Extremal effective curves and non-semiample line bundles on $M_{g,n}$

In this work, we develop a new method for establishing extremality in the closed cone of 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 that extends Keel’s argument, we demonstrate that a substantial portion of the cone of nef divisors on $M_{g,n}$ 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 $M_{3,n}$. Moreover, we show that this extremal ray is contractible in characteristic $p$. Our method relies on two main ingredients: (1) the construction of a new collection of nef divisors on $M_{g,n}$, and (2) the identification of a tractable inductive structure on the Picard group, arising from Knudsen’s construction of $M_{g,n}$.