 Birational geometry

Birational geometry studies algebraic varieties up to isomorphism of dense Zariski-open subsets. For irreducible varieties, this is equivalent to isomorphism of their function fields over the base field. For example, $t\mapsto (t,t^2)$ identifies the affine line with the parabola $y=x^2$. The rational parametrization $t\mapsto ((1-t^2)/(1+t^2),2t/(1+t^2))$ identifies suitable open subsets of the affine conic $x^2+y^2=1$ and the affine line over a field of characteristic different from two. A blowup changes a variety along a centre while preserving a dense open subset, so it is a central birational operation.

Birational geometry is a branch of algebraic geometry that studies the relationships between algebraic varieties through birational equivalences. These are equivalences that allow the objects in question to be related by rational maps, which can typically be viewed as fewer-dimensional representations of the varieties.