Intersections with divisors on moduli of curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:11:09Z http://mathoverflow.net/feeds/question/96673 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96673/intersections-with-divisors-on-moduli-of-curves Intersections with divisors on moduli of curves OldMacdonaldHadaForm 2012-05-11T14:40:28Z 2012-05-11T18:31:21Z <p>Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.</p> <p>Consider </p> <p>$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$ </p> <p>the first Chern class of a line bundle (note that the Picard coincides with $H^2(\overline{\mathcal{M}}_{g,n}, \mathbb{Z})$ and with the Neron-Severi group in this case)</p> <p>It is well-known from Arbarello-Cornalba-Harer that the group of first Chern classes is generated by $\kappa_1$ and by $\delta$-classes and $\psi$-classes.</p> <p>My question: is the multiplication by $\gamma$ an isomorphism:</p> <p>$H^{3g-3+n-1} \to H^{3g-3+n+1} \ ?$</p> <p>(as it happens in the Hard Lefschetz theorem when $\gamma$ is an hyperplane section). Are there $\gamma$s for which it is not an isomorphism?</p> <p>This is of course true when $\gamma$ is in the ample cone or in the antiample cone. </p>