Catriona Maclean

The aim of this note is to establish a somewhat surprising connection between functions on Newton–Okounkov bodies and Seshadri constants of line bundles on algebraic surfaces.

We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nystr\"om. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, relying on the work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.

Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of $\R$-divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral.

Based on a recent work by Thomas Bauer reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.