Cycles on curves and Jacobians: a tale of two tautological rings
arXiv:1407.2216
Abstract
We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties may not hold.
31 pages, 4 figures. This is part of the author's PhD thesis, which greatly expands an earlier preprint arXiv:1206.3783