2. Schemes🔗

The objects and operations in terms of which the algebraic theorems are stated: algebraic k-schemes, regularity, orders and normal crossings; the blow-up of a scheme along an ideal sheaf; projective morphisms; and successions of blow-ups with their transforms and boundaries, divisor families with simple normal crossings, the operations on successions, and the blow-up sequence functors of the functorial theorems.

1. 2.1. Algebraic k-schemes, singularities and normal crossings
   1. 2.1.1. Algebraic k-schemes
   2. 2.1.2. Regular points and the singular locus
   3. 2.1.3. Stalks and orders of ideal sheaves
   4. 2.1.4. Invertible ideal sheaves
   5. 2.1.5. Normal crossings
2. 2.2. The blow-up along an ideal sheaf
   1. 2.2.1. The Rees algebra and the affine blow-up
   2. 2.2.2. Gluing the affine blow-ups
   3. 2.2.3. The universal property
   4. 2.2.4. Transforms along a blow-up
   5. 2.2.5. The exceptional divisor and the push-forward of a blow-up
   6. 2.2.6. Example: resolving the cusp
3. 2.3. Projective morphisms
   1. 2.3.1. The symmetric algebra and the projective bundle of a module
   2. 2.3.2. The projective bundle of a quasi-coherent sheaf
   3. 2.3.3. Projective morphisms
4. 2.4. Successions of blow-ups and functors
   1. 2.4.1. Finite successions of blow-ups
   2. 2.4.2. Divisor families and simple normal crossings
   3. 2.4.3. Operations on successions
   4. 2.4.4. Blow-up sequence functors