principalisointi

Principalisointi, or principalization, is a concept in algebraic geometry describing the process of turning a given ideal sheaf on a smooth variety into a principal ideal on a higher model. Given a smooth variety X over a field of characteristic zero and a coherent ideal I ⊂ O_X, principalization seeks a finite sequence of blowups with smooth centers f: X' → X such that the total transform I·O_{X'} is principal, i.e., generated by a single function, and the exceptional divisor E = ∑ a_i E_i has simple normal crossings. In this setting, one can express I·O_{X'} = O_{X'}(-D) for an effective Cartier divisor D with SNC (simple normal crossings) support.

The construction is designed so that the centers of the blowups are chosen to manage the singularities or the complexity of the ideal, ensuring that eventually the transform becomes invertible. The result is a birational model X' in which the original ideal corresponds to a divisor with controlled intersections, encoding the data of the original singularities.

Principalisointi is a key tool in the embedded resolution of singularities, providing a mechanism to reduce questions about singularities to questions about divisors with simple combinatorial structure. It underpins algorithmic approaches to resolution, multiplier ideals, and other aspects of birational geometry. Variants include principalization of subschemes and log-principalization, where the goal is to control the divisor associated to several ideals.

Notes: Foundational work on principalization appears in the resolution of singularities program, with developments by Hironaka, Bierstone–Milnor, Villamayor, Encinas and others showing existence in characteristic zero. In positive characteristic, principalization remains more subtle and is established only in special cases.