nLab
projective variety

Contents

Idea

An projective algebraic variety (over an algebraically closed field kk) is the 0-locus of a homogeneous ideal of polynomials in (n+1)(n+1) variables over kk in the projective n-space n\mathbb{P}^n.

Examples

The archetypical example is projective space itself. In direct generalization of this but less evident: every Grassmannian is naturally a projective variety, even a smooth variety.

This generalizes to quiver representations. Every Grassmannian of a Quiver representation is a projective variety. In fact, every projective variety arises this way (Reineke, ‘12).

References

Last revised on September 12, 2012 at 17:44:42. See the history of this page for a list of all contributions to it.