Fujita-Zariski decompositions on some product threefolds

TL;DR

This paper explicitly computes Fujita-Zariski decompositions and Newton-Okounkov bodies for certain product threefolds, providing detailed volume calculations for big divisors on these complex algebraic varieties.

Contribution

It introduces explicit methods for Fujita-Zariski decompositions on specific product threefolds and computes their Newton-Okounkov bodies and volumes, advancing understanding of their birational geometry.

Findings

Computed Fujita-Zariski decompositions for specific threefolds.

Determined Newton-Okounkov bodies for certain ample polarizations.

Calculated volumes of big divisors on blow-ups of these threefolds.

Abstract

We use explicit blow-ups and computations of birational Fujita-Zariski decompositions to determine generic infinitesimal Newton-Okounkov bodies for box-product ample polarizations on three classes of spaces: product between a curve and the projective plane, products of three curves, and the product between a curve and a Jacobian surface. In particular we compute the volume of many big but non-nef divisors on blow-ups of these threefolds.