{ hyperelliptic},\\p_i\text{ Weierstrass points},\\ q_j, q_j' \text{ hyperelliptic conjugate}\end{array} \right\} \subset \mathcal{M}_{g,n+2m} \end{equation*}

Then, [Faber, Pandharipande - 2005] showed that the fundamental classes of the closures

[Hypg,n,2m]H(Mg,n+2m),\begin{align*} [\overline{\mathrm{Hyp}}_{g,n,2m}] \in H^*(\overline{\mathcal{M}}_{g,n+2m}), \end{align*}

are always tautological. In [Schmitt, van Zelm - 2018] with Jason van Zelm we give an algorithm to compute intersection numbers of the hyperelliptic cycles with tautological classes. Using these numbers, the package admcycles can compute them (in many cases) via the function Hyperell(g,n,m).

Example: Strata of differentials

Given a=(a1,,an)Zn\textbf{a}=(a_1, \ldots, a_n) \in \mathbb{Z}^n with iai=2g2\sum_i a_i = 2g-2, we consider the locus Hg(a)={(C,p1,,pn):ωCOC(i=1naipi)}={(C,p1,,pn):η meromorphic differential on C with div(η)=i=1naipi}Mg,n.\begin{align*} \mathcal{H}_g(\textbf{a}) &= \left \{(C,p_1, \ldots, p_n) : \omega_C \cong \mathcal{O}_C \left( \sum_{i=1}^n a_i p_i \right) \right \} \\ &=\left \{(C,p_1, \ldots, p_n) : \begin{array}{c} \exists\, \eta \text{ meromorphic differential on $C$}\\\text{ with }\mathrm{div}(\eta) =\sum_{i=1}^n a_i p_i \end{array} \right \}\subseteq \mathcal{M}_{g,n}. \end{align*} Then Hg(a)\mathcal{H}_g(\textbf{a}) is a closed algebraic subset of Mg,n\mathcal{M}_{g,n} and taking the fundamental class of the closure Hg(a)\overline{\mathcal{H}}_g(\textbf{a}), we obtain cohomology classes [Hg(a)]H(Mg,n).[\overline{\mathcal{H}}_g(\textbf{a})] \in H^*(\overline{\mathcal{M}}_{g,n}).

These classes have been studied intensely in the last couple of years. Some highlights:

  • In the appendix of the paper [Farkas, Pandharipande - 2015], Janda, Pandharipande, Pixton and Zvonkine wrote down a conjectural formula [Hg(a1,,an)]+(boundary corrections)=DRg(a1+1,,an+1)[\overline{\mathcal{H}}_g(a_1, \ldots, a_n)] + \sum \text{(boundary corrections)} = \mathrm{DR}_g(a_1+1, \ldots, a_n+1) relating the classes of strata of meromorphic differentials to the so-called double ramification cycles. These DR cycles are computed by an explicit formula in the tautological ring proposed by Pixton. This conjecture was recently proven by combining the results of papers [Holmes, Schmitt - 2019], [Bae, Holmes, Pandharipande, Schmitt, Schwarz - 2020]. The above formula can be used to recursively compute all cycles [Hg(a1,,an)][\overline{\mathcal{H}}_g(a_1, \ldots, a_n)], both in the holomorphic and meromorphic case. This has been implemented in the function Strataclass(g,1,(a1, ..., an)).

In [0]:
H = Strataclass(2,1,(2,)) # computes closure of {(C,p) : omega_C = O_C(2p)} Hpush = H.forgetful_pushforward([1]) # computes pushforward under pi : Mbar_{2,1} -> Mbar_2 Hpush.simplify()
In [0]:
H2 = Hyperell(2,1) # computes closure of {(C,p) : p Weierstrass point} (H - H2).is_zero()
  • The strata Hg(a)\overline{\mathcal{H}}_g(\textbf{a}) themselves were studied in a series of papers by Bainbridge, Chen, Gendron, Grushevsky, Möller ([BCGGM - 2018], [BCGGM - 2019], [BCGGM - 2020]). In particular, the authors define a smooth compact moduli space MSg(a)Hg(a), \mathcal{MS}_g(\textbf{a}) \to \overline{\mathcal{H}}_g(\textbf{a}), sitting proper, birationally over Hg(a)\overline{\mathcal{H}}_g(\textbf{a}), called the space of multiscale differentials. They describe the boundary strata of MSg(a)\mathcal{MS}_g(\textbf{a}) (and thus of Hg(a)\overline{\mathcal{H}}_g(\textbf{a})) in terms of certain enhanced level graphs.

