Intersection theory on (using computers)
Last week:
Tuesday: introduction of the moduli space of stable curves
Thursday: cohomology groups and the subset of tautological classes
Today:
How to do computations in 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
Additional references for the tautological ring
An introduction to the tautological ring and double ramification cycles
- Felix Janda (secret lecture notes, 13 pages)The moduli space of curves
- Johannes Schmitt (non-secret lecture notes, 106 pages)
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/Deligne-Mumford stack 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 .
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.
Let's see how to understand the output for the example [0, 1] [[1, 2, 3], [4]] [(3, 4)]
. Every stable graph is encoded with three lists:
[0, 1]
is a list of genera of the vertices,[[1, 2, 3], [4]]
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], [4]] [(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 with precisely 2 edges. Match them to the picture above.
Digression : the number of stable graphs
Exercise* (homework)
Show that for any , the number of isomorphism classes of stable graphs of genus with legs satisfies where is the natural logarithm. (Hint: Google "Cayley's formula")
Exercise** (homework)
Let be the number of isomorphism classes of stable graphs of genus with legs.
Is it true that , i.e. that there exist such thatExercise (homework)
Program the function
T(g,n)
inadmcycles
and crash this worksheet by computingT(2,6)
.
Gluing morphisms
Assume we are given a stable graph like the following
Then associated to we have a gluing map
Proposition
The map is finite with image equal to the closure of
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 such tautological classes in admcycles
, in our favorite example of . Looking back at the picture from above, there is something we should be able to verify:
The classes and are supported on the green and blue boundary divisors. Their intersection product should be supported on the red locus associated to the graph .
Looking back at the picture, we can see that this problem can be approached combinatorially: given stable graphs , the stratum is contained in if and only if can be obtained from by "contracting some edges".
In our example, is the only graph such that we can obtain both and by contracting some edges. Thus (by basic intersection theory), the product must be supported on . 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 .
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 .
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 . Use it in the cell below to compute .
To do this calculation by hand, you would use the projection formula for the morphism :
Well, that was nice, but also a bit boring. Let's look at something more interesting, namely the intersection of the class :
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 . These arise from the excess intersection formula. For a class like which is a pushforward under a closed, regular embedding of codimension , it tells you that you get the pushforward of the top Chern class of the normal bundle , and in the case above a short computation shows
Intersection numbers
Above we saw how to express products of classes as linear combinations of . However, we can also compute actual intersection numbers. For this, given a tautological class t
contained in , we can compute its degree 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
This is the first time that we really see that 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 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 -classes? Using that such classes are pushforwards of -classes under forgetful maps, such integrals can be converted into (linear combinations of) integrals involving only -classes on spaces with more markings:
Finally, integrals over arbitrary decorated stratum classes are just given by products of integrals of and -classes from each of the vertices:
Tautological relations
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
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.
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 the kernel vanishes (and thus that indeed give all relations)? We can use the following strategy: there exists an intersection pairing
As we have seen in the last sections, we can explicitly compute all pairings appearing above using admcycles
and obtain a matrix whose entries are and this matrix satisfies Then if we check it follows that in indeed the kernel must vanish.
Example :
Let's see this in practice: for the classes in considered above, we saw that toTautbasis()
gave out vectors of length , so in this case. Since in this case, the relevant pairing is
Let's compute the matrix and its rank:
So indeed we get the equality and thus we can be sure that the system is complete in this case.
Application (ongoing research with R. Pandharipande, S. Molcho)
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)
.
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 of curves such that the stable graph of is a tree (i.e. contains no circular path). Then there exists a class such that So, up to a scalar factor, the class is a power of a divisor class! However, it turns out that this equality does not hold in general on the whole of .
What we can show now is that such a formula cannot work at all if is sufficiently large! To be slightly more precise, denote by the sub--algebra of generated by elements of cohomological degree at most . In particular, 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 we have and for we have assuming that the generalized Faber-Zagier relations give all the relations in the spaces Idea of proof Useadmcycles
to check the statement in and . The assumption tells us that in the corresponding spacestoTautbasis()
really does express as well as elements of in a basis of , so the statement is just linear algebra. The case of larger can then be shown using a small argument applying boundary gluing maps like using the fact that .
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 is , so the matrix we have to compute is a matrix of size at least of intersection numbers on the space of complex dimension . This will take a bit of time and effort, but doesn't seem impossible.
Appendix
Functoriality under forgetful maps
A nice feature of the tautological ring is that it is closed under pushforwards and pullbacks by forgetful maps . Let's try it out:
Exercise
Let be the stable graph of the boundary point of . Compute the pullback of under the forgetful map . 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])
.
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.
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 .