On Newton-Okounkov bodies of Graded Linear Series

TL;DR

This paper extends the theory of Newton-Okounkov bodies from big divisors to graded linear series, providing new formulas, existence results, and characterizations related to volume and base loci.

Contribution

It generalizes Newton-Okounkov bodies to graded linear series and establishes slice formulas, existence of generic bodies, and volume characterizations.

Findings

Generalized Newton-Okounkov bodies for graded linear series

Proved slice formulas and existence of generic bodies

Characterized graded linear series with full volume

Abstract

We generalize the theory of Newton-Okounkov bodies of big divisors to the case of graded linear series. One of the results is the generalization of slice formulas and the existence of generic Newton-Okounkov bodies for birational graded linear series. We also give a characterization of graded linear series which have full volume in terms of their base locus.