Effective divisors on moduli spaces

The pseudo-effective cone Eff(X) of a smooth projective variety X is a fundamental, yet elusive invariant. On one hand, a few general facts are known: the interior of the effective cone is the cone of big divisors so, in particular, X is of general type if and only if KX ∈ int(Eff(X)); less obviously [4], a variety X is uniruled if and only if KX is not pseudo-effective and the dual of Eff(X) is the cone of movable curves; and, the effective cone is known to be polyhedral for Fano varieties. For further background, see [60]. On the other hand, no general structure theorem is known and the calculation of Eff(X) is a daunting task even in some of the simplest cases. For instance, the problem of computing the cone Eff(C) for a very general curve C of genus g is known to be equivalent to Nagata’s Conjecture, see [16]. The aim of this paper is to survey what is known about the effective cones of moduli spaces, with a focus on the moduli spacesMg,n of stable curves, Ag of principally polarized abelian varieties andMg,n(X,β) of stable maps. Because related moduli spaces often have an inductive combinatorial structure and the associated families provide a rich cycle theory, the study of effective cones of moduli spaces has often proven more tractable and more applicable than that of general algebraic varieties. For example, in the case of Mg, we may define, following [49], the slope s(D) of a divisor class D of the form aλ − bδ − cirrδirr − ∑ i ciδi, with a and b positive,