Meaning of projective variety | Babel Free

Definitions

A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.

Equivalents

Examples

“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.”

“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 is a complex submanifold of #92;mathbb#123;C#125;Pⁿ.”

“The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic.”

CEFR level