  • The intersection theory and tautological ring of MSg(a)\mathcal{MS}_g(\textbf{a}) have been implemented by [Costantini, Möller, Zachhuber - 2020] in an extension/sub-package of admcycles called diffstrata. They use this in a second paper [Costantini, Möller, Zachhuber - 2020] to compute the (orbifold) Euler characteristics of the open strata Hg(a){\mathcal{H}}_g(\textbf{a}) in a range of examples.

In [0]:
from admcycles.diffstrata import Stratum X = Stratum((2,)) X.euler_characteristic()

B) Lambda classes (and generalizations)

On Mg,n\overline{\mathcal{M}}_{g,n} there exists a vector bundle Eg\mathbb{E}_g, called the Hodge bundle, with fibres Eg(C,p1,,pn)=H0(C,ωC).\mathbb{E}_g|_{(C,p_1, \ldots, p_n)} = H^0(C, \omega_C). The Chern classes λi=ci(Eg)H2i(Mg,n)\lambda_i = c_i(\mathbb{E}_g) \in H^{2i}(\overline{\mathcal{M}}_{g,n}) are called the λ\lambda-classes. They are tautological, with an explicit formula first computed by [Mumford - 1983], and can be computed in terms of generators [Γ,α][\Gamma, \alpha] using lambdaclass(i,g,n).

Exercise

A special case of the main result of [Faber, Pandharipande - 2000] is that the generating series F(t)=1+g>0t2gMg,1ψ12g2λgF(t) = 1 + \sum_{g > 0} t^{2g} \int_{\overline{\mathcal{M}}_{g,1}} \psi_1^{2g-2} \lambda_g is given by F(t)=t/2sin(t/2).F(t) = \frac{t/2}{\sin(t/2)}. Check their result for g=2g=2.

Hint: To get you started, I compute the expansion of FF at t=0t=0 below.

In [0]:
R.<t> = PowerSeriesRing(QQ) F = (t/2)/sin(t/2); F
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Note that for the universal curve π:Cg,nMg,n\pi : \overline{\mathcal{C}}_{g,n} \to \overline{\mathcal{M}}_{g,n} and the relative dualizing line bundle ωπ\omega_\pi of this morphism, we have Eg=R0πωπ.\mathbb{E}_g = R^0 \pi_* \omega_\pi. The paper [Pagani, Ricolfi, van Zelm - 2019] generalizes the λ\lambda-classes and computes, for an arbitrary line bundle L\mathcal{L} on Cg,n\overline{\mathcal{C}}_{g,n} the Chern classes ci(RπL)c_i(R^\bullet \pi_* \mathcal{L}) of the derived pushforward of L\mathcal{L} by π\pi. In fact, admcycles had a hand in this: the original version [Pagani, Ricolfi - 2018] of the paper missed some terms in the computation, which was discovered by van Zelm using admcycles.

Formulas for lambda-classes

(ongoing research with R. Pandharipande, S. Molcho)

As mentioned before, there is a computation of Mumford, giving a formula for the λ\lambda-classes λi=ci(Eg)H2i(Mg,n).\lambda_i = c_i(\mathbb{E}_g) \in H^{2i}(\overline{\mathcal{M}}_{g,n})\,. The formula following by Mumford's computation is reasonably nice, but it does feature some slightly complicated terms. However, we could hope for something better! Consider the open subset Mg,nct\mathcal{M}_{g,n}^{\mathrm{ct}} of curves (C,p1,,pn)(C,p_1, \ldots, p_n) such that the stable graph of CC is a tree (i.e. contains no circular path). Then there exists a class θRH2(Mg,n)\theta \in RH^2(\overline{\mathcal{M}}_{g,n}) such that λg=1g!θgH2g(Mg,nct).\lambda_g = \frac{1}{g!} \theta^g \in H^{2g}(\mathcal{M}_{g,n}^{\mathrm{ct}})\,. So, up to a scalar factor, the class λg\lambda_g is a power of a divisor class! However, it turns out that this equality does not hold in general on the whole of Mg,n\overline{\mathcal{M}}_{g,n}.

