Algebarski varijeteti

Algebarski varijeteti su centralni objekti izučavanja u algebarskoj geometriji. Klasično, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

Upleteni kubni objekat je projektivni algebarski varijetet.

Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.

The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry.

Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.

Literatura

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• Mumford, David (1999). The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians (2nd изд.). Springer-Verlag. ISBN 978-3-540-63293-1. Zbl 0945.14001. 
• Shafarevich, Igor (1995). Basic Algebraic Geometry II Schemes and complex manifolds (2nd изд.). Springer-Verlag. ISBN 978-3-540-57554-2. Zbl 0797.14002. 

Spoljašnje veze

• Milne, James S. (2008). „Algebraic Geometry”. Приступљено 2009-09-01.