 Path: Lecture-AJ.ipynb
Views: 359
Visibility: Unlisted (only visible to those who know the link)
Image: ubuntu2004
Kernel: SageMath 9.1

# Intersection theory on $\overline{\mathcal{M}}_{g,n}$ (using computers)

### Last week:

• Tuesday: introduction of the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves

• Thursday: cohomology groups $H^*(\overline{\mathcal{M}}_{g,n})$ and the subset $RH^*(\overline{\mathcal{M}}_{g,n})$ of tautological classes

### Today:

How to do computations in $RH^*(\overline{\mathcal{M}}_{g,n})$ in practice :

• lots of operations (cup product, intersection numbers, etc) have purely combinatorial description (good)

• the combinatorics is much too difficult to do by hand for large g and n (bad)

• we can use the help of computers (in the form of the SageMath package admcycles) to do the combinatorics for us (good)

### How to get access to admcycles

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 $g,n \geq 0$ be integers (with $2g-2+n>0$). \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 • $\overline{\mathcal{M}}_{g,n}$ is a smooth, irreducible, compact complex orbifold/Deligne-Mumford stack of dimension $\dim_\mathbb{C} \overline{\mathcal{M}}_{g,n} = 3g-3+n$. • The subset $\mathcal{M}_{g,n}$ where the curve $C$ is smooth is dense and open and the complement $\partial \overline{\mathcal{M}}_{g,n}$ is a normal-crossing divisor. Below we illustrate the space $\overline{\mathcal{M}}_{1,2}$. Instead of looking at the curves themselves, we can draw the corresponding dual (or stable) graphs. Let's start using admcycles to enumerate these stable graphs for us. In : from admcycles import * # list_strata(g,n,e) returns the list of stable graphs in Mbar_g,n with e edges L = list_strata(1,2,1) L  Let's see how to understand the output for the example [0, 1] [[1, 2, 3], ] [(3, 4)]. Every stable graph is encoded with three lists: • [0, 1] is a list of genera of the vertices, • [[1, 2, 3], ] is a list of the half-edges and legs at the vertices, • [(3, 4)] is a list of pairs of half-edges that are connected to form edges. Therefore, the output [0, 1] [[1, 2, 3], ] [(3, 4)] encodes the following stable graph: Using the command StableGraph(genera, legs, edges) we can create such a stable graph by hand, where genera, legs and edges are the lists above (we'll see it further below). ### 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 stable graphs in $\overline{\mathcal{M}}_{1,2}$ with precisely 2 edges. Match them to the picture above. In :   ## Digression : the number of stable graphs ### Exercise* (homework) Show that for any $\epsilon>0$, the number $T_n$ of isomorphism classes of stable graphs of genus $0$ with $n$ legs satisfies $e^{(1/2-\epsilon)n \log n} \leq T_n \leq e^{(2+\epsilon) n \log n} \text{ for }n \gg 0\ ,$ where $\log$ is the natural logarithm. (Hint: Google "Cayley's formula") ### Exercise** (homework) Let $T_{g,n}$ be the number of isomorphism classes of stable graphs of genus $g$ with $n$ legs. Is it true that $\log T_{g,n} \in \theta((g+n) \log(g+n))$, i.e. that there exist $c,C>0$ such that $e^{c (g+n) \log(g+n)} \leq T_{g,n} \leq e^{C (g+n) \log(g+n)} \text{ for } g+n \gg 0\ ?