What we can show now is that such a formula cannot work at all if gg is sufficiently large! To be slightly more precise, denote by RHk(Mg,n)RH(Mg,n)RH^*_{\leq k}(\overline{\mathcal{M}}_{g,n}) \subseteq RH^*(\overline{\mathcal{M}}_{g,n}) the sub-Q\mathbb{Q}-algebra of RH(Mg,n)RH^*(\overline{\mathcal{M}}_{g,n}) generated by elements of cohomological degree at most 2k2k. In particular, RH1(Mg,n)RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n}) is the set of classes which can be written as linear combinations of products of divisor classes.

Then we have the following:

Theorem (Molcho, S, Pandharipande)
For g4g \geq 4 we have λgRH1(Mg,n)\lambda_g \notin RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n}) and for g5g \geq 5 we have λgRH2(Mg,n),\lambda_g \notin RH^*_{\leq 2}(\overline{\mathcal{M}}_{g,n})\, , assuming that the generalized Faber-Zagier relations RFZR^{\text{FZ}} give all the relations in the spaces RH8(M4,1)  and  RH10(M5,1).RH^{8}(\overline{\mathcal{M}}_{4,1}) \ \text{ and }\ RH^{10}(\overline{\mathcal{M}}_{5,1}). Idea of proof Use admcycles to check the statement in M4,1\overline{\mathcal{M}}_{4,1} and M5,1\overline{\mathcal{M}}_{5,1}. The assumption tells us that in the corresponding spaces toTautbasis() really does express λg\lambda_g as well as elements of RH1(Mg,n)RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n}) in a basis of RH(Mg,n)RH^*(\overline{\mathcal{M}}_{g,n}), so the statement is just linear algebra. The case of larger g,ng,n can then be shown using a small argument applying boundary gluing maps like ξ:M5,1×M1,2M6,1,\xi : \overline{\mathcal{M}}_{5,1} \times \overline{\mathcal{M}}_{1,2} \to \overline{\mathcal{M}}_{6,1}\,, using the fact that ξλ6=λ5λ1\xi^* \lambda_6 = \lambda_5 \otimes \lambda_1.

With the strategy discussed before, we can hopefully get rid of the assumption of the Faber-Zagier relations. The only problem is that the expected dimension of RH10(M5,1)RH^{10}(\overline{\mathcal{M}}_{5,1}) is 13711371, so the matrix MM we have to compute is a matrix of size at least 1371×13711371 \times 1371 of intersection numbers on the space M5,1\overline{\mathcal{M}}_{5,1} of complex dimension 1313. This will take a bit of time and effort, but doesn't seem impossible.



Thanks for your attention!

















Appendix

Connection to moduli spaces of abelian varieties

Ongoing developments

  • Johannes Schwab (PhD student of M. Möller) - Expand diffstrata for spaces of quadratic differentials

  • Javier Sendra (Master student of G. Oberdieck and myself) - Implement algorithm for computing Gromow-Witten invariants of K3 surfaces described by [Maulik, Pandharipande, Thomas - 2010]

  • Danilo Lewanski (Postdoc, IPhT Paris), Zekun Ji (Master student of myself) - Exploring ELSV type formulas for various kinds of Hurwitz numbers (double Hurwitz numbers, r-Hurwitz numbers)

Exercise

Verify one of the most important tautological relations, called the WDVV-relation on the space M0,4\overline{\mathcal{M}}_{0,4}:

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D) Double ramification cycles

When discussing strata of differentials, we saw that double ramification cycles played an important role for computing them. Again, we'll mostly treat them as a black box, but we'll open the box a little bit to be able to show some cool conjecture.

