logprincipalization

Logprincipalization is a mathematical process within algebraic geometry and commutative algebra that involves transforming a given ideal into a principal ideal through a sequence of modifications, often called blow-ups, in a controlled manner. This technique is especially pertinent when working with complex varieties or schemes where simplifying the structure of ideals facilitates resolution of singularities and analysis of algebraic structures.

The core idea of logprincipalization is to systematically modify a variety or scheme so that the ideal in question becomes generated by a single element—i.e., principal—while respecting certain logarithmic structures. This is achieved through a series of logarithmically admissible blow-ups, which are carefully chosen to preserve the underlying geometric and algebraic properties. The process ensures that the transformed ideal is monomial with respect to a chosen divisor, aligning with the concept of logarithmic structures in algebraic geometry.

Logprincipalization plays a significant role in the resolution of singularities, enabling the simplification of complex algebraic or geometric configurations by reducing ideals to their simplest forms. It also assists in the study of toroidal embeddings and in the development of canonical log resolutions, providing tools to analyze how ideals behave under birational transformations.

The technique has implications across several areas of mathematics, including the study of divisor class groups, model theory, and the desingularization of algebraic varieties. Its development builds on foundational work by Hironaka and subsequent advancements in logarithmic geometry, contributing to the broader goal of understanding and manipulating the structure of algebraic objects through systematic modifications.

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