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 be integers (with ). $$\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
is a smooth, irreducible, compact complex orbifold of dimension .
The subset where the curve is smooth is dense and open and the complement is a normal-crossing divisor.
Below we illustrate the space .
Recursive boundary structure
To we can associate a stable graph
Conversely, given a stable graph we have a gluing map
Proposition
The map is finite with image equal to the closure of
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 with precisely 2 edges.
(Hint: given a list L
you can compute its length using len(L)
)
The cohomology of
compact space the singular cohomology is a finite-dimensional -algebra
Definition (-classes)
complex line bundle,
Definition (-classes)
Forgetful morphism [ smooth]
Definition (The tautological ring)
The tautological ring is spanned as a -vector subspace by elements called decorated strata classes.
Let's start playing around with tautological classes. We can display a list of all generators using the function list_tautgens(g,n,d)
and get access to this list using tautgens(g,n,d)
.
Theorem (Graber, Pandharipande - 2003)
The set of tautological classes is closed under the intersection/cup product, and there exists an explicit formula with running over graphs which are simultaneous specializations of and .
We can look at some examples involving the generators above.
We can also compute actual intersection numbers. For this, given a tautological class t
contained in , we can compute its degree using t.evaluate()
. This is based on [Witten - 1991; Kontsevich - 1992].
Exercise
Use the cell below to compute the integral Hint: You can get access to the classes either from the list T
above, or using the function psiclass(i,g,n)
.
An important aspect of the tautological ring is that the generators 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 have been implemented by Pixton in the predecessor of admcycles
and are accessible in admcycles
in two ways: given a tautological class t
in you can
check if it is contained in the system (and thus zero) by calling
t.is_zero()
,express it in a basis of the vector space which is conjecturally isomorphic to , by calling
t.toTautbasis()
.
Exercise
Verify one of the most important tautological relations, called the WDVV-relation on the space :
Natural cycle classes on the moduli space of stable curves
Using the fact that 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 there exists a vector bundle , called the Hodge bundle, with fibres The Chern classes are called the -classes. They are tautological, with an explicit formula first computed by Mumford, and can be computed in terms of generators using lambdaclass(i,g,n)
.
Exercise
A special case of the main result of [Faber, Pandharipande - 1998] is that the generating series is given by Check their result for .
Hint: To get you started, I compute the expansion of at below.
Note that for the universal curve and the relative dualizing line bundle of this morphism, we have The paper [Pagani, Ricolfi, van Zelm - 2019] generalizes the -classes and computes, for an arbitrary line bundle on the Chern classes of the derived pushforward of by . 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 a closed, algebraic subset of -codimension , there exists a fundamental class where is the isomorphism from Poincaré duality.
In joint work with Jason van Zelm, we studied admissible cover cycles - fundamental classes of loci of curves in admitting finite covers to some curve of genus such that are ramification points of the cover.
An important special case are the loci of hyperelliptic curves
Then, [Faber, Pandharipande - 2013] showed that the fundamental classes of the closures
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 with , we consider the locus Then is a closed algebraic subset of with Taking the fundamental class of the closure , we obtain cohomology classes
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 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 , 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.
The strata themselves were studied in a series of papers [Bainbridge, Chen, Gendron, Grushevsky, Möller]. In particular, the authors define a smooth compact moduli space sitting proper, birationally over , called the space of multiscale differentials. In particular, they describe the boundary strata of (and thus of ) in terms of certain enhanced level graphs.
The intersection theory and tautological ring of have been implemented by [Costantini, Möller, Zachhuber - 2020] in an extension/sub-package of
admcycles
calleddiffstrata
. They use this in a second paper [Costantini, Möller, Zachhuber - 2020] to compute the (orbifold) Euler characteristics of the open strata in a range of examples.
