Tautological classes on the moduli space of hyperelliptic curves with rational tails
We study tautological classes on the moduli space of stable -pointed hyperelliptic curves of genus with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. The connection with recent conjectures by Pixton is also discussed.
- 1 Tautological classes on the space of hyperelliptic curves
- 2 Tautological classes on the universal Jacobian
- 3 The Faber-Pandharipande cycle
- 4 The Gross-Schoen cycle
- 5 The degree relation
- 6 Products of the universal curve over
- 7 The Fulton-MacPherson compactification
- 8 The tautological ring of
- 9 The tautological ring of
- 10 Tautological cohomology via monodromy
- 11 A remark on Pixton’s conjecture
- 12 Concluding remarks and further directions
In this article we study tautological classes on the moduli space of stable -pointed hyperelliptic curves of genus with rational tails. Tautological classes are natural algebraic cycles reflecting the nature of the generic object parameterized by the moduli space. The set of generators consists of an explicit collection of cycles. In particular, tautological groups are finite dimensional vector spaces. This distinguishes remarkably the tautological ring from the out of reach space of all algebraic cycles. A basic question regarding tautological algebras is to give a meaningful class of tautological relations.
Our strategy in studying tautological classes on is to look at the fibers of the projection . The reduced fiber of over a moduli point corresponding to a smooth hyperelliptic curve is the Fulton-MacPherson compactification of the configuration space of points on . There is a natural way to view the tautological ring of as an algebra over the tautological ring of the cartesian product of . Basic relations among tautological classes on are obtained. These relations are divided into 3 parts:
The vanishing of the Faber-Pandharipande cycle,
The vanishing of the Gross-Schoen cycle,
A relation of degree involving points.
We show that all these relations can be obtained from relations on the universal Jacobian over the space . Our argument in proving the first two vanishings depends strongly on the fact that we are working with hyperelliptic curves. The last relation has a different nature and holds over any family of smooth curves of genus . We will give two independent arguments to prove this relation. Yin has pointed out that the degree relation can be obtained from the vanishing of a certain Chow motive as well.
From our results we get a complete description of tautological relations on the moduli space . We will see that the structure of the tautological algebra is determined by studying tautological classes on the fiber :
Let be a fixed hyperelliptic curve of genus . The tautological ring of the moduli space is naturally isomorphic to the tautological ring of the fiber . In particular, the intersection pairings are perfect in all degrees.
Everything mentioned above concerns tautological classes in Chow. The Gorenstein property of implies the same results in cohomology. This shows that there is no difference between Chow and cohomology as long as we restrict to tautological classes. Using a result of Petersen and Tommasi, which was our motivation for this project, we prove the following:
The cycle class map induces an isomorphism between the tautological ring of the moduli space in Chow and monodromy invariant classes in cohomology.
At the end we discuss the connection between the relations on the space and Pixton’s relations on .
We consider algebraic cycles modulo rational equivalence. All Chow groups are taken with -coefficients.
Acknowledgments. I am grateful to all my colleagues who made suggestions and corrections on the preliminary version of this note.
I would like to thank Carel Faber, Gerard van der Geer, Richard Hain, Robin de Jong, Nicola Pagani, Aaron Pixton and Orsola Tommasi for the valuable discussions and their comments. Thanks to Felix Janda for sending me the notes on Pixton’s relations on products of the universal curve, answering many questions in that direction and his comments. Special thanks are due to Qizheng Yin for explaining several aspects of the theory developed in his thesis and his comments and corrections. Several parts of this research was carried out during my stay at the Max-Planck-Institut für Mathematik at Bonn in 2013. This research was completed in the group of Sergey Shadrin at the KdV Instituut voor Wiskunde at the university of Amsterdam. Thanks to Sergey Shadrin for his interest in this project and his supports. I would like to thank both institutes for their supports.
1. Tautological classes on the space of hyperelliptic curves
Let be the space of stable -pointed hyperelliptic curves of genus with rational tails. It is a quasi-projective variety of dimension . It parameterizes objects of the form , where is stable hyperelliptic with distinct smooth points for . We assume that is a reduced nodal curve of arithmetic genus with exactly one component of genus . For each rational component of the curve the markings and nodes are called special points. By the stability condition we require that all rational components have at least 3 special points. As a result each object of the corresponding moduli problem has finitely many automorphisms and the resulting stack is of Deligne-Mumford type.
