The last condition requires in other words is not among,,,. The stack then has dimension. Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes:
Forgetful maps which act by removing a given point xk from the set of marked points, then restabilizing the marked curved if it is not stable anymore.
Gluing maps that identify the k-th marked point of a curve to the l-th marked point of the other. Another set of gluing maps is that identify the k-th and l-th marked points, thus increasing the genus by creating a closed loop.
The tautological rings are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps. The tautological cohomology ring is the image of under the cycle map. As of 2016, it is not known whether the tautological and tautological cohomology rings are isomorphic.
Generating set
For we define the class as follows. Let be the pushforward of 1 along the gluing map which identifies the marked point xk of the first curve to one of the three marked points yi on the sphere. For definiteness order the resulting points as x1,..., xk−1, y1, y2, xk+1,..., xn. Then is defined as the pushforward of along the forgetful map that forgets the point y2. This class coincides with the first Chern class of a certain line bundle. For we also define be the pushforward of along the forgetful map that forgets the k-th point. This is independent of k. These pushforwards of monomials do not form a basis. The set of relations is not fully known.
Faber conjectures
The tautological ring on the moduli space of smooth n-pointed genus g curves simply consists of restrictions of classes in. We omit n when it is zero. In the case of curves with no marked point, Mumford conjectured, and Madsen and Weiss proved, that for any the map is an isomorphism in degreed for large enoughg. In this case all classes are tautological. Although trivially vanishes for because of the dimension of, the conjectured bound is much lower. The conjecture would completely determine the structure of the ring: a polynomial in the of cohomological degree d vanishes if and only if its pairing with all polynomials of cohomological degree vanishes. Parts and of the conjecture were proven. Part, also called the Gorenstein conjecture, was only checked for. For and higher genus, several methods of constructing relations between classes find the same set of relations which suggest that the dimensions of and are different. If the set of relations found by these methods is complete then the Gorenstein conjecture is wrong. Besides Faber's original non-systematic computer search based on classical maps between vector bundles over, the d-th fiber power of the universal curve, the following methods have been used to find relations:
Virtual classes of the moduli space of stable quotients by Pandharipande and Pixton.
Witten's r-spin class and Givental-Telemann's classification of cohomological field theories, used by Pandharipande, Pixton, Zvonkine.
Geometry of the universal Jacobian over, by Yin.
Powers of theta-divisor on the universal abelian variety, by Grushevsky and Zakharov.
These four methods are proven to give the same set of relations. Similar conjectures were formulated for moduli spaces of stable curves and of compact-type stable curves. However, Petersen-Tommasi proved that and fail to obey the Gorenstein conjecture. On the other hand, Tavakol proved that for genus 2 the moduli space of rational-tails stable curves obeys the Gorenstein condition for every n.
