The locus of cyclic covers in the moduli space of curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:22:29Z http://mathoverflow.net/feeds/question/75943 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75943/the-locus-of-cyclic-covers-in-the-moduli-space-of-curves The locus of cyclic covers in the moduli space of curves Ariyan Javanpeykar 2011-09-20T13:10:45Z 2011-09-20T16:21:43Z <p>Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. Let $Z$ be the closure in $\mathcal{M}_g$ of the set of smooth curves of genus $g$ which are a cyclic cover of the projective line. </p> <p><strong>Question.</strong> Is $Z$ irreducible?</p> <p><strong>Question.</strong> What is the dimension of $Z$? Do we have non-trivial bounds?</p> <p><strong>Question.</strong> Is $Z$ affine?</p> <p><em>Remark.</em> Let $W$ be the closure of the set of smooth curves which are a cyclic cover of the projective line of prime degree. Then it is known that $W$ is affine. Note that $W\subset Z$.</p> http://mathoverflow.net/questions/75943/the-locus-of-cyclic-covers-in-the-moduli-space-of-curves/75972#75972 Answer by rita for The locus of cyclic covers in the moduli space of curves rita 2011-09-20T16:21:43Z 2011-09-20T16:21:43Z <p>The following paper: </p> <p>M. Cornalba, On the locus of curves with automorphisms, Annali di Matematica pura ed applicata (4) 149 (1987), 135-151. Erratum, Annali di Matematica pura ed applicata (4) 187 (2008), 185-186. (A revised version incorporating the changes described in the Erratum is available on the author's web page) </p> <p>contains an explicit description of the irreducible components of the locus of curves with an automorphism, including computation of the dimensions. </p>