Extremal effective curves and non-semiample line bundles on $\overline{\rm{M}}_{g,n}$

Abstract

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for non-semiampleness which extends Keel's argument, we demonstrate that a substantial portion of the cone of nef divisors of $\overline{\mathrm{M}}_{g,n}$ is not semiample. As an application, we construct the first explicit example of a non-contractible extremal ray of the closed cone of effective curves on $\overline{\mathrm{M}}_{3,n}$. Our method relies on two main ingredients: (1) the construction of a new collection of nef divisors on $\overline{\mathrm{M}}_{g,n}$, and (2) the identification of a tractable inductive structure on the Picard group, arising from Knudsen's construction of $\overline{\mathrm{M}}_{g,n}$.