Tautological classes on are defined as natural algebraic cycles on the moduli space. Let be the universal hyperelliptic curve of genus and denote by its relative dualizing sheaf. Its class in the Picard group of is denoted by . Denote by the -fold fiber product of over . We define , where is the projection onto the factor for every . Its pull-back via the contraction map is denoted by the same letter. Tautological classes on come from the classes and those supported on the boundary of the partial compactification . Recall that for each subset of the marking set having at least 2 elements there is a boundary divisor class in Pic. It corresponds to those nodal curves having 2 components. According to our definition one component is of genus and the other component is rational. The index set refers to the markings on the rational component. We now define the tautological ring of the moduli space:
The tautological ring of is the -subalgebra of the rational Chow ring of generated by the divisor classes for and the boundary divisors for subsets with at least 2 elements.
There is another, equivalent, way to define tautological classes. We consider the system of Chow rings for all stable . The system of tautological rings are defined as the smallest collection of -subalgebras of the Chow rings having the identity element and stable under all natural maps among these spaces. It is straightforward to see that all classes we defined above belong to the tautological ring with this definition and they are generators.
Recall that the line bundle , whose fiber over the moduli point is the cotangent space of at , gives tautological classes. Its first Chern class is called the -class. It is related to the classes considered before via the following equality:
Other natural classes on the moduli space are kappa classes. Recall that the kappa class is defined as . Notice that the kappa class vanishes when since has trivial Chow groups.
We study the connection between tautological classes on moduli of curves and the universal Jacobian. There are many natural maps from a curve into its Jacobian. We find it more convenient to use a Weierstraß point on the curve to define such a map. We work with the moduli space of Weierstraß pointed hyperellipctic curves of genus . This space parameterizes objects of the form , where is a hyperelliptic curve of genus and is a Weierstraß point on . The universal family admits a section . It associates the Weierstraß point to the pair . The space is a finite cover of of degree . In a similar way we consider a pointed version of this moduli space and define the space . In an analogous manner we define tautological rings of the moduli spaces parameterizing Weierstraß pointed curves. In each case there is a map to a moduli of curves by ignoring the Weierstraß point. Tautological classes are defined as pull-backs of tautological classes on the moduli of curves via these maps.
2. Tautological classes on the universal Jacobian
In this section we review basic notions about tautological classes on the universal Jacobian. Relations among these classes give interesting results on moduli of curves. The tautological ring of a fixed Jacobian variety is defined by Beauville. In  he studies tautological classes on the Jacobian of a fixed curve under algebraic equivalence. The idea is to consider the class of a curve of genus inside its Jacobian and apply all natural operators to it induced from the group structure on the Jacobian and the intersection product in the Chow ring. He shows that the resulting algebra becomes stable under the Fourier transform. In fact, if one applies the Fourier transform to the class of the curve, all components in different degrees belong to the tautological algebra. Beauville shows that these components yield a set of generators with elements. When the curve admits a degree map into the projective line the tautological ring is generated by elements.
Let be a family of smooth curves of genus which admits a section . Denote by the relative Picard scheme of divisors of degree zero. It is an abelian scheme over the base of relative dimension . The section induces an injection from into the universal Jacobian . The geometric point on a curve is sent to the line bundle via the morphism . The abelian scheme is equipped with the Beauville decomposition defined in . Components of this decomosition are eigenspaces of the natural maps corresponding to multiplication with integers. More precisely, for an integer consider the associated endomorphism on . The subgroup is defined as all degree classes on which the morphism acts via multiplication with . Equivalently, the action of the morphism on is multiplication by . The Beauville decomposition has the following form:
Beside the intersection product on Chow groups there is another multiplication on . The Pontryagin product defined in terms of the addition
on . Let be the natural projections and be elements of the Chow ring of . The Pontryagin product of and is defined as .
