Complete Subvarieties of $\rm{M}_{g,n}$ and a Lifting Problem

Abstract

Finding the maximal dimension of complete subvarieties of the moduli space of smooth $n$-pointed curves of genus $g$ is a long-standing open problem. Here we show that for $g\ge 3\cdot 2^{d-1}$, if the characteristic of the base field is greater than $2$, then $\rm{M}_g$ contains a complete subvariety of dimension $d$. Furthermore, in positive characteristic, we construct a complete surface in $\rm{M}_{g,n}$ for $g\ge 3$ and $n\ge 1$, which contain a general point. These results follow from the proofs of the lifting conjectures, introduced here. In particular, we translate the existence of complete subvarieties to properties of line bundles on $\rm{M}_{g,n}$. Our method reframes Zaal's approach, with increased efficiency via Keel's results on semi-ample line bundles in positive characteristic. This method demonstrates the difference in the geometry of moduli spaces between characteristic $0$ and characteristic $p$.