What is $M_g$ over a finite field, really? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:44:29Z http://mathoverflow.net/feeds/question/72903 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72903/what-is-m-g-over-a-finite-field-really What is $M_g$ over a finite field, really? jlk 2011-08-15T04:29:18Z 2011-08-15T15:24:04Z <p>Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \text{Sch} \to \text{Sets}$ whose Yoneda points are (flat) families of non-singular, genus $g$ curves. </p> <p><strong>What is $M_{g} \times \mathbb{Z}/p$?</strong></p> <p>One would expect that $M_{g} \times \mathbb{Z}/p$ co-represents $M^{\sharp}_{g} \times \mathbb{Z}/p$, but in general, the formation of GIT quotients can <a href="http://mathoverflow.net/questions/38529/uniform-quotient-vs-universal-quotient" rel="nofollow">fail</a> to commute with passing to fibers. Does that failure occur here?</p> <p>In other words: <strong>does $M_{g} \times \mathbb{Z}/p$ co-represent $M^{\sharp}_{g} \times \mathbb{Z}/p$?</strong></p> <p>Some references for the construction of $M_g$ over $\mathbb{Z}$ can be found <a href="http://mathoverflow.net/questions/65932/moduli-space-of-curves-over-mathbb-z" rel="nofollow">here</a>.</p> <p><em>Added:</em> Torsten Ekedahl had concerns about the use of the term co-represents." I intended it to mean that there is a natural transformation $M^{\sharp}_{g} \to M_g$ that is universal with respect to transformations into a scheme. Of course, this should be the same as asking that $M_{g}$ is the coarse space of the moduli stack $\mathcal{M}_{g}$, for a suitable notion of coarse space.</p>