Extremal Effective Cycles and Nef Line Bundles on \(\overline{\rm{M}}_{g,n}\)

Abstract

There has been a growing body of work devoted to the study of effective cones of codimension-\(k\) cycles \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) on \(\overline{\rm{M}}_{g,n}\), the moduli space of \(n\) pointed stable curves of genus \(g\). In this paper, we remove the genus-dependence present in previous bounds on the number of marked points, and prove the following results: (1) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) has infinitely many extremal rays for \(k\ge 2\), \(g\ge 3\) and \(n\ge 2k-2\), and (2) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) is non-polyhedral for \(k\ge 2\), \(g\ge 1\) and \(n\ge k+5\). Moreover, we show that (3) every rational tails boundary stratum spans an extremal ray. Our method refines that of Chen and Coskun by extending arguments based on morphisms, or equivalently semiample divisors, to a setting that also allows for the use of nef divisors. Certain non-semiample nef divisors on \(\overline{\rm{M}}_{g,n}\), namely so-called semigroup kappa divisors of a particular kind, play a crucial role.