Logarithmic pair
Encyclopedia
In algebraic geometry
, a logarithmic pair consists of a variety
, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D.
The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X.
A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form
d log(z) = dz/z along components of the divisor given locally by z=0.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a logarithmic pair consists of a variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
Definition
A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1.A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D.
The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X.
A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form
d log(z) = dz/z along components of the divisor given locally by z=0.