Cones and Asymptotic Invariants of Multigraded Systems of Ideals

Recent work has discovered perhaps unexpectedly rich structures for base loci and asymptotic invariants on the cone of big divisors of smooth complex projective varieties. One may ask what sorts of cones and functions can occur. In general this question is not currently well-understood. However, a key feature of this work (implicit in [7] and explicit in [3]) is that the structures depend on algebraic relations between base ideals, namely, the fact that they form multigraded systems of ideals. The question then arises: what can occur in this abstract setting? In this paper, we give examples showing that the only restrictions obeyed by the cones and asymptotic invariants of general multigraded systems of ideals are ones of convexity, which are imposed formally. Turning to a detailed introduction, let X be a smooth, irreducible complex projective variety, and N(X)R be the finite-dimensional vector space of real numerical equivalence classes of divisors on X . This vector space contains interesting cones reflecting the geometry of X . For example, one has the cone of nef divisors Nef(X) ⊂ N(X)R, the closure of the cone spanned by ample divisors on X , and Eff(X) ⊂ N(X)R the closure of the cone spanned by effective divisors, which contains the cone of big divisors Big(X) as its interior. These cones have attracted a great deal of attention and provide many examples of interesting behavior. Moreover, it has recently been realized that there are naturally defined functions on these cones that reflect the behavior of the linear series in question. For example, fixing x ∈ X and an effective divisor D on X , set