$ ### Exercise (homework) Program the function T(g,n) in admcycles and crash this worksheet by computing T(2,6). ### Gluing morphisms Assume we are given a stable graph $\Gamma$ like the following Then associated to $\Gamma$ we have a gluing map $\xi_\Gamma : \prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g(v),n(v)} = \overline{\mathcal{M}}_{0,3} \times \overline{\mathcal{M}}_{1,1}\to \overline{\mathcal{M}}_{1,2}$ Proposition The map $\xi_\Gamma$ is finite with image equal to the closure $\overline{\mathcal{M}}^\Gamma$ of $\mathcal{M}^\Gamma = \{(C,p_1, \ldots, p_n) : \Gamma_{(C,p_1, \ldots, p_n)} = \Gamma\}$ ### The cohomology $H^*(\overline{\mathcal{M}}_{g,n})$ of $\overline{\mathcal{M}}_{g,n}$ $\overline{\mathcal{M}}_{g,n}$ compact space $\implies$ the singular cohomology $H^*(\overline{\mathcal{M}}_{g,n})$ is a finite-dimensional $\mathbb{Q}$-algebra Definition ($\psi$-classes) $\mathbb{L}_i \to \overline{\mathcal{M}}_{g,n}$ complex line bundle, $\mathbb{L}_i|_{(C,p_1, \ldots, p_n)} = T_{p_i}^* C$ $\psi_i = c_1(\mathbb{L}_i) \in H^2(\overline{\mathcal{M}}_{g,n}).$ Definition ($\kappa$-classes) Forgetful morphism $\pi: \overline{\mathcal{M}}_{g,n+1} \to \overline{\mathcal{M}}_{g,n}, (C,p_1, \ldots, p_n, p_{n+1}) \mapsto (C,p_1, \ldots, p_n)$ [$C$ smooth] $\kappa_a = \pi_* \left( (\psi_{n+1})^{a+1}\right) \in H^{2a}(\overline{\mathcal{M}}_{g,n}).$ Definition (The tautological ring) The tautological ring $RH^*(\overline{\mathcal{M}}_{g,n}) \subset H^*(\overline{\mathcal{M}}_{g,n})$ is spanned as a $\mathbb{Q}$-vector subspace by elements $[\Gamma,\alpha]=(\xi_\Gamma)_* \left(\underbrace{\text{product of }\kappa,\psi\text{-classes}}_{\alpha}\text{ on }\prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g(v),n(v)} \right),$ called decorated strata classes. Let's start playing around with such tautological classes in admcycles, in our favorite example of $\overline{\mathcal{M}}_{1,2}$. Looking back at the picture from above, there is something we should be able to verify: The classes $A = [\Gamma_1,1]$ and $B = [\Gamma_2,1]$ are supported on the green and blue boundary divisors. Their intersection product $A \cdot B$ should be supported on the red locus associated to the graph $\Gamma_3$. In : Gamma1 = StableGraph([0, 1], [[1, 2, 3], ], [(3, 4)]) Gamma2 = StableGraph(, [[1, 2, 3, 4]], [(3, 4)]) A = Gamma1.boundary_pushforward() B = Gamma2.boundary_pushforward() A*B  Looking back at the picture, we can see that this problem can be approached combinatorially: given stable graphs $\Gamma, \Gamma'$, the stratum $\overline{\mathcal{M}}^{\Gamma'}$ is contained in $\overline{\mathcal{M}}^{\Gamma}$ if and only if $\Gamma$ can be obtained from $\Gamma'$ by "contracting some edges". In our example, $\Gamma_3$ is the only graph such that we can obtain both $\Gamma_1$ and $\Gamma_2$ by contracting some edges. Thus (by basic intersection theory), the product $A \cdot B$ must be supported on $\overline{\mathcal{M}}^{\Gamma_3}$. But clearly, such combinatorics can be performed by computers. This intuitive argument was made precise by Graber and Pandharipande, and extended to decorated strata classes $[\Gamma, \alpha]$. Theorem (Graber, Pandharipande - 2003) The set of tautological classes $RH^*(\overline{\mathcal{M}}_{g,n})$ is closed under the intersection/cup product, and there exists an explicit formula $[\Gamma,\alpha] \cdot [\Gamma',\alpha'] = \sum \mu_i [\Gamma_i, \alpha_i]$ with $\Gamma_i$ running over graphs which are simultaneous specializations of $\Gamma$ and $\Gamma'$. Instead of giving you the full formula (see e.g. my lecture notes for the gory details), let's try it out in some examples. ### Exercise The function psiclass(i,g,n) creates the class $\psi_i \in RH^2(\overline{\mathcal{M}}_{g,n})$. Use it in the cell below to compute $B \cdot \psi_1 = [\Gamma_2, 1] \cdot \psi_1$. In : Gamma1 = StableGraph([0, 1], [[1, 2, 3], ], [(3, 4)]) Gamma2 = StableGraph(, [[1, 2, 3, 4]], [(3, 4)]) A = Gamma1.boundary_pushforward() B = Gamma2.boundary_pushforward() # write code below  To do this calculation by hand, you would use the projection formula for the morphism $\xi_{\Gamma_2} : \overline{\mathcal{M}}_{0,4} \to \overline{\mathcal{M}}_{1,2}$: $[\Gamma_2, 1] \cdot \psi_1 = (\xi_{\Gamma_2})_*(1) \cdot \psi_1 = (\xi_{\Gamma_2})_*(1 \cdot (\xi_{\Gamma_2})^*\psi_1) = [\Gamma_2, \psi_1]\,.$ Well, that was nice, but also a bit boring. Let's look at something more interesting, namely the intersection of the class $A \cdot A$: In : A^2  When you intersect a class with itself (or more generally, if you intersect classes which meet in unexpectedly low codimension), you start seeing terms of the form $-\psi_h -\psi_{h'}$. These arise from the excess intersection formula. For a class like $A = (\xi_{\Gamma_1})_*(1)$ which is a pushforward under a closed, regular embedding $\xi_{\Gamma_1} : \overline{\mathcal{M}}_{0,3} \times \overline{\mathcal{M}}_{1,1} \to \overline{\mathcal{M}}_{1,2}$ of codimension $d=1$, it tells you that you get the pushforward $(\xi_{\Gamma_1})_*(1)^2 = (\xi_{\Gamma_1})_*(c_d(\mathcal{N}_{\xi_{\Gamma_1}}))$ of the top Chern class of the normal bundle $\mathcal{N}_{\xi_{\Gamma_1}}$, and in the case above a short computation shows $c_1(\mathcal{N}_{\xi_{\Gamma_1}})= -\psi_h -\psi_{h'} \in H^2(\overline{\mathcal{M}}_{0,3} \times \overline{\mathcal{M}}_{1,1})\,.$ ## Intersection numbers Above we saw how to express products of classes $[\Gamma, \alpha]$ as linear combinations of $[\Gamma_i, \alpha_i]$. However, we can also compute actual intersection numbers. For this, given a tautological class t contained in $RH^{2(3g-3+n)}(\overline{\mathcal{M}}_{g,n})$, we can compute its degree $\int_{[\overline{\mathcal{M}}_{g,n}]} \mathrm{t} = \mathrm{deg}(\mathrm{t} \frown [\overline{\mathcal{M}}_{g,n}])$ using t.evaluate(). Let's start with a famous computation (that you'll see in class). ### Exercise Use the cell below to compute the integral $\int_{[\overline{\mathcal{M}}_{1,1}]} \psi_1.$ In :   This is the first time that we really see that $\overline{\mathcal{M}}_{1,1}$ is a stack and not an algebraic variety. Otherwise, the intersection number would have to be an integer. How does admcycles compute such integrals/intersection numbers? This is based on a conjecture Witten (1991), proved in Kontsevich (1992) (and explained beautifully in lecture notes by Joachim Kock). It boils down to a large set of recursive equations between the intersection numbers $\langle \tau_{k_1} \cdots \tau_{k_n} \rangle = \int_{[\overline{\mathcal{M}}_{g,n}]} \psi_1^{k_1} \cdots \psi_n^{k_n}$ which can be seen to completely determine all of these numbers. This recursion was programmed by Aaron Pixton and integrated into admcycles. What about integrals involving $\kappa$-classes? Using that such classes are pushforwards of $\psi$-classes under forgetful maps, such integrals can be converted into (linear combinations of) integrals involving only $\psi$-classes on spaces with more markings: $\int_{[\overline{\mathcal{M}}_{1,1}]} \kappa_1 = \int_{[\overline{\mathcal{M}}_{1,1}]} \pi_* \psi_2^2 = \int_{[\overline{\mathcal{M}}_{1,2}]} \psi_2^2\,.