Conventions for the resolution development

This page follows the source conventions of Field and A field's prime subfield is isomorphic to $\mathbb{Q}$ in characteristic zero and to ${\mathbb{F}}_p$ in characteristic $p$: $K$ is a field of characteristic zero, i.e. its prime subfield is $\mathbb{Q}$. A $K$-variety is an integral, separated scheme of finite type over $K$ (Integral schemes, Separated morphism of schemes, Locally finite type and finite type morphisms); a smooth $K$-scheme is a scheme of finite type over $K$ whose structure morphism is smooth (Smooth morphism of schemes); in characteristic zero a scheme of finite type over $K$ is smooth over $K$ exactly when all its local rings are regular, by the perfect-field geometric-regularity criterion (Field tests for geometric regularity, Locally standard smooth iff flat with geometrically regular fibres). The core resolution construction is over an algebraically closed field of characteristic zero, as in the source. Items with an explicitly broader field or characteristic range retain their stated hypotheses; the descent item and final theorems apply over arbitrary characteristic-zero fields. The positive-characteristic derivative converses require the perfect-field qualifications stated in their items. All blowups are blowups of regular closed subschemes of smooth $K$-schemes, and all divisors are effective Cartier divisors; SNC' abbreviates simple normal crossings' in the sense of Simple normal crossings divisors and simultaneous normal crossings position. Resolutions are stated for reduced or integral base schemes and are constructed from an ambient smooth scheme. The Axiom of Choice is assumed throughout this resolution development and is inherited from the published blowup and relative-Proj suppliers (The Axiom of Choice); no dependent-choice or other choice principle is used by this page's new arguments beyond what those suppliers already assume.