The universal theta divisor trivialized along the zero section is defined in the rational Picard group of . It defines a principal polarization on . The first Chern class of the Poincaré bundle is defined as . The Fourier Mukai transform gives an isomorphism between and . It is defined as follows:
We now recall the definition of the tautological ring of from . It is defined as the smallest -subalgebra of the Chow ring which contains the class of and is stable under the Fourier transform and all maps for integers . It follows that for an integer it becomes stable under as well. From this definition we get infinitely many tautological classes. But one can see that the tautological algebra is finitely generated. In particular, it has finite dimensions in each degree. The generators are expressed in terms of the components of the curve class in the Beauville decomposition. Define the following classes:
We have that and . The class vanishes for or or . The tautological class comes from the class of relative dualizing sheaf of . It is defined as
It is proven in  that the tautological ring of is generated by the classes and . A crucial feature of the tautological ring of the universal Jacobian is the Lefschetz decomposition. In the classic case the action on Chow groups of an abelian variety was studied by Künnemann . Polishchuk  has studied the action for abelian schemes. We follow the standard convention that is generated by elements satisfying:
In this notation the action of on Chow groups of is defined as
The operators have simple forms. The operator is given by the following differential operator:
The differential operator is a powerful tool to produce tautological relations. The idea is to start from obvious relations and apply the operator to it several times. This procedure yields a large class of tautological relations. A surprising fact is that one can get highly non-trivial relations from this method.
Yin  studies tautological classes on the universal Jacobian in his recent thesis. There are basic relations among tautological classes coming from the action on its Chow ring. Interpreting these relations on the Jacobian side gives an interesting class on relations on moduli of curves. All tautological relations on the universal curve for and on for are recovered using this method.
As another application we will see that all components of the curve class in positive degrees vanish for families of hypereliptic curves. These vanishings will be used in this article to find all tautological relations on . Let be the universal curve over the space of Weierstraß pointed hyperelliptic curves of genus . Geometric points of correspond to objects of the form where is a smooth hyperelliptic curve of genus , is a Weierestraß point of and is arbitrary. The degree zero divisor belongs to the Jacobian of . This association defines the map
The component of the Beauville decomposition of image of via is denoted by as usual.
Let and be as above. The component vanishes for all .
The proof is known to the expert. For example in  Beauville proves a similar statement when is a fixed hyperelliptic curve. He shows that the component is algebraically equivalent to zero when . Here we want to prove the same vanishings for families of hyperelliptic curves under rational equivalence.
Notice that all components vanish when is odd. To see this consider the Ceresa cycle . This class is zero according to our definition of the morphism . This shows the vanishing of for odd . We now use this fact to show the vanishing of other components with positive indices. Equivalently, we show that the class is zero when . The vanishing of is immediate since we know that . Consider the following equations:
which is the same as . Notice that the class vanishes since the Picard group of is trivial. This proves the claim since the coefficient of is not zero. ∎
In  Beauville shows that the tautological ring of the Jacobian of a fixed hyperelliptic curve of genus under algebraic equivalence is isomorphic to . The vanishing proved above gives the same presentation for the tautological ring of over under rational equivalence.
Let and be as before. We have the relation in Pic.
Consider the map defined before. We will show that the desired relation follows from the vanishing of the divisor . We need to calculate the pull-back of to via . The method for calculating the pull-back of tautological classes on the universal Jacobian to families of curves is explained in the thesis of Yin. We briefly recall the procedure from . Let be the projections onto the factors of . The section induces two sections . Denote by the class of the diagonal inside . According to the definition the class is equal to the degree one component of the following expression:
An argument based on chasing through cartesian squares shows that the expression above is the same as:
We therefore have the following:
The result follows since the classes and vanish. ∎
3. The Faber-Pandharipande cycle
There is a natural way to define the tautological ring of products of a fixed smooth curve for positive integers . It is the -algebra generated by canonical classes and diagonals. More precisely, denote by the canonical class on the curve and consider the natural projections and for . From this collection of maps one gets the divisor classes and , where is the diagonal class. The tautological ring is defined to be the -subalgebra of generated by . One could simply restrict tautological cycles on the product of the universal curve of genus to the fiber and recover the same set of generators.