$ Finally, integrals over arbitrary decorated stratum classes $[\Gamma, \alpha]$ are just given by products of integrals of $\kappa$ and $\psi$-classes from each of the vertices: ## Tautological relations An important aspect of the tautological ring is that the generators $[\Gamma, \alpha]$ 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 $R^{\text{FZ}}$ have been implemented by Pixton in the predecessor of admcycles and are accessible in admcycles in two ways: given a tautological class t in $RH^{2d}(\overline{\mathcal{M}}_{g,n})$ you can • check if it is contained in the system $R^{\text{FZ}}$ (and thus zero) by calling t.is_zero(), • express it in a basis of the vector space $\mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}},$ which is conjecturally isomorphic to $RH^{2d}(\overline{\mathcal{M}}_{g,n})$, by calling t.toTautbasis(). ### Exercise There exists a linear relation between the classes $\kappa_1,\, \psi = \psi_1 + \psi_2,\, A = [\Gamma_1, 1] \in RH^2(\overline{\mathcal{M}}_{1,2})\,.$ 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. In : 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],],[(3,4)]).boundary_pushforward()  In :   In :   In :   In :   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 $0 \to K \to \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} \to RH^{2d}(\overline{\mathcal{M}}_{g,n}) \to 0 \quad (*)$ the kernel $K$ vanishes (and thus that $R^{\text{FZ}}$ indeed give all relations)? We can use the following strategy: there exists an intersection pairing \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 $M$ whose entries are $M_{[\Gamma_1, \alpha_1], [\Gamma_2, \alpha_2]} = \int_{[\overline{\mathcal{M}}_{g,n}]} [\Gamma_1, \alpha_1] \cdot [\Gamma_2, \alpha_2]\,,$ and this matrix satisfies $\mathrm{rank}\, M \leq \dim RH^{2d}(\overline{\mathcal{M}}_{g,n}).$ Then if we check $\mathrm{rank}\, M = \dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}}$ it follows that in $(*)$ indeed the kernel $K$ must vanish. ### Example : $RH^2(\overline{\mathcal{M}}_{1,2})$ Let's see this in practice: for the classes in $RH^2(\overline{\mathcal{M}}_{1,2})$ considered above, we saw that toTautbasis() gave out vectors of length $2$, so $\dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} = 2$ in this case. Since $3g-3+n = 2$ in this case, the relevant pairing is $RH^{2}(\overline{\mathcal{M}}_{g,n}) \otimes RH^{2}(\overline{\mathcal{M}}_{g,n}) \to \mathbb{Q}\,.$ Let's compute the matrix $M$ and its rank: In : 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  In : M.rank()  So indeed we get the equality $\mathrm{rank}\, M = \dim \mathrm{span}([\Gamma, \alpha] : \Gamma, \alpha) / R^{\text{FZ}} = 2$ and thus we can be sure that the system $R^{\text{FZ}}$ is complete in this case. ## Application (ongoing research with R. Pandharipande, S. Molcho) On $\overline{\mathcal{M}}_{g,n}$ there exists a vector bundle $\mathbb{E}_g$, called the Hodge bundle, with fibres $\mathbb{E}_g|_{(C,p_1, \ldots, p_n)} = H^0(C, \omega_C).