Construction (Pixton - 2014)
Let g,k,dZ0g,k,d \in \mathbb{Z}_{\geq 0} and A=(a1,,an)ZnA=(a_1, \ldots, a_n) \in \mathbb{Z}^n with iai=k(2g2+n)\sum_i a_i = k(2g-2+n). Then for any integer r1r \geq 1 Pixton gave an explicit formula $$\mathrm{DR}_g^{\,d,k,r}(A) = \sum_{\Gamma,w} \left[\Gamma, \text{(polynomial in $\kappa,\psiclassesdependingon-classes depending on wParseError: KaTeX parse error: Expected 'EOF', got '}' at position 2: )}̲ \right]\in RH^…wherethesumrunsoverstablegraphs where the sum runs over stable graphs \Gammaandadmissibleweightings and *admissible weightings wmod mod ron* on \Gamma(someadditionalcombinatorialgadget).Thecoefficientofeachindividualterm (some additional combinatorial gadget). The coefficient of each individual term [\Gamma, \alpha]aboveexpressionturnsouttobeapolynomialin above expression turns out to be a polynomial in rfor for r \gg 0$ and we define the DR cycle as the value DRgd,k(A)=DRgd,k,r(A)r=0RH2d(Mg,n)\mathrm{DR}_g^{\,d,k}(A) = \mathrm{DR}_g^{\,d,k,r}(A)|_{r=0} \in RH^{2d}(\overline{\mathcal{M}}_{g,n}) of this polynomial at r=0r=0.

The tautological class DRgd,k(A)\mathrm{DR}_g^{\,d,k}(A) is accessible in admcycles by the function DR_cycle(g,A,d) and we can even compute DRgd,k,r(A)\mathrm{DR}_g^{\,d,k,r}(A) using DR_cycle(g,A,d,rpoly=True).

We can use admcycles to verify theoretical results about Double ramification cycles in special cases. Take the following result, proving a conjecture by Pixton.

Theorem (Clader, Janda - Jan 2016)
Let g,k,dZ0g,k,d \in \mathbb{Z}_{\geq 0} and A=(a1,,an)ZnA=(a_1, \ldots, a_n) \in \mathbb{Z}^n with iai=k(2g2+n)\sum_i a_i = k(2g-2+n). Then DRgd,k(A)\mathrm{DR}_g^{\,d,k}(A) vanishes in degree d>gd>g.

Let us check the above vanishing in a special case in genus g=2g=2.

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DR=DR_cycle(2,(2,3,-5),d=2) DR.is_zero()
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DR2=DR_cycle(2,(2,3,-5),d=3) DR2.is_zero()

Admissible cover cycles in admcycles

In joint work with Jason van Zelm, we studied admissible cover cycles - fundamental classes of loci of curves (C,p1,,pr)(C,p_1, \ldots, p_r) in Mg,r\overline{\mathcal{M}}_{g,r} admitting finite covers CDC \to D to some curve DD of genus ggg'\leq g such that p1,,prp_1, \ldots, p_r are ramification points of the cover.

For simplicity, let's restrict ourselves to the case of degree 22 covers with g=0,1g'=0,1, then we have

Hypg,n,2m={(C,p1,,pn,q1,q1,,qm,qm):C hyperelliptic,pi ramification points,qj,qj conjugate}Mg,n+2mBg,n,2m={(C,p1,,pn,q1,q1,,qm,qm):C bielliptic,pi ramification points,qj,qj conjugate}Mg,n+2m\begin{align*} \mathrm{Hyp}_{g,n,2m} &= \{(C,p_1,\ldots, p_n, q_1, q_1', \ldots, q_m, q_m') : C \text{ hyperelliptic}, p_i\text{ ramification points}, q_j, q_j' \text{ conjugate}\} &\subset M_{g,n+2m}\\ \mathrm{B}_{g,n,2m} &= \{(C,p_1,\ldots, p_n, q_1, q_1', \ldots, q_m, q_m') : C \text{ bielliptic}, p_i\text{ ramification points}, q_j, q_j' \text{ conjugate}\} &\subset M_{g,n+2m} \end{align*}

Then, in many cases, the package admcycles can compute the fundamental classes

[Hypg,n,2m],[Bg,n,2m]H(Mg,n+2m),\begin{align*} [\overline{\mathrm{Hyp}}_{g,n,2m}], [\overline{\mathrm{B}}_{g,n,2m}] \in H^*(\overline{\mathcal{M}}_{g,n+2m}), \end{align*}

accessible via the functions Hyperell(g,n,m) and Biell(g,n,m).

