Tautological ring

In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring).
Definition.
Let formula_1 be the moduli stack of stable marked curves formula_2, such that
The last condition requires formula_3 in other words ("g","n") is not among (0,0), (0,1), (0,2), (1,0). The stack formula_1 then has dimension formula_5. Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes:
The tautological rings formula_9 are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps.
The tautological cohomology ring formula_10 is the image of formula_11 under the cycle map. As of 2016, it is not known whether the tautological and tautological cohomology rings are isomorphic.
Generating set.
For formula_12 we define the class formula_13 as follows. Let formula_14 be the pushforward of 1 along the gluing map formula_15 which identifies the marked point "xk" of the first curve to one of the three marked points "yi" on the sphere (the latter choice is unimportant thanks to automorphisms). For definiteness order the resulting points as "x"1, ..., "x""k"−1, "y"1, "y"2, "x""k"+1, ..., "xn". Then formula_16 is defined as the pushforward of formula_17 along the forgetful map that forgets the point "y"2. This class coincides with the first Chern class of a certain line bundle.
For formula_18 we also define formula_19 be the pushforward of formula_20 along the forgetful map formula_21 that forgets the "k"-th point. This is independent of "k" (simply permute points).
These pushforwards of monomials (hereafter called basic classes) do not form a basis. The set of relations is not fully known.
Faber conjectures.
The tautological ring formula_25 on the moduli space of smooth "n"-pointed genus "g" curves simply consists of restrictions of classes in formula_9. We omit "n" when it is zero (when there is no marked point).
In the case formula_27 of curves with no marked point, Mumford conjectured, and Madsen and Weiss proved, that for any formula_28 the map formula_29 is an isomorphism in degree "d" for large enough "g". In this case all classes are tautological.
Although formula_34 trivially vanishes for formula_35 because of the dimension of formula_36, the conjectured bound is much lower. The conjecture would completely determine the structure of the ring: a polynomial in the formula_37 of cohomological degree "d" vanishes if and only if its pairing with all polynomials of cohomological degree formula_38 vanishes.
Parts (1) and (2) of the conjecture were proven. Part (3), also called the Gorenstein conjecture, was only checked for formula_39. For formula_40 and higher genus, several methods of constructing relations between formula_24 classes find the same set of relations which suggest that the dimensions of formula_34 and formula_43 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 formula_44, the "d"-th fiber power of the universal curve formula_45, the following methods have been used to find relations:
These four methods are proven to give the same set of relations.
Similar conjectures were formulated for moduli spaces formula_1 of stable curves and formula_49 of compact-type stable curves. However, Petersen-Tommasi proved that formula_50 and formula_51 fail to obey the (analogous) Gorenstein conjecture. On the other hand, Tavakol proved that for genus "2" the moduli space of rational-tails stable curves formula_52 obeys the Gorenstein condition for every "n".