page_title: Algebraic variety
section_title: Example 1
hierarchical_section_title: Algebraic variety / Examples / Projective variety / Example 1
caption_attribution_description: English: Graph of an elliptic curve
context_page_description: Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, 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.
  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 in one variable with complex number coefficients is determined by the set of its roots in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets.
context_section_description: A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The projective line P¹ is an example of a projective curve; it can be viewed as the curve in the projective plane P² = {[x, y, z]} defined by x = 0. For another example, first consider the affine cubic curve
  in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation:
  which defines a curve in P² called an elliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P¹, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves).
caption: The affine plane curve y2 = x3 − x. The corresponding projective curve is called an elliptic curve.
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