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.
be the moduli stack of stable marked curves
Besides permutations of the marked points, the following morphisms between these moduli stacks play an important role in defining tautological classes: The tautological rings
are simultaneously defined as the smallest subrings of the Chow rings closed under pushforward by forgetful and gluing maps.
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 x1, ..., xk−1, y1, y2, xk+1, ..., xn.
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.
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 (when there is no marked point).
of curves with no marked point, Mumford conjectured, and Madsen and Weiss proved, that for any
is an isomorphism in degree d for large enough g. In this case all classes are tautological.
, 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
Parts (1) and (2) of the conjecture were proven.
Part (3), also called the Gorenstein conjecture, was only checked for
and higher genus, several methods of constructing relations between
classes find the same set of relations which suggest that the dimensions of
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: These four methods are proven to give the same set of relations.
Similar conjectures were formulated for moduli spaces
fail to obey the (analogous) Gorenstein conjecture.
On the other hand, Tavakol[6] proved that for genus 2 the moduli space of rational-tails stable curves
obeys the Gorenstein condition for every n.