Part of Speech

Noun

Phonetic Transcription

/bɪˈreɪʃənl vəˈrɛti/

Meaning and Usage

A birational variety is a concept in algebraic geometry referring to a type of algebraic variety that is not necessarily isomorphic, but has a rational correspondence with another variety. In simpler terms, it describes a relationship between two varieties where they can be related through rational maps, despite not being identical in structure. This concept is important in understanding transformations and relationships between different algebraic structures.

Birational varieties are frequently discussed in written mathematical literature, particularly in contexts involving algebraic geometry, and are essential for advanced studies in this field.

Example Sentences

1. The mathematician presented a birational variety in her thesis, demonstrating the intricate relationships between various algebraic structures.

2. Математик представила бирциональную разновидность в своей диссертации, демонстрируя сложные взаимосвязи между различными алгебраическими структурами.

3. In exploring the concept of birational variety, students learn how to manipulate and connect different algebraic spaces.

4. Изучая концепцию бирциональной разновидности, студенты учатся манипулировать и соединять различные алгебраические пространства.

5. The study of birational variety has led to significant advancements in our understanding of algebraic geometry.

6. Изучение бирциональных разновидностей привело к значительным достижениям в нашем понимании алгебраической геометрии.

Idiomatic Expressions

The term birational variety itself is not commonly found in idiomatic expressions due to its specialized nature within mathematics. However, the concept encompasses several essential terms related to its framework such as "birational map" and "rational variety". Here are some example sentences illustrating these terms:

1. Understanding a birational map allows mathematicians to simplify complex equations in birational variety studies.

2. Понимание бирциональной функции позволяет математикам упрощать сложные уравнения в исследованиях бирциональных разновидностей.

3. The notion of a rational variety is crucial when examining the properties of birational varieties.

4. Понятие рациональной разновидности имеет решающее значение при исследовании свойств бирциональных разновидностей.

5. To connect the two birational varieties, the researcher constructed a detailed birational map.

6. Для связи двух бирциональных разновидностей исследователь создал подробную бирциональную карту.

Etymology

The term birational derives from a combination of the prefix "bi-" meaning two, and "rational," which refers to rational numbers or functions. The idea of relating two varieties through rational maps gives rise to this term, indicating a two-way correspondence or relationship in the context of varieties in algebraic geometry.

Synonyms and Antonyms

Synonyms: - Rational variety - Algebraic variety (in a broader sense)

Antonyms: - Non-rational variety (context-dependent, typically in contrast to rational varieties)

This structured overview provides a comprehensive understanding of the term birational variety and its relevance within the domain of algebraic geometry.