The Gorenstein conjecture fails for the tautological ring of M_(2,n)

摘要

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring R?(M_(2,N)) is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on M_(2,N). By work of Graber and Pandharipande, such a class exists on M_(2,20), and we present some evidence indicating that N is in fact 20.