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 and with . Then for any integer Pixton gave an explicit formula $$\mathrm{DR}_g^{\,d,k,r}(A) = \sum_{\Gamma,w} \left[\Gamma, \text{(polynomial in $\kappa,\psiwParseError: KaTeX parse error: Expected 'EOF', got '}' at position 2: )}̲ \right]\in RH^…\Gammawr\Gamma[\Gamma, \alpha]rr \gg 0$ and we define the DR cycle as the value of this polynomial at .
The tautological class is accessible in admcycles
by the function DR_cycle(g,A,d)
and we can even compute 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 and with . Then vanishes in degree .
Let us check the above vanishing in a special case in genus .
The following is a conjecture told to me by Longting Wu, which would generalize the theorem above.
Conjecture (Wu)
Let and with . Then for , the -th derivative in at of the class vanishes in degree .
The theorem by Clader and Janda is the case of the conjecture above, since we defined the DR cycle as the value of at .
Let's check Longting's conjecture in an example.
Relationship between strata of differentials and Witten's r-spin class
In the following, I'll discuss an ongoing project together with Q. Chen, F. Janda, R. Pandharipande, A. Pixton, Y. Ruan, A. Sauvaget and D. Zvonkine, threatening to eventually result in a paper abbreviated [CJPPRSZ].
The project concerns relationships between cycles of strata of differentials and Witten's -spin class . This class is a cohomological field theory studied by [Witten - 1991], [Polishchuk, Vaintrob - 2001], [Chiodo - 2006], [Mochizuki - 2006], [Fan, Jarvis, Ruan - 2013]. For our purposes, it suffices to say that given a positive integer , Witten's -spin class is a function and the (complex) codimension of the class is given by The following conjecture is the starting point of our investigation relating Witten's class to the strata of differentials.
Conjecture (Janda, Pandharipande, Pixton, Zvonkine - Jul 2016)
Let be nonnegative integers with , then the expression becomes a polynomial in for and we have
When exploring this Conjecture, F. Janda implemented a formula for the Witten class in a Sage program by A. Pixton (which is the predecessor of admcycles
). On the other hand, the cycles and had previously been computed in [Belorousski, Pandharipande - 2010]. Pixon's program was then able to verify the equality of the conjecture in these two cases.
Recently, Janda's implementation of Witten's class has been incorporated into admcycles
as the function Wittenrspin
. It follows the convention that
Exercise
Check the conjecture above for for and .
Hint: Recall that you can express a tautological class t
in a basis by t.toTautbasis()
.
But now that we have Witten's class implemented, we can start exploring further conjectures! To do so, we should allow more general insertions than just the (constant) positive integers . It turns out, that the right thing to try is linear insertions $$\lambda_i r + b_i\text{, for }\begin{cases}\lambda_i \in \mathbb{Q}, \text{ and $rParseError: KaTeX parse error: Expected 'EOF', got '}' at position 25: …ntly divisible,}̲\\ b_i \in \mat…\lambda_i, b_i\sum_i \lambda_i = e \in \mathbb{Z}rr \gg 0$ sufficiently divisible), and the value of this polynomial at is computed by Wittenrspin
.
The simplest example of this is , concentrated in a single insertion with , which results in a class of complex codimension .
Exercise
Make an educated guess what kind of class W2
could compute, and check it.
Don't scroll down if you want to avoid spoilers.
So, as we've seen a single insertion seems to correspond to computing a stratum of differentials with a pole of order at the corresponding marking. However, if we want to have a stratum with two distinct poles, we run into a problem: two insertions , raise the codimension to , so we cannot get a stratum of differentials this way (which is codimension even with two poles).
It turns out that such insertions still compute something interesting.
Theorem (S.)
Assume the conjecture about strata of holomorphic differentials above. Then for nonnegative integers and positive integers with the classWittenrspin(g,(a1, ..., an, r-b1, ..., r-bm))
computing the value at of equals the fundamental class of the closure of $$\mathcal{H}_g(\textbf{a}, -\textbf{b})^{\text{res}=0} = \left \{(C,p_1, \ldots, p_n, q_1, \ldots, q_m) : \begin{array}{c} \exists\, \eta \text{ meromorphic differential on $CParseError: KaTeX parse error: Expected 'EOF', got '}' at position 1: }̲\\\text{ with }…$
Idea of proof:
Below we can check the equality using admcycles
. Since the computation takes a long time, I prepared it before and just show you the output.
