What this page does and does not prove about contractions and resolution

The page's commissioned claims comprise the intrinsic exceptional-curve and contraction definition, universal property and uniqueness of a contraction, the point-blowup direction, negativity of contracted curves, factorization of proper birational morphisms of regular surfaces, and resolution of normal integral finite-type surfaces over every field and characteristic (Exceptional curves of the first kind and their contractions, Universal property and uniqueness of a contraction, Blowing up a regular point is a contraction, Negativity of contracted curves on regular surfaces, Factorization of birational morphisms of regular surfaces into point blowups, Resolution of normal surface singularities). The surface resolution concludes regularity over an arbitrary field and smoothness only over a perfect field.

The page does not claim Castelnuovo's existence criterion for contracting every exceptional curve on a proper regular surface, negative definiteness of the complete intersection matrix of a reducible exceptional divisor, the arbitrary-Noetherian alteration/resolution equivalence of Stacks 54.14.5, or a resolution theorem in dimension at least three. These are outside the commissioned scope. The corrected local proof of the commissioned surface theorem includes the triple-cubic and stable-cubic square-conic cases, without adding these stronger statements.

• The list of commissioned claims is exhaustive for this page; omitted classical statements are deliberately not asserted anywhere on it.
• The corrected local proof of the surface resolution theorem includes the triple-cubic and stable-cubic square-conic cases, which is what lets the scope note exclude the stronger classical statements without weakening the commissioned one.