Line bundles L on projective toric varieties can be understood as formal differences $(\Delta^+ − \Delta^−)$ of convex polyhedra in the character lattice. We show how it is possible to use this language for understanding the cohomology of L by studying the set-theoretic difference

$(\Delta^− \setminus \Delta^+)$.

Moreover, when interpreting these cohomology groups as certain Ext-groups, we demonstrate how the approach via

$(\Delta^− \setminus \Delta^+)$ leads

to a direct description of the associated extensions.

(The first part is joint work with Jarek Buczinski, Lars Kastner, David Ploog, and Anna-Lena Winz.)