$ The Chern classes $\lambda_i = c_i(\mathbb{E}_g) \in H^{2i}(\overline{\mathcal{M}}_{g,n})$ are called the $\lambda$-classes. They are tautological, with an explicit formula first computed by Mumford, and can be computed in terms of generators $[\Gamma, \alpha]$ using lambdaclass(i,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 $\mathcal{M}_{g,n}^{\mathrm{ct}}$ of curves $(C,p_1, \ldots, p_n)$ such that the stable graph of $C$ is a tree (i.e. contains no circular path). Then there exists a class $\theta \in RH^2(\overline{\mathcal{M}}_{g,n})$ such that $\lambda_g = \frac{1}{g!} \theta^g \in H^{2g}(\mathcal{M}_{g,n}^{\mathrm{ct}})\,.$ So, up to a scalar factor, the class $\lambda_g$ is a power of a divisor class! However, it turns out that this equality does not hold in general on the whole of $\overline{\mathcal{M}}_{g,n}$. What we can show now is that such a formula cannot work at all if $g$ is sufficiently large! To be slightly more precise, denote by $RH^*_{\leq k}(\overline{\mathcal{M}}_{g,n}) \subseteq RH^*(\overline{\mathcal{M}}_{g,n})$ the sub-$\mathbb{Q}$-algebra of $RH^*(\overline{\mathcal{M}}_{g,n})$ generated by elements of cohomological degree at most $2k$. In particular, $RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n})$ 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 - last Friday) For $g \geq 4$ we have $\lambda_g \notin RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n})$ and for $g \geq 5$ we have $\lambda_g \notin RH^*_{\leq 2}(\overline{\mathcal{M}}_{g,n})\, ,$ assuming that the generalized Faber-Zagier relations $R^{\text{FZ}}$ give all the relations in the spaces $RH^{8}(\overline{\mathcal{M}}_{4,1}) \ \text{ and }\ RH^{10}(\overline{\mathcal{M}}_{5,1}).$ Idea of proof Use admcycles to check the statement in $\overline{\mathcal{M}}_{4,1}$ and $\overline{\mathcal{M}}_{5,1}$. The assumption tells us that in the corresponding spaces toTautbasis() really does express $\lambda_g$ as well as elements of $RH^*_{\leq 1}(\overline{\mathcal{M}}_{g,n})$ in a basis of $RH^*(\overline{\mathcal{M}}_{g,n})$, so the statement is just linear algebra. The case of larger $g,n$ can then be shown using a small argument applying boundary gluing maps like $\xi : \overline{\mathcal{M}}_{5,1} \times \overline{\mathcal{M}}_{1,2} \to \overline{\mathcal{M}}_{6,1}\,,$ using the fact that $\xi^* \lambda_6 = \lambda_5 \otimes \lambda_1$. 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 $RH^{10}(\overline{\mathcal{M}}_{5,1})$ is $1371$, so the matrix $M$ we have to compute is a matrix of size at least $1371 \times 1371$ of intersection numbers on the space $\overline{\mathcal{M}}_{5,1}$ of complex dimension $13$. This will take a bit of time and effort, but doesn't seem impossible. Thanks for your attention! # Appendix ## Functoriality under forgetful maps A nice feature of the tautological ring is that it is closed under pushforwards and pullbacks by forgetful maps $\pi: \overline{\mathcal{M}}_{g,n+1} \to \overline{\mathcal{M}}_{g,n}$. Let's try it out: In : P = psiclass(2,1,2)^2 # psi_2^2 on Mbar_{1,2} P.forgetful_pushforward() # pushforward of P under pi : Mbar_{1,2} -> Mbar_{1,1} forgetting marking 2  ### Exercise Let $\Gamma_0$ be the stable graph of the boundary point of $\overline{\mathcal{M}}_{1,1}$. Compute the pullback of $[\Gamma_0, 1]$ under the forgetful map $\pi: \overline{\mathcal{M}}_{1,2} \to \overline{\mathcal{M}}_{1,1}$. For