Note that in Strataclass
we use the option res_cond=[2]
to impose vanishing residue at the second marking (the residue at the third marking then vanishes automatically due to the residue theorem). With method='diffstrata'
we specify that we use the package diffstrata
for this computation. Indeed, on the multiscale space one can use tautological relations to convert the residue condition on the differential, defining a divisor class on , into a sum of -classes and boundary strata, and then push forward to .
One question remains: how do we get strata of meromorphic differentials with multiple poles but without extra residue conditions? The following computations (first done in June 2020) suggest that the right kind of insertions are of the form for .
Above we took , but a combinatorial argument based on the formula for Witten's class shows that it gives the same answer for any rational with .
So to summarize, we have that from Witten's class, we get strata of differentials where insertions
give zeros of the corresponding order,
with give poles of order ,
give poles of order with vanishing residues.
After this conjecture was formulated based on the experiments in admcycles
, it took Felix Janda about a week to announce a sketch of proof that the conjecture is true (based on the methods developed to prove the original conjecture, about which he and Dimitri Zvonkine talked in the seminar last semester).
Open questions
What about insertions for ? I have a small data point in this direction: for consider the stratum in . There is a well-defined map sending the curve admitting a differential with zero at and poles at to the point In [Gendron, Tahar - 2020] the authors compute the number of preimages of a generic point in . I checked in many nontrivial examples, that this number agrees with the degree of the zero cycle
Wittenrspin(0,(a, (n-1)/n*r-b1, ..., (n-1)/n*r-bn))
so it seems that insertions with could give strata of differentials with more complicated (linear) conditions on the residues.If all entries of the vector are divisible by two, the strata of differentials decompose into a disjoint union according to whether the line bundle , which is a square root of , is an odd or an even theta characteristic. Can we find a way to compute both summands separately?
Is there a version of Witten's class that produces strata of -differentials?
Ongoing developments
Johannes Schwab (PhD student of M. Möller) - Expand
diffstrata
for spaces of quadratic differentialsJavier 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)
Thanks for your attention!
Appendix/Additional material
Admissible cover cycles in admcycles
In joint work with Jason van Zelm, we studied admissible cover cycles - fundamental classes of loci of curves in admitting finite covers to some curve of genus such that are ramification points of the cover.
For simplicity, let's restrict ourselves to the case of degree covers with , then we have
Then, in many cases, the package admcycles
can compute the fundamental classes
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 , consider the gluing morphism
Then the pullback of the cycle is given by
In particular, the intersection of with is given by . 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.
The class can also be constructed using boundary_pushforward
.
Relationship between DR-cycles and admissible cover cycles
There is a natural connection between Double ramification and admissible cover cycles. The cycle is defined as the pushforward of the moduli space of stable maps to rubber . The components of this rubber moduli space, where generically the curve is smooth, map exactly to the admissible cover cycle of curves mapping to with marked ramification points over .
For covers of degree , the admissible cover cycles above are exactly the hyperelliptic cycles. The possible partitions are
: two marked Weierstrass point
: one marked Weierstrass point, one pair of conjugate points
: two pairs of conjugate points
As a proof of concept, the above approach can be applied in genus . Here we are looking at the codimension hyperelliptic cycles
So, in the first case, we know that the cycles and agree away from the boundary. Let's use admcycles
to identify the correction - which must be a combination of the classes of boundary divisors.
By similar experiments, one concludes the following formulas:
On Tuesday, we started exploring the case and found/verified a more complicated formula
As a final example, let us verify the following result
Theorem (Holmes, Pixton, S. - Nov 2017)
Let be vectors of integers with and , then we have but the same relation is not in general true on all of .
in the case .