nLab projective variety

Idea

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

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).

Related concepts

• projective manifold

References

• Markus Reineke, Every projective variety is a quiver Grassmannian (arXiv:1204.5730, blog discussion)