this, use that if T is a tautological class, the pullback of T under a forgetful map of markings [m1,...,mk] is computed by T.forgetful_pullback([m1,...,mk]). In :   ### Exercise Verify one of the most important tautological relations, called the WDVV-relation on the space $\overline{\mathcal{M}}_{0,4}$: ## Natural cycle classes on the moduli space of stable curves Using the fact that $\overline{\mathcal{M}}_{g,n}$ 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) Lambda classes (and generalizations) On $\overline{\mathcal{M}}_{g,n}$ there exists a vector bundle $\mathbb{E}_g$, called the Hodge bundle, with fibres $\mathbb{E}_g|_{(C,p_1, \ldots, p_n)} = H^0(C, \omega_C).$ The Chern classes $\lambda_i = c_i(\mathbb{E}_g) \in H^{2i}(\overline{\mathcal{M}}_{g,n})$ are called the $\lambda$-classes. They are tautological, with an explicit formula first computed by Mumford, and can be computed in terms of generators $[\Gamma, \alpha]$ using lambdaclass(i,g,n). ### Exercise A special case of the main result of [Faber, Pandharipande - 1998] is that the generating series $F(t) = 1 + \sum_{g > 0} t^{2g} \int_{\overline{\mathcal{M}}_{g,1}} \psi_1^{2g-2} \lambda_g$ is given by $F(t) = \frac{t/2}{\sin(t/2)}.$ Check their result for $g=2$. Hint: To get you started, I compute the expansion of $F$ at $t=0$ below. In : R.<t> = PowerSeriesRing(QQ) F = (t/2)/sin(t/2); F  In : (psiclass(1,2,1)^2 * lambdaclass(2,2,1)).evaluate()  Note that for the universal curve $\pi : \overline{\mathcal{C}}_{g,n} \to \overline{\mathcal{M}}_{g,n}$ and the relative dualizing line bundle $\omega_\pi$ of this morphism, we have $\mathbb{E}_g = R^0 \pi_* \omega_\pi.$ The paper [Pagani, Ricolfi, van Zelm - 2019] generalizes the $\lambda$-classes and computes, for an arbitrary line bundle $\mathcal{L}$ on $\overline{\mathcal{C}}_{g,n}$ the Chern classes $c_i(R^\bullet \pi_* \mathcal{L})$ of the derived pushforward of $\mathcal{L}$ by $\pi$. 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. ### B) Admissible cover cycles For $S \subset \overline{\mathcal{M}}_{g,n}$ a closed, algebraic subset of $\mathbb{C}$-codimension $d$, there exists a fundamental class $[S] \in H_{\dim - 2d}(\overline{\mathcal{M}}_{g,n}) \cong H^{\dim - 2d}(\overline{\mathcal{M}}_{g,n})^\vee \underset{\mathrm{PD}}{\cong} H^{2d}(\overline{\mathcal{M}}_{g,n}),$ where $\mathrm{PD}$ is the isomorphism from Poincaré duality. In joint work with Jason van Zelm, we studied admissible cover cycles - fundamental classes of loci of curves $(C,p_1, \ldots, p_r)$ in $\overline{\mathcal{M}}_{g,r}$ admitting finite covers $C \to D$ to some curve $D$ of genus $g'\leq g$ such that $p_1, \ldots, p_r$ are ramification points of the cover. An important special case are the loci of hyperelliptic curves $\begin{equation*} \mathrm{Hyp}_{g,n,2m} = \left\{(C,p_1,\ldots, p_n, q_1, q_1', \ldots, q_m, q_m') : \begin{array}{c} C \text{ hyperelliptic},\\p_i\text{ Weierstrass points},\\ q_j, q_j' \text{ hyperelliptic conjugate}\end{array} \right\} \subset \mathcal{M}_{g,n+2m} \end{equation*}$ Then, [Faber, Pandharipande - 2013] showed that the fundamental classes of the closures \begin{align*} [\overline{\mathrm{Hyp}}_{g,n,2m}] \in H^*(\overline{\mathcal{M}}_{g,n+2m}), \end{align*} are always tautological, and in many cases, the package admcycles can compute them via the function Hyperell(g,n,m). ### C) Strata of differentials Given $\textbf{a}=(a_1, \ldots, a_n) \in \mathbb{Z}^n$ with $\sum_i a_i = 2g-2$, we consider the locus \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 $\mathcal{H}_g(\textbf{a})$ is a closed algebraic subset of $\mathcal{M}_{g,n}$ with $\mathrm{codim}_\mathbb{C} \mathcal{H}_g(\textbf{a}) = \begin{cases} g-1 & \text{ for }a_1, \ldots, a_n \geq 0,&\text{ (strata of holomorphic differentials)}\\ g & \text{ otherwise.