Complete Subvarieties of ${\mathrm{M}}_{\mathrm{g},\mathrm{n}}$ 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 ${\mathrm{M}}_{\mathrm{g}}$ contains a complete subvariety of dimension $d$. Furthermore, in positive characteristic, we construct a complete surface in ${\mathrm{M}}_{\mathrm{g},\mathrm{n}}$ 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 ${\mathrm{M}}_{\mathrm{g},\mathrm{n}}$. 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 $p$.

22 pages