projective variety

Noun

projective variety (plural projective varieties)

1. A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.
   • Varieties are sometimes called closed sets and some authors call an open subset of a projective variety a quasiprojective variety. The latter term is in an attempt to unify the concept of affine and projective variety.2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724
   • A closed subset V#92;subset#92;mathbb#123;C#125;Pⁿ is called a (complex) projective variety if, locally, V is defined by a set of complex polynomial equations. Outside of its singular locus, that is, away from the subset where the defining equations do not have maximal rank, the projective variety...2006, Werner Ballmann, Lectures on Kähler Manifolds, European Mathematical Society, page 16