Resolution theorems and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Principalization — Theorem 3.1. Triples (X, I, E) with X smooth of finite type over a field of characteristic zero, I an ideal sheaf nonzero on every component and E an snc divisor; compatible with smooth morphisms. The isomorphism locus is the complement of the cosupport of I together with the singular locus of E.
• Elimination of indeterminacy — Corollary 3.4. X smooth, U dense open in X, Y proper over the base field, f: U -> Y a morphism; the composite of blow-ups is an isomorphism over U.
• Resolution of singularities — Theorem 4.8. Reduced schemes of finite type over a field of characteristic zero; proper, an isomorphism over the smooth locus, snc exceptional divisor, compatible with smooth morphisms; projective for quasi-projective schemes.
• Proper smooth models — Corollary 5.1. Reduced schemes of finite type in characteristic zero; a proper morphism from a smooth scheme that is an isomorphism over a dense open subset.
• Smooth compactification — Corollary 5.2. Smooth separated schemes of finite type in characteristic zero; an open dense immersion into a smooth proper scheme whose boundary is the support of an snc divisor; projective for quasi-projective input.

Reading order

Begin with the commutative algebra and the smooth morphisms listed under prerequisites. Then follow the ten lessons in order: the first two set up blow-ups and the form of the main theorems, the next six develop the induction on dimension, and the last proves principalization, resolution and their consequences.