The cycles are computed by using information from their pullbacks via boundary gluing morphisms. For instance, for g=g1+g2g=g_1+g_2, consider the gluing morphism

ξ:Mg1,1×Mg2,1Mg.\xi: \overline{\mathcal{M}}_{g_1,1} \times \overline{\mathcal{M}}_{g_2,1} \to \overline{\mathcal{M}}_g.

Then the pullback of the cycle [Hypg]H(Mg)[\overline{\mathrm{Hyp}}_g] \in H^*( \overline{\mathcal{M}}_g ) is given by

ξ[Hypg]=[Hypg1,1][Hypg2,1].\xi^* [\overline{\mathrm{Hyp}}_g] = [\overline{\mathrm{Hyp}}_{g_1,1}] \otimes [\overline{\mathrm{Hyp}}_{g_2,1}].

In particular, the intersection of Dg1=ξ[Mg1,1×Mg2,1]D_{g_1} = \xi_* [ \overline{\mathcal{M}}_{g_1,1} \times \overline{\mathcal{M}}_{g_2,1} ] with [Hypg][\overline{\mathrm{Hyp}}_g] is given by α=ξ[Hypg1,1][Hypg2,1]\alpha = \xi_* [\overline{\mathrm{Hyp}}_{g_1,1}] \otimes [\overline{\mathrm{Hyp}}_{g_2,1}]. This is something we can again verify in an example, showing how such boundary pushforwards can be constructed.

First we construct the stable graph associated to the boundary divisor.

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g1 = 1; g2 = 2; g = g1 + g2; bdrygraph = StableGraph([g1,g2], [[1],[2]], [(1,2)]) Dg1 = bdrygraph.boundary_pushforward(); Dg1

The class α=ξ[Hypg1,1][Hypg2,1]\alpha = \xi_* [\overline{\mathrm{Hyp}}_{g_1,1}] \otimes [\overline{\mathrm{Hyp}}_{g_2,1}] can also be constructed using boundary_pushforward.

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Hyp_g = Hyperell(g) alpha = bdrygraph.boundary_pushforward([Hyperell(g1,1), Hyperell(g2,1)])
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(Hyp_g * Dg1).toTautbasis()
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alpha.toTautbasis()

Relationship between DR-cycles and admissible cover cycles

There is a natural connection between Double ramification and admissible cover cycles. The cycle DRg(A)\mathrm{DR}_{g}(A) is defined as the pushforward of the moduli space of stable maps to rubber P1\mathbb{P}^1. The components of this rubber moduli space, where generically the curve is smooth, map exactly to the admissible cover cycle of curves CC mapping to P1\mathbb{P}^1 with marked ramification points over 0,0,\infty.

For covers of degree 22, the admissible cover cycles above are exactly the hyperelliptic cycles. The possible partitions AA are

  • (2,2)(2,-2) : two marked Weierstrass point

  • (2,1,1)(2,-1,-1) : one marked Weierstrass point, one pair of conjugate points

  • (1,1,1,1)(1,1,-1,-1) : two pairs of conjugate points

As a proof of concept, the above approach can be applied in genus g=1g=1. Here we are looking at the codimension 11 hyperelliptic cycles