Faber and Pandharipande in an unpublished work have studied this ring in cohomology. From their analysis one gets a complete description of this ring. In particular, there is an explicit presentation of all relations. The resulting algebra becomes Gorenstein.
It is natural to ask whether has the Gorenstein property as well. The situation becomes difficult already when we consider the surface . There are 4 cycles in degree 2 with the following relations:
The proportionality holds in cohomology. One wonders whether this relation is true in Chow as well. This was shown by Faber and Pandharipande for . Green and Griffiths  study the zero cycle for generic curves defined over complex numbers. Their Hodge theoretic analysis is based on an infinitesimal invariant. In particular, they show that the Faber-Pandharipande cycle doesn’t vanish when is a generic curve of . In  Yin shows that the same statement is true in arbitrary characteristic. The idea of the proof is to write the Faber-Pandharipande cycle as the pull-back of a tautological class from the Jacobian of the curve. He observes that the corresponding tautological class on the Jacobian doesn’t vanish for a generic curve of genus . Yin proves that the same is true for its pull-back. This shows the non triviality of this cycle for such curves. We will prove the vanishing of the Faber-Pandharipande cycle on the locus of hyperelliptic curves with the same idea:
Let be the universal hyperelliptic curve of genus . The cycle vanishes on .
Let be the universal curve over the space together with the section . Denote by the induced sections from to the space . We have the relations
from the statement proven in Corollary 2.3. This gives the desired vanishing since and .
There is another way to see this: Consider the following map
Notice that a geometric point on the space has the form , where is a hyperelliptic curve with a Weierstraß point and are arbitrary. The image of this point under is the divisor on the Jacobian of . After calculating the pull-back of the classes to we obtain the following relation:
The result follows from the vanishing of on the universal Jacobian proved in Proposition 2.1. ∎
The vanishing of the Faber-Pandharipande cycle can be used to show another vanishing on the universal hyperelliptic curve:
The cycle vanishes on the universal curve .
We have the relations
on the space . Intersect these relations with the divisor classes and compute their push-forwards to under the natural projection , onto the first factor. The vanishing of follows. ∎
In  the Faber-Pandharipande cycle is shown to be the pull-back of the class , for a fixed curve of genus . It is possible to prove Proposition 3.1 and Corollary 3.2 using the class with a similar method. The only difference is that the pull-back of has extra terms involving and , which both vanish after all.
4. The Gross-Schoen cycle
In  Gross and Schoen considered a smooth and projective curve defined over a field together with a -rational point . The codimension 2 cycle on the product is defined in terms of the diagonal classes and the point . The authors call this class the modified diagonal cycle and study some of its properties. The basic fact about is that it vanishes in cohomology. It is proven that the class in the second Griffiths group , measuring homologically trivial cycles modulo algebraically trivial cycles, is independent of the choice of the point . When is a rational curve, an elliptic curve or a hyperelliptic curve the cycle is shown to be zero in Chow. In the first two cases the point can be arbitrary but for hyperelliptic curves it has to be a Weierstraß point. In this article we want to show the same result in the relative setting.
Let us recall the definition of the Gross-Schoen cycle in the classical case. Let be a smooth curve with a point as above. Consider the following subvarieties:
The degree 2 cycle is defined on the product as follows:
There is another version of this cycle with respect to the canonical class. It has the following form:
where is the diagonal class as usual. Recall that a curve has a subcanonical point if the equality holds. In this situation the classes and defined above coincide with each other. Notice that the cycle is symmetric.
The cycle vanishes on the locus of hyperelliptic curves.
Let be the family of Weierstraß pointed hyperelliptic curves of genus as before. We define a map
For a pointed curve and points the associated divisor on the Jacobian of is the divisor . The cycle is the pull back of the following class via :
In  Zhang studies the connection between the triviality of the Gross-Schoen cycle and the Ceresa cycle in the Chow ring of for a fixed curve . From his result one can see their equivalence assuming the triviality of the Faber-Pandharipande cycle. The vanishing of the Ceresa cycle for families of Weierstraß pointed hyperelliptic curves is obvious from its definition. It would be interesting to see whether the result proven above can be obtained from this vanishing by the approach of Zhang. That might give insight into the following natural question about the torsion cycles found in this article:
What are the orders of the Faber-Pandharipande and Gross-Schoen cycle in the integral Chow ring of the moduli space of pointed hyperelliptic curves?
