Birational Contractions on Moduli of pointed Curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:30:19Z http://mathoverflow.net/feeds/question/65418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65418/birational-contractions-on-moduli-of-pointed-curves Birational Contractions on Moduli of pointed Curves AM 2011-05-19T11:27:08Z 2011-10-11T17:40:46Z <p>On the Moduli space $\overline{M}_{g}$ of genus $g$ stable curves the <em>Hodge class</em> $\lambda$ induces a birational morphism $f$ on a projective variety contracting the boundary, that is the exceptional locus of $f$ coincides with the boundary of the moduli space.</p> <p><strong>Is there a line bundle $L$ on the moduli space of pointed curves (as instance on $\overline{M}_{2,1}$ and on the moduli space of $2$-pointed elliptic curves) with the same property (i.e. a line bundle which induces a birational morphism whose exceptional locus coincides with the boundary) ?</strong></p> http://mathoverflow.net/questions/65418/birational-contractions-on-moduli-of-pointed-curves/65445#65445 Answer by ulrich for Birational Contractions on Moduli of pointed Curves ulrich 2011-05-19T14:54:46Z 2011-05-19T14:54:46Z <p>A detailed description of the nef cones of the examples you mention is given in the thesis of William Rulla (The birational geometry of <code>$\overline{M}_3$</code> and <code>$\overline{M}_{2,1}$</code>, The University of Texas at Austin, 2001.) From this, it follows that there are no such line bundles for the two examples that you mention. I would guess that the same holds in general as well.</p> http://mathoverflow.net/questions/65418/birational-contractions-on-moduli-of-pointed-curves/77842#77842 Answer by AM for Birational Contractions on Moduli of pointed Curves AM 2011-10-11T17:40:46Z 2011-10-11T17:40:46Z <p>The Hodge class $\lambda_{2,1}$ on $\overline{M}_{2,1}$ </p> <p>is the pull-back of the Hodge class $\lambda$ on $\overline{M}_{2}$ via the forgetful morphism</p> <p>$$\pi:\overline{M}_{2,1}\rightarrow\overline{M}_2,$$</p> <p>that is $\lambda_{2,1} = \pi^{*}\lambda$. </p> <p><strong>So $\lambda_{1,2}$ is nef but not big on $\overline{M}_{2,1}$.</strong></p>