}&\text{ (strata of meromorphic differentials)} \end{cases}$ Taking the fundamental class of the closure $\overline{\mathcal{H}}_g(\textbf{a})$, we obtain cohomology classes $[\overline{\mathcal{H}}_g(\textbf{a})] \in H^*(\overline{\mathcal{M}}_{g,n}).$ 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 $[\overline{\mathcal{H}}_g(a_1, \ldots, a_n)] + \sum \text{(boundary corrections)} = \mathrm{DR}_g(a_1+1, \ldots, a_n+1)$ relating the classes of strata of meromorphic differentials to the double ramification cycles. These DR cycles (discussed below) 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 $[\overline{\mathcal{H}}_g(a_1, \ldots, a_n)]$, both in the holomorphic and meromorphic case. This has been implemented in the function Strataclass(g,1,(a1, ..., an)), and for the purpose of this presentation, we'll treat it as a black box. In : H = Strataclass(2,1,(2,)) # computes closure of {(C,p) : omega_C = O_C(2p)} Hpush = H.forgetful_pushforward() # computes pushforward under pi : Mbar_{2,1} -> Mbar_2 Hpush.simplify()  • The strata $\overline{\mathcal{H}}_g(\textbf{a})$ themselves were studied in a series of papers [Bainbridge, Chen, Gendron, Grushevsky, Möller]. In particular, the authors define a smooth compact moduli space $\mathcal{MS}_g(\textbf{a}) \to \overline{\mathcal{H}}_g(\textbf{a}),$ sitting proper, birationally over $\overline{\mathcal{H}}_g(\textbf{a})$, called the space of multiscale differentials. In particular, they describe the boundary strata of $\mathcal{MS}_g(\textbf{a})$ (and thus of $\overline{\mathcal{H}}_g(\textbf{a})$) in terms of certain enhanced level graphs. • The intersection theory and tautological ring of $\mathcal{MS}_g(\textbf{a})$ 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 ${\mathcal{H}}_g(\textbf{a})$ in a range of examples. In : from admcycles.diffstrata import Stratum X = Stratum((2,)) X.euler_characteristic()  We'll come back to some ongoing investigations about strata of differentials and some open questions in a later section. ### 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 $g,k,d \in \mathbb{Z}_{\geq 0}$ and $A=(a_1, \ldots, a_n) \in \mathbb{Z}^n$ with $\sum_i a_i = k(2g-2+n)$. Then for any integer $r \geq 1$ Pixton gave an explicit formula\mathrm{DR}_g^{\,d,k,r}(A) = \sum_{\Gamma,w} \left[\Gamma, \text{(polynomial in $\kappa,\psi$-classes depending on$wParseError: KaTeX parse error: Expected 'EOF', got '}' at position 2: )}̲ \right]\in RH^…$where the sum runs over stable graphs$\Gamma$and *admissible weightings$w$mod$r$* on$\Gamma$(some additional combinatorial gadget). The coefficient of each individual term$[\Gamma, \alpha]$above expression turns out to be a polynomial in$r$for$r \gg 0$ and we define the DR cycle as the value $\mathrm{DR}_g^{\,d,k}(A) = \mathrm{DR}_g^{\,d,k,r}(A)|_{r=0} \in RH^{2d}(\overline{\mathcal{M}}_{g,n})$ of this polynomial at $r=0$.

