Dan Petersen: The structure of the tautological ring in genus one

The tautological ring of the moduli space of (stable n-pointed) curves is defined roughly as the smallest subring of the Chow ring containing all "geometrically natural" classes. A rather complete description of this ring in case of genus zero is given by the results in Sean Keel's 1992 paper "Intersection theory on moduli space of stable n-pointed curves of genus zero": the tautological ring, the cohomology ring and the Chow ring all coincide; the ring is spanned by cycle classes of boundary strata and all relations between the strata classes follow from the basic 4-point (also known as WDVV) relation.

In this talk, I discuss the situation in genus one. It has been known for a long time that such a simple statement can no longer be true, as there will in general be odd cohomology, and the Chow ring will in general be infinite-dimensional. Nevertheless, it turns out that the tautological ring is still as simple as we could hope for: the tautological ring and the even cohomology ring coincide; the ring is spanned by boundary strata classes, and all relations follow from the 4-point relation in genus zero and Getzler's relation on M_{1,4}. (These assertions were announced by Ezra Getzler in 1996, but his proof never appeared.)