Hyp1,2={(E,p,q):O(2p2q)O}M1,2,Hyp1,1,2={(E,p,q1,q2):O(2pq1q2)O}M1,3,Hyp1,0,4={(E,p1,p2,q1,q2):O(p1+p2q1q2)O}M1,4.\begin{align*} \mathrm{Hyp}_{1,2} &= \{(E,p,q) : \mathcal{O}(2p-2q)\cong \mathcal{O}\} &\subset \mathcal{M}_{1,2},\\ \mathrm{Hyp}_{1,1,2} &= \{(E,p,q_1,q_2) : \mathcal{O}(2p-q_1-q_2) \cong \mathcal{O}\} &\subset \mathcal{M}_{1,3},\\ \mathrm{Hyp}_{1,0,4} &= \{(E,p_1,p_2,q_1,q_2) : \mathcal{O}(p_1+p_2-q_1-q_2) \cong \mathcal{O}\} &\subset \mathcal{M}_{1,4}. \end{align*}

So, in the first case, we know that the cycles DR1(2,2)\mathrm{DR}_1(2,-2) and [Hyp1,2][\overline{\mathrm{Hyp}}_{1,2}] agree away from the boundary. Let's use admcycles to identify the correction - which must be a combination of the classes of boundary divisors.

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DR = DR_cycle(1,(2,-2)) Hyp_1_2 = Hyperell(1,2) (DR-Hyp_1_2).toTautbasis()
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bgraphs = list_strata(1,2,1); bgraphs
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[b.boundary_pushforward().toTautbasis() for b in bgraphs]
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D_0_2=stgraph([0,1],[[1,2,4],[5]],[(4,5)]).boundary_pushforward() # B = Boundary divisor [(g=0;1,2)-(g=1)] relation1=DR-(Hyp_1_2+D_0_2) relation1.is_zero()

By similar experiments, one concludes the following formulas:

DR1(2,2)=[Hyp1,2]+[D0,2]RH2(M1,2),DR1(2,1,1)=[Hyp1,1,2]+[D0,3]RH2(M1,3),DR1(1,1,1,1)=[Hyp1,0,4]+[D0,4]RH2(M1,4)\begin{align*} \mathrm{DR}_1(2,-2) &= [\overline{\mathrm{Hyp}}_{1,2}] + [D_{0,2}] &&\in RH^2(\overline{\mathcal{M}}_{1,2}),\\ \mathrm{DR}_1(2,-1,-1) &= [\overline{\mathrm{Hyp}}_{1,1,2}] + [D_{0,3}] &&\in RH^2(\overline{\mathcal{M}}_{1,3}),\\ \mathrm{DR}_1(1,1,-1,-1) &= [\overline{\mathrm{Hyp}}_{1,0,4}] + [D_{0,4}] &&\in RH^2(\overline{\mathcal{M}}_{1,4}) \end{align*}

On Tuesday, we started exploring the case g=2g=2 and found/verified a more complicated formula

DR2(2,2)=[Hyp2,2]+ boundary corrections.\mathrm{DR}_2(2,-2) = [\overline{\mathrm{Hyp}}_{2,2}] + \text{ boundary corrections.}

As a final example, let us verify the following result

Theorem (Holmes, Pixton, S. - Nov 2017)
Let a,b\textbf a, \textbf b be vectors of nn integers with iai=k1(2g2+n)\sum_i a_i = k_1 (2g-2+n) and ibi=k2(2g2+n)\sum_i b_i = k_2 (2g-2+n), then we have DRg(a)DRg(b)=DRg(a)DRg(a+b)H4g(Mg,nct),\mathrm{DR}_g(\textbf{a}) \cdot \mathrm{DR}_g(\textbf{b}) = \mathrm{DR}_g(\textbf{a}) \cdot \mathrm{DR}_g(\textbf{a}+\textbf{b}) \in H^{4g}({\mathcal{M}}_{g,n}^{ct}), but the same relation is not in general true on all of Mg,n\overline{\mathcal{M}}_{g,n}.

in the case g=1,n=3g=1, n=3.

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R.<a1,a2,a3,b1,b2,b3> = PolynomialRing(QQ,6) diff = DR_cycle(1,(a1,a2,a3)) * ( DR_cycle(1,(b1,b2,b3)) - DR_cycle(1,(a1+b1,a2+b2,a3+b3)) )
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diff.toTautbasis()
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diff.is_zero()
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diff.toTautbasis(moduli='ct')