Modified diagonals can be defined in a more general settings. The following formulation is due to O’Grady. Let be any -dimensional algebraic variety over a field and be a -rational point. The modified cycle can be defined on the product . For any subset of let
The modified cycle associated to the point is the -cycle on given by
In  it is conjectured that for a hyperkähler variety of dimension there exists a point such that the cycle vanishes. It is also conjectured that the modified diagonal vanishes for a point on an abelian variety of dimension . A recent result of Moonen and Yin  establishes the second conjecture. In  the same authors among other things give a motivic description of modified diagonals.
5. The degree relation
We have proved that the Faber-Pandharipande cycle and Gross-Schoen cycle vanish on families of hyperelliptic curves. In this section we obtain a degree relation which plays an essential role in our study. We will see that there are two different ways to get this relation. The first source of this relation is again the universal Jacobian. We use the formula given by Grushevsky and Zakharov for the pull-back of the theta divisor to the moduli of curves. The second method is based on studying linear systems for generic curves of genus . This method produces relations in more general settings. Restricting to the locus of hyperelliptic curves we obtain a tautological relation of degree involving points.
5.1. Relations coming from the theta divisor
The geometry of the theta divisor on the universal Jacobian gives a very simple way to prove our relation. The pull-back of the theta divisor to the space of curves is investigated by several authors. Hain  gives a formula for this class in terms of standard boundary cycles in cohomology. Hain uses this formula to compute the pull-back of the zero section of the universal Jacobian to the space of curves of compact type. Hain’s formula answers Eliashberg’s question. Analogue results are proven by Müller . Grushevsky and Zakharov ,  give a formula for the pull-back of the theta divisor to the spaces classifying pointed curves of compact type and the space of stable pointed curves. Here we follow the notation in . Let be the universal Jacobian of degree zero divisors over . We denote its pull-back under the natural projection by the same letter. Consider a collection of integers satisfying . For any moduli point
one gets a degree zero divisor on the Jacobian of . This association defines a map
Let be the universal symmetric theta divisor trivialized along the zero section as before. In  Grushevsky and Zakharov compute the pull-back in terms of standard divisor classes on . We recall the definition of the divisor class for and a subset of the marking set . The generic point on this divisor corresponds to a singular curve having 2 irreducible components. One of the components has genus whose set of markings is . For any such subset the number is defined as the sum .
For deg , the class of the pull-back of the universal symmetric theta divisor trivialized along the zero section is equal to
In  a similar formula is proven when deg . For hyperelliptic curves these relations give equivalent results.
The vanishing of the class in the Chow ring of the universal Jacobian gives a relation among tautological classes on . We restrict this relation to the locus of hyperelliptic curves and get a relation on . One can show that the relations coming from the vanishing of the class follow from the relations found on for as long as . When one gets one new relation. More precisely, for all choices of parameters , the resulting relations on are multiples of each other up to a linear combination of the relations involving points. As we will see in Section 6 there will be no new relations afterwards.
5.2. Relations from higher jets of differentials
The method is similar to the method introduced by Faber in  with slight changes. This gives tautological relations on products of the universal curve over . The resulting relation holds for a general family of curves.
Let be the universal curve of genus with the relative dualizing sheaf . We also make the usual convention . The -fold fibered product of the curve over the is denoted by . We consider two natural locally free sheaves on this space. Let be the projection onto the first factors. Its relative dualizing sheaf is denoted by . The sum of the diagonal classes on defines the divisor class :
The locally free sheaf defined for every as follows:
This is the usual Hodge bundle of rank when . The fiber of at a point is the vector space , where is the dualizing sheaf of the curve . For it is of rank . Another natural bundle is obtained from evaluating differential forms on divisors. We define the following locally free sheaf of rank on :
The fiber of the sheaf at a point is
Consider the natural evaluation map:
For a general the morphism doesn’t behave well. Its kernel has the fiber
which depends on the curve and the points . However the situation becomes simpler when is large enough. More precisely, when the morphism is injective. The quotient bundle has rank . This means that for all we get the following vanishing:
The class vanishes for all .
