From Wikipedia:
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well.
What is an example of two varieties $X$ and $Y$ over the field $k=\mathbb{Q}$ that are birational, but not isomorphic?
just a few more details if possible?
E.g. $X = \{ y^2 - x^3 = 0 \} \subset \mathbb{A}^2 $ and $Y = \{ y -x^2 = 0 \} \subset \mathbb{A}^2 $.
What would be the maps $f : X \to Y$ and $g: Y \to X$ ? Why are they birational and not inverse.