We discuss the relationship between global sections of divisors and birational geometry. Our starting point is the observation that fixed components of a divisor do not contribute to its space of global sections. This leads to two complementary operations that preserve sections: subtraction of divisorial fixed components and contraction to a suitable D-minimal model. We develop these ideas for Mori dream spaces and Calabi-Yau varieties with controlled singularities. In these settings, the geometry of D-minimal models governs how the computation of global sections decomposes across the effective cone. More precisely, we show that global sections can be reduced to Euler characteristic computations on suitable birational models, where subtraction and contraction replace an arbitrary effective divisor by a movable or nef one. As a consequence, previously observed piecewise polynomial and quasipolynomial formulae admit a natural birational interpretation, with their domains governed by the Mori chamber decomposition. Conversely, these formulae encode information about the birational geometry itself: finite computations of $h^0$ can be used to extract information about the Mori chamber decomposition and thereby infer properties of the underlying birational models. In particular, cohomological signatures can detect singularities, class group torsion, and other aspects of the birational geometry.