The tautological class $\mathrm{DR}_g^{\,d,k}(A)$ is accessible in admcycles by the function DR_cycle(g,A,d) and we can even compute $\mathrm{DR}_g^{\,d,k,r}(A)$ 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 $g,k,d \in \mathbb{Z}_{\geq 0}$ and $A=(a_1, \ldots, a_n) \in \mathbb{Z}^n$ with $\sum_i a_i = k(2g-2+n)$. Then $\mathrm{DR}_g^{\,d,k}(A)$ vanishes in degree $d>g$.

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

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

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


The following is a conjecture told to me by Longting Wu, which would generalize the theorem above.

Conjecture (Wu)
Let $g,k,d \in \mathbb{Z}_{\geq 0}$ and $A=(a_1, \ldots, a_n) \in \mathbb{Z}^n$ with $\sum_i a_i = k(2g-2+n)$. Then for $\ell \geq 0$, the $l$-th derivative in $r$ at $r=0$ of the class $\mathrm{DR}_g^{\,d,k,r}(A)$ vanishes in degree $d>g+\ell$.

The theorem by Clader and Janda is the case $\ell=0$ of the conjecture above, since we defined the DR cycle as the value of $\mathrm{DR}_g^{\,d,k,r}(A)$ at $r=0$.

Let's check Longting's conjecture in an example.

In :
D = DR_cycle(1,(1,2,3,-6),3,rpoly=True)
D.toTautbasis()


## Admissible cover cycles in admcycles

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

For simplicity, let's restrict ourselves to the case of degree $2$ covers with $g'=0,1$, then we have

\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

\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 $g=g_1+g_2$, consider the gluing morphism

$\xi: \overline{\mathcal{M}}_{g_1,1} \times \overline{\mathcal{M}}_{g_2,1} \to \overline{\mathcal{M}}_g.$

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

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

In particular, the intersection of $D_{g_1} = \xi_* [ \overline{\mathcal{M}}_{g_1,1} \times \overline{\mathcal{M}}_{g_2,1} ]$ with $[\overline{\mathrm{Hyp}}_g]$ is given by $\alpha = \xi_* [\overline{\mathrm{Hyp}}_{g_1,1}] \otimes [\overline{\mathrm{Hyp}}_{g_2,1}]$. 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.

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


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

In :
Hyp_g = Hyperell(g)
alpha = bdrygraph.boundary_pushforward([Hyperell(g1,1), Hyperell(g2,1)])

In :
(Hyp_g * Dg1).toTautbasis()

In :
alpha.toTautbasis()


## Relationship between DR-cycles and admissible cover cycles

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

For covers of degree $2$, the admissible cover cycles above are exactly the hyperelliptic cycles. The possible partitions $A$ are

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

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

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

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

\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 $\mathrm{DR}_1(2,-2)$ and $[\overline{\mathrm{Hyp}}_{1,2}]$ agree away from the boundary. Let's use admcycles to identify the correction - which must be a combination of the classes of boundary divisors.

In :
DR = DR_cycle(1,(2,-2))
Hyp_1_2 = Hyperell(1,2)
(DR-Hyp_1_2).toTautbasis()

In :
bgraphs = list_strata(1,2,1); bgraphs

In :
[b.boundary_pushforward().toTautbasis() for b in bgraphs]

In :
D_0_2=stgraph([0,1],[[1,2,4],],[(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:

\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 $g=2$ and found/verified a more complicated formula

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

As a final example, let us verify the following result

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

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

In :
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)) )

In :
diff.toTautbasis()

In :
diff.is_zero()

In :
diff.toTautbasis(moduli='ct')