The Chern classes of the bundle can be calculated with the same method as in  using Grothendieck-Riemann-Roch. We have the following formula for its total Chern class:
We recover Faber’s relations  when . As we will see there are tautological relations on which don’t come from Faber’s relations.
As an example let and take . In this case we get the relation . After multiplying this relation with the divisor class on and pushing it forward to via the projection we get a degree 3 relation involving 6 points. This relation was found in  and was used to study the tautological ring of .
As another example take for . From Proposition 5.3 we get the following relation involving points:
Notice that by our assumption. From this relation we get a degree relation involving points. There are many ways to do this. All give equivalent results for hyperelliptic curves. Here is one example of doing this: Multiply the relation above with the monomial and push it forward to . The resulting relation is symmetric with respect to markings.
6. Products of the universal curve over
In this section we give a description of the tautological rings for products of the universal hyperelliptic curve . We will see that the relations found in previous sections can be used to find all tautological relations. This is based on explicit computations of the intersection pairings. To simplify the computations we work with a different set of generators.
Let be an integer. For every and define the following classes:
It is straightforward to see that the collection of elements generate the tautological algebra of . The relations we found in previous sections become simpler in terms of these variables. The relation translates into . The following relations
are equivalent to the vanishings . The vanishing of the Gross-Schoen cycle is equivalent to . The degree relation comes from the vanishing of the following symmetric expression:
Each term of the sum corresponds to a partition of into subsets with 2 elements.
While we consider only hyperelliptic curves our presentation works for elliptic curves as well. In this case the origin of the elliptic curve plays the role of a Weierstraß point.
Let be the universal elliptic curve over . Geometric points of are elliptic curves , where is a smooth curve of genus one and denotes its origin. The morphism admits a natural section . It associates to the moduli point . The image of the section is denoted by the same letter. Consider the -fold fiber product . Notice that is birational to the moduli space ! The divisor class defines the divisor in the Picard group of for . We also have the diagonal class for . The class is defined as . In  the vanishing of the Faber-Pandharipande cycle on is obtained from a tautological relation on . The vanishing of the Gross-Schoen cycle gives . The connection between this relation and Getzler’s relation on is explained in . The next case deals with . There are 10 generators in degree 1,3 and 21 generators in degree 2. The intersection matrix of the pairing is invertible. This shows that tautological groups are of dimension 10 in degrees 1 and 3. The resulting intersection matrix for has the form
This matrix has rank 20. The kernel is one dimensional and corresponds to the relation
This relation can be obtained from Getzler’s relation via a pull-back. It is proven in  that every relation in the tautological ring of follows from these relations. This was used to find all tautological relations for the moduli space .
Let and consider . The vanishing of the Faber-Pandharipande cycle gives the relation . This is the restriction of a tautological relation found on by Getzler . When the vanishing of the Gross-Schoen cycle corresponds to the relation . This is the restriction of the relation on found by Belorousski and Pandharipande . The last relation contains 15 terms and has the following form:
In  we proved that these relations determine the structure of the tautological ring for every .
The relations involving points can be used to generate the tautological group of a given degree with elements of the form
In this situation such element is said to be a standard monomial. We define and . What we mentioned before means that the tautological group of is generated by standard monomials. Intersection pairings have a simple form if one works with standard monomials. There is a natural way to associate a standard monomial to every standard monomial . It is simply defined as the following product
We say that and are dual to each other and write . An elementary argument shows that for standard monomials of the same degree the product vanishes unless they have the same -parts. This means that interesting blocks of intersection pairings come from matrices having the following form: Let be an integer and consider all standard monomials of the form . These belong to the tautological group . Denote by the -vector space generated by these elements. The permutation group acts on via its natural action on indices. This makes into a representation of the symmetric group . The decomposition of into irreducible components has the following form:
where is the representation associated to the partition