CANONICAL DIVISOR Meaning and
Definition
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A canonical divisor is a term commonly used in algebraic geometry to refer to a specific class of divisors defined on an algebraic variety. Divisors, in this context, are essentially certain subvarieties with codimension one.
More formally, a canonical divisor on an algebraic variety is a divisor class that arises from the notion of differential forms on the variety. It can be thought of as the divisor associated with the "singularities" of these differential forms.
In terms of properties, a canonical divisor has a unique canonical representative within its divisor class. This representative has certain standard properties and is often chosen to be the divisor with minimal degree. The relation with differential forms also gives rise to the property that the sum of the coefficients of a canonical divisor equals the dimension of the variety.
Canonical divisors have proven to be very valuable tools in algebraic geometry, allowing for the study of important geometric and geometric-algebraic invariants. They provide insights into the structure of a variety, its singular points, and its birational maps. They also serve as a fundamental building block for various other concepts and constructions in the field.
In summary, a canonical divisor is a class of divisors defined on an algebraic variety that captures the singularities of differential forms. They are valuable tools in the study of geometric and geometric-algebraic invariants and provide insights into the structure of the variety.
Etymology of CANONICAL DIVISOR
The word "canonical" comes from the Greek word "kanōn", meaning rule or law. It was introduced into English through the Latin word "canonicus", which means "according to rule or principle". In mathematics, "canonical" is used to describe something that is standard, fundamental, or universal.
The word "divisor" comes from the Latin word "divisorius", meaning "one who divides". In mathematics, a divisor refers to a number that divides another number evenly, leaving no remainder.
Therefore, the term "canonical divisor" in mathematics combines these two words. It describes a divisor that is considered fundamental, standard, or following a particular rule or principle.
