LINEAR SYSTEM OF DIVISORS Meaning and
Definition
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A linear system of divisors refers to a mathematical concept within algebraic geometry. In this context, a divisor is a formal sum of points on an algebraic curve, where each point is assigned a multiplicity. A linear system of divisors, on the other hand, is a collection of divisors satisfying certain conditions.
More specifically, a linear system of divisors on an algebraic curve is a set of divisors such that any element of the set can be written as the sum of a fixed divisor called the base divisor and a "free" divisor. The free divisor can be any divisor that is linearly equivalent to the base divisor, meaning they differ by a principal divisor (a divisor of the form "div(f)" for some rational function f on the curve).
The linear system is typically denoted by "L(D)", where "D" is the base divisor. The dimension of a linear system L(D) represents the number of free parameters involved in the divisors within that system.
Linear systems of divisors play a crucial role in algebraic geometry as they provide a way to study and understand properties of algebraic curves. They allow mathematicians to investigate the existence of specific divisors on curves and their interactions, which can be used to understand the geometry and structure of the curves themselves. Moreover, linear systems are fundamental in topics like the Riemann-Roch theorem, which relates the divisor theory and the topology of an algebraic curve.
