A log minimal model is a generalization of the minimal model program in algebraic geometry, adapted to the context of pairs consisting of a variety and a divisor. It aims to simplify the structure of these pairs while preserving their essential properties, and is crucial for understanding the birational geometry of varieties with mild singularities.

Log Minimal Model

log minimal model

The Minimal Model Program is a central part of the field of algebraic geometry, aiming to classify algebraic varieties by simplifying their structure through birational transformations. It focuses on constructing models that are 'minimal' in the sense that they have the simplest possible canonical divisors, which helps in understanding the geometry and classification of higher-dimensional varieties.

minimal model program

Algebraic geometry is a branch of mathematics that studies the solutions of systems of polynomial equations using abstract algebraic techniques, primarily focusing on the properties and structures of algebraic varieties. It serves as a bridge between algebra and geometry, providing a deep understanding of both geometric shapes and algebraic equations through the lens of modern mathematics.

algebraic geometry

context of pairs

variety and a divisor

Structure of pairs

Essential properties are the fundamental characteristics that define the identity and existence of an entity, distinguishing it from others. These properties are intrinsic and necessary, meaning that without them, the entity would not be what it is.

essential properties

Birational geometry is a branch of algebraic geometry that studies the properties of algebraic varieties that are preserved under birational equivalence, which is a relation where varieties are considered equivalent if they can be transformed into each other by rational maps. It plays a crucial role in the classification of algebraic varieties, aiming to understand their structure by simplifying them through birational transformations, such as blow-ups and blow-downs